AF Geometry, Unit #3
UNIT OVERVIEW
Unit Title
Unit Designer
Geometric relationships & properties
Approximate Length
IA Period
18 teaching days
2
Stage 1: Desired Results
ENDURING UNDERSTANDINGS AND BIG IDEAS
What do you want students to know in 10 years about this topic?
Enduring understanding/Big idea
What it looks like in this unit
Scholars will apply algebra when solving equations involving
simple angle pairs (vertical angles/linear pairs), parallel lines
Geometric and algebraic procedures are interconnected and
and transversals, the angle sum of triangles, and many other
build on one another.
sets of angles. Scholars will use algebraic expressions to
calculate angle measures by writing and solving equations.
Scholars will classify angles, first by using properties learned
in middle school math (such as linear pairs and
complementary/supplementary relationships); then, they will
extend their knowledge of angles to classify angles formed by
parallel lines and transversals.
Two- and three-dimensional objects can be classified, described,
and analyzed by their geometric attributes using a variety of
strategies, tools, and technologies.
Geometric figures can be mapped onto one another by a single
transformation or by a sequence of consecutive transformations.
These transformations can occur on or off the coordinate plane.
Scholars will classify triangles based on their angles, and use
properties of isosceles and equilateral triangles to solve
problems. They will also investigate the differences between
acute, obtuse, and equilateral triangles when performing
constructions of triangle medians and angle/side bisectors.
Last, scholars will use properties of quadrilaterals to classify
them based on relationships between their angles and sides.
Relationships between alternate interior, alternate exterior,
and same-side interior angles all follow from properties of
corresponding angles and vertical angles. By exploring these
relationships geometrically using transformations, scholars will
gain a deeper understanding of how geometric theorems can
follow from postulates.
AF Geometry, Unit #3
Proof is a justification that is logically valid based on definitions,
postulates, and theorems. Analyzing geometric relationships
through the writing of proofs develops reasoning and justification
skills.
Effective mathematical arguments involve both concise language
and clear reasoning, in the form of closely related steps justified
by relevant evidence.
All constructions are based on geometric properties of
congruence.
Scholars will informally prove relationships between angles
formed by parallel lines and transversals (see above).
Scholars will extend their understanding of proofs to a third
type (flow proofs), and to a new type of shape (quadrilaterals).
Basic constructions (segment bisectors, perpendicular
bisectors, and angle bisectors) are extended in this unit to
apply to triangles; scholars will investigate the concurrency of
these lines to gain an understanding of inscribed and
circumscribed circles.
Scholars will apply the corresponding angles theorem to
construct parallel lines.
Measurements (both direct and indirect) can be made to
describe, compare, and make sense of real-life objects.
Geometric measurements can be represented in algebraic
expressions and equations.
Indirect measurements of segments and angles are performed
in this unit by applying geometric properties and theorems.
ESSENTIAL QUESTIONS
What question(s) will guide this unit and focus learning and thinking?
(1) How can we use relationships between special angle pairs to find missing measurements?
 When two lines intersect, they form vertical angles (which lie across from each other, and are congruent) and linear pairs
(which lie next to each other, and are supplementary)
 By knowing the relationship between two angle pairs (complementary, supplementary, or congruent), it is possible to
write (and, if necessary, solve) an algebraic equation to find variables and/or angle measurements
 When two parallel lines are intersected by a third nonparallel line (“transversal”), four types of angle pairs result:
o Corresponding angles are congruent
o Alternate exterior angles are congruent
o Alternate interior angles are congruent
o Same-side (consecutive) interior angles are supplementary
AF Geometry, Unit #3
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The relationships between the last three sets of angles can be derived from congruent corresponding angles and
properties of vertical angles and/or linear pairs. Just as with other special angle pair relationships, once the relation has
been determined it is possible to write and solve algebraic equations to find missing quantities.
The interior angles of a triangle always sum to 180o.
The sum of the interior angles of any n-gon is given by 𝑆 = (𝑛 − 2) ∙ 180; each interior angle of a polygon forms a linear
pair with its corresponding exterior angle. The exterior angles of any polygon always sum to 360o.
(2) What relationships exist between the sides and angles of triangles?
 In any triangle, the longest side of the triangle is always located opposite (across from) the largest angle; the shortest
side of the triangle is always located opposite the smallest angle.
 In isosceles triangles, two of the sides are congruent. These sides are referred to as the “legs” of the isosceles triangle.
The angles located across from the legs, called the “base angles”, are congruent.
 The third (non-congruent) side of an isosceles triangle is called the “base” of the triangle. The third (non-congruent)
angle of an isosceles triangle is located opposite the base, and is called the “vertex angle”.
(3) What special segments exist within triangles? What are points of concurrency? How are they used?
 When three or more lines, rays, or segments intersect at the same point, they are called concurrent. The point of
intersection is called the point of concurrency. Points of concurrency are useful in geometric constructions.
 Any point that lies on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.
o The three perpendicular bisectors of a triangle intersect at a point, called the circumcenter, which is equidistant
from the vertices of the triangle (the endpoints of the sides).
o The circumcenter of a triangle can be used to construct a circle passing through all three vertices of the triangle
(the circumscribed circle of the triangle).
o The circumcenter of a triangle may lie inside the triangle (if the triangle is acute), on one side of the triangle (if the
triangle is right), or outside the triangle (if the triangle is obtuse).
o For a right triangle, the circumcenter will always lie on the triangle’s hypotenuse.
 Any point that lies on the bisector of an angle is equidistant from the two sides of the angle.
o The three angle bisectors of a triangle intersect at a point, called the incenter, which is equidistant from the sides
of the triangle.
o The incenter of a triangle can be used to construct a circle that is tangent to the three sides of the triangle (the
inscribed circle of the triangle).
o The incenter of a triangle always lies in the interior of the triangle.
 A median of a triangle is a segment from a vertex to the midpoint of the opposite side. A median bisects one side of the
triangle.
o The three medians of a triangle are concurrent. The point of concurrency, called the centroid, always lies in the
AF Geometry, Unit #3
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interior of the triangle.
An altitude of a triangle is the perpendicular segment from a vertex to the opposite side, or to a line that contains the
opposite side. An altitude does not necessarily bisect the angle or side it contains.
o The lines containing the altitudes of a triangle are concurrent.
o The three altitudes of a triangle are concurrent. The point of concurrency, called the orthocenter, may lie inside
the triangle (if the triangle is acute), on one side of the triangle (if the triangle is right), or outside the triangle (if
the triangle is obtuse).
For isosceles triangles, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base
are all the same segment. For equilateral triangles, this is true for the special segment from any vertex.
(4) What are the special properties of parallelograms?
 Parallelograms are quadrilaterals in which both pairs of opposite sides are parallel. Several important theorems follow
from this fact:
o If a quadrilateral is a parallelogram, then its opposite sides are congruent.
o If a quadrilateral is a parallelogram, then its opposite angles are congruent.
o If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
o If a quadrilateral is a parallelogram, then its diagonals bisect each other.
 A rectangle is a parallelogram in which the adjacent sides are perpendicular to each other – that is, a parallelogram in
which all four angles are right angles.
 A parallelogram is a rectangle if and only if its diagonals are congruent. This follows from the SSS congruence postulate
and the fact that consecutive angles of a parallelogram are supplementary (by using CPCTC to show that the two
consecutive angles must be congruent, we can show that they must each equal 90o).
UNIT NARRATIVE
What is the purpose of this unit? How does it fit into the broader context of the course?
Unit 3 extends knowledge of angle pair relationships from 7th and 8th grade. Scholars may be familiar with vertical angles and terms
such as “complementary” and “supplementary”, but have never seen them in the context of parallel lines with a transversal.
Additionally, this unit bridges the gap between those kinds of angle pairs and the “Rule of 180” for triangles – a key concept, tested
frequently on standardized tests such as the SAT. An emphasis is placed here on integration with algebra as well as formal and
informal proofs.
The second half of the unit builds upon knowledge of basic 2-dimensional shapes (triangles and quadrilaterals) by introducing
AF Geometry, Unit #3
geometric postulates and theorems used to classify them. Scholars build precise definitions for shapes, then use these definitions to
discover and/or prove other important properties. Constructions, introduced in the previous unit, are a vital tool in this process.
STANDARDS FOR MATHEMATICAL PRACTICE
Which are the primary SMPs for this unit? How will they be incorporated and reinforced throughout this unit?
Complex, SAT-type problems are first introduced in this unit, many of which involve
MP.1 Make sense of problems and
multiple steps and information from multiple units. Scholars will learn to find an “entry
persevere in solving them
point” of a problem to try and find an unknown quantity, then use this new (known)
quantity to calculate additional information.
MP.2 Reason abstractly and
quantitatively
Scholars build upon their knowledge of proof, introduced in the first unit, to write and
MP.3 Construct viable arguments and
analyze proofs involving geometric properties of triangles and quadrilaterals. They will
critique the reasoning of others
also use definitions (most notably, definitions of different types of quadrilaterals) to
classify 2-dimensional objects according to their properties.
MP.4 Model with mathematics
Scholars will complete constructions using compasses and straightedges, patty paper
MP.5 Use appropriate tools strategically
and/or geometric software.
Particular emphasis is placed in this unit on precise geometric vocabulary and notation.
Classification of 2-dimensional figures according to their properties introduces scholars
to the importance of unambiguous definitions. Additionally, scholars precisely
MP.6 Attend to precision
articulate steps in proofs involving parallel lines, triangles, and quadrilaterals.
MP.7 Look for and make use of structure
MP.8 Look for and express regularity in
repeated reasoning
Scholars are continuing to build and develop in this skill; by unit 3, they should be
expected to accurately write full proofs on their own.
When investigating the interior angle sum of polygons, scholars will use inductive
reasoning and patterns to show that the equation is derived from the fact that every
polygon of degree n contains exactly (n – 2) non-overlapping triangles.
AF Geometry, Unit #3
CONTENT STANDARDS
Prove geometric theorems.
G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector
of a line segment are exactly those equidistant from the segment’s endpoints.
G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and
half the length; the medians of a triangle meet at a point.
**NOTE: The midsegment theorem will be incorporated into unit 4: similarity, and thus is not present in this unit.**
Major:
G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Additional:
Understand and apply theorems about circles.
G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
ESSENTIAL SKILLS AND PROCEDURAL KNOWLEDGE
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What do you want students to be able to do comfortably, accurately, and with flexibility?
Identify relationships between various angle pairs, and find missing measurements by writing and solving algebraic
equations.
Describe relationships within triangles and quadrilaterals, and use these relationships to (1) derive theorems about these
shapes, and (2) classify the shapes based on their characteristics.
Construct geometric figures using a compass and straightedge.
Justify constructions using complete sentences and correct geometric vocabulary.
Construct proofs in many different formats; translate between proofs written in varying formats
AF Geometry, Unit #3
UNIT VOCABULARY (PART 1: ANGLE PAIR RELATIONSHIPS)
Vocabulary to review
(from previous units, or previous mathematics classes)
Complementary angles: a set of angles whose sum is 90o.
Supplementary angles: a set of angles whose sum is 180o.
Linear pair: a pair of adjacent, supplementary angles formed
by two intersecting lines.
Vertical angles: a pair of opposite angles formed by two
intersecting lines. Vertical angles are congruent to each other.
Parallel lines: two lines that do not intersect. Parallel lines have
equal slopes, but different y-intercepts.
Perpendicular lines: two lines that intersect to form a right
(90o) angle. Perpendicular lines have negative reciprocal
slopes.
Isosceles triangle: a triangle in which two of the sides are
congruent. The congruent sides are called the “legs” of the
triangle. The angles located opposite the legs are called the
base angles, and are congruent.
Equilateral triangle: a triangle which has three congruent sides
and three congruent (60o) angles.
New vocabulary:
Transversal: a line that intersects two or more parallel lines.
Corresponding angles: a pair of angles formed by the
intersection of a transversal with two parallel lines.
Corresponding angles occupy the same relative position at each
intersection; they are congruent.
Alternate exterior angles: a pair of angles formed by the
intersection of a transversal with two parallel lines. Alternate
exterior angles are located on opposite sides of the transversal
and on the outside of the two parallel lines; they are congruent.
Alternate interior angles: a pair of angles formed by the
intersection of a transversal with two parallel lines. Alternate
interior angles are located on opposite sides of the transversal
and between the two parallel lines; they are congruent.
Consecutive interior (same-side interior) angles: a pair of
angles formed by the intersection of a transversal with two
parallel lines. Consecutive interior angles are located on the
same side of the transversal and between the two parallel lines;
they are supplementary.
Interior angle: any angle located inside of a triangle or polygon
(an angle formed between the two sides of a triangle or
polygon). The interior angles of every triangle sum to 180o, and
the interior angles of any n-gon sum to (n – 2)·180o.
Exterior angle: any angle located on the outside of a triangle or
polygon (an angle formed between a side and an extension of
an adjacent side of a triangle or polygon). Any exterior angle
forms a linear pair with its adjacent interior angle. The exterior
angles of any triangle or polygon sum to 360o.
AF Geometry, Unit #3
UNIT VOCABULARY (PART 2: PROPERTIES OF TRIANGLES AND PARALLELOGRAMS)
Vocabulary to review
(from previous units, or previous mathematics classes)
Perpendicular bisector: a line or segment that is perpendicular
to a given segment and contains the midpoint of that segment.
Angle bisector: a ray that divides an angle into two smaller congruent
angles.
Equidistant: a point is equidistant from two figures if the point is
the same distance from each figure. For example, the midpoint
of a segment is equidistant from the two endpoints.
New vocabulary:
Concurrent lines, rays, or segments: three or more lines,
rays, or segments that intersect at the same point.
Point of concurrency: the point of intersection of three or more
concurrent lines, rays, or segments.
Circumcenter: the point of intersection (concurrency) of the
three perpendicular bisectors of the sides of any triangle. The
circumcenter of a triangle may lie inside the triangle (if the
triangle is acute), on one side of the triangle (if the triangle is
right), or outside the triangle (if the triangle is obtuse).
Incenter: the point of intersection (concurrency) of the three
angle bisectors of any triangle. The incenter always lies in the
interior of a triangle.
Median of a triangle: any segment from a vertex of a triangle to
the midpoint of the opposite side.
Centroid: the point of intersection (concurrency) of the three
medians of any triangle. The centroid always lies in the interior
of a triangle. The centroid is called the “center of mass” or
“center of gravity” of a triangle, because it is the point at which
a triangle can be balanced.
Altitude of a triangle: the perpendicular segment from a vertex
of a triangle to the opposite side or (in obtuse triangles, for the
altitudes from the acute angles) the perpendicular segment from
a vertex to the line that contains the opposite side.
Orthocenter: the point of intersection (concurrency) of the three
altitudes of any triangle. The orthocenter of a triangle may lie
inside the triangle (if the triangle is acute), on one side of the
triangle (if the triangle is right), or outside the triangle (if the
triangle is obtuse).
Parallelogram: a quadrilateral in which both pairs of opposite
sides are parallel.
AF Geometry, Unit #3
Rectangle: a parallelogram with four right angles.
IMPORTANT FORMULAS, POSTULATES, AND THEOREMS
Angle pair
relationships:
theorems about
angles formed by
two or more
intersecting lines
When two lines intersect,
 Vertical angles are congruent. (ex. ∠𝐴 ≅ ∠𝐷)
 Two angles that form a linear pair are supplementary.
(ex. 𝑚∠𝐴 + 𝑚∠𝐶 = 1800 )
When two parallel lines are cut by a transversal,
Angle pair
relationships:
theorems about
angles formed by
parallel lines and
transversals
 Corresponding angles are congruent. (ex. ∠𝐴 ≅ ∠𝐸)
 Alternate interior angles are congruent. (ex. ∠𝐶 ≅ ∠𝐹)
 Alternate exterior angles are congruent. (ex. ∠𝐵 ≅ ∠𝐺)
 Same-side interior angles are supplementary.
(ex. 𝑚∠𝐷 + 𝑚∠𝐹 = 1800 )
When two lines are cut by a transversal, the lines are parallel if any of the following are true:
Converses of angle
pair relationships:
theorems
 Corresponding angles are congruent.
 Alternate interior angles are congruent.
 Alternate exterior angles are congruent.
 Same-side interior angles are supplementary.
Perpendicular transversal theorem: If a transversal is perpendicular to one of two parallel lines, then it is
Additional theorems perpendicular to the other.
about parallel lines
Lines perpendicular to a transversal theorem: In a plane, if two lines are perpendicular to the same line, then they
and transversals
are parallel to each other.
Angle Sum
Theorems for
Triangle Interior Angle Sum Theorem: the sum of the interior angles of any triangle is 180o.
AF Geometry, Unit #3
triangles and
polygons
Corollary: The acute angles of a right triangle are complementary.
Quadrilateral Interior Angle Sum Theorem: the sum of the interior angles of any quadrilateral is 360o.
Polygon Interior Angle Sum Theorem: the sum of the interior angles of a convex n-gon is given by the equation 𝑆 =
(𝑛 − 2) ∙ 1800.
Polygon Exterior Angle Sum Theorem: the sum of the exterior angles of a convex n-gon is 360o.
Theorems about
isosceles and
equilateral
triangles
Base angles theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.
Corollary: If a triangle is equilateral, then it is equiangular.
Converse of the base angles theorem: If two angles of a triangle are congruent, then the sides opposite them are
congruent.
Corollary: If a triangle is equiangular, then it is equilateral.
Perpendicular bisector theorem: If a point is on a perpendicular bisector of a segment, then it is equidistant from
the endpoints of the segment.
Converse: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the
segment.
Theorems about
special segments in
triangles
Angle bisector theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Converse: If a point is on the interior of an angle and is equidistant from the sides of the angle, then it lies on the
bisector of the angle.
Median area division theorem: The median of a triangle divides the triangle into two smaller triangles with equal
area (and each with an area equal to one-half the area of the original triangle).
Corollary: The three medians of a triangle divide the triangle into six smaller triangles of equal area (and each
with an area equal to one-sixth the area of the original triangle).
Concurrency of perpendicular bisectors theorem: The perpendicular bisectors of a triangle intersect at a point
that is equidistant from the vertices of the triangle, called the circumcenter.
Theorems about the
concurrency of
special segments in
triangles
Triangle angle-side
relationships
Concurrency of angle bisectors theorem: The angle bisectors of a triangle intersect at a point that is equidistant
from the sides of the triangle, called the incenter.
Concurrency of medians theorem: The medians of a triangle intersect at a point, called the centroid, that is twothirds the distance from each vertex to the midpoint of the opposite side (the centroid divides a median into two
segments whose lengths are in the ratio 2:1, with the longer segment nearest the vertex.
Concurrency of altitudes theorem: The lines containing the altitudes of a triangle are concurrent; the altitudes of a
triangle are also concurrent and intersect at a point called the orthocenter.
Triangle inequality theorem: The sum of the lengths of the two shorter sides of a triangle is always greater than the
AF Geometry, Unit #3
length of the third (longest) side.
Triangle angle-side relationship: In any triangle, the angle opposite the longer side is larger than the angle
opposite the shorter side.
Corollary: In any triangle, the side opposite the larger angle is longer than the side opposite the smaller angle.
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Converse: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
Corollary: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral
is a parallelogram.
Theorems about
quadrilaterals
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Converse: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Rectangle corollary: a quadrilateral is a rectangle if and only if it has four right angles.
A parallelogram is a rectangle if and only if its diagonals are congruent.
PREREQUISITE SKILLS
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What skills will scholars need in order to be successful with this unit?
Name a point, line, or angle using proper geometric notation; distinguish between proper and improper angle names.
Solve a multi-step algebraic equation (including those with multiple sets of like terms and/or variables on both sides of the
equation).
Substitute a value for a variable; evaluate an expression using order of operations.
Classify geometric figures based on defined properties. For example:
o Classify angles as acute, right, or obtuse based on their angle measures.
o Classify triangles as acute, right, or obtuse based on their angle measures and as scalene, isosceles, or equilateral
based on their side lengths.
o Classify quadrilaterals based on their defined properties – parallel/congruent sides, right angles.
Construct the following, using a compass and straightedge, paper-folding techniques, and/or technology software:
AF Geometry, Unit #3
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o Perpendicular bisectors
o Angle bisectors
o Lines perpendicular to a given line, passing through a given point.
Graph the lines 𝑥 = 𝑎, 𝑦 = 𝑎, 𝑦 = 𝑥, and 𝑦 = −𝑥 on the coordinate plane.
Stage 2: Acceptable Evidence
WRITTEN ASSESSMENTS
Quiz 3.1
Unit 3 exam
Include file names and descriptions of unit test and quizzes
Quiz on lessons 1 – 8 from unit 1, covering the following topics:
 Angle pair relationships (vertical angles, linear pairs)
 Angles formed by parallel lines and transversals
 Angle sum of triangles
 Relationships between angles and sides in triangles (including isosceles triangles and the
base angles theorem)
Exam on all lessons from unit 3. Questions should be pulled from SAT/Regents/MCAS where
applicable, and written to at least the rigor level of the IA.
PERFORMANCE ASSESSMENTS
Angle pair relationships:
Include file name/link or task and description
In the diagram shown at right, 𝑝||𝑞. Find the values of x,
y, and z.
Show all of your work and justify your answers using the
theorems discussed in class.
AF Geometry, Unit #3
Constructions, part 1: Placing
a Fire Hydrant (Link:
Illustrative Mathematics)
You have been asked to place a fire hydrant so that it
is an equal distance form three locations indicated on
the following map.
a) Show how to fold your paper to physically
construct this point as an intersection of two
creases.
b) Explain why the above construction works,
and in particular why you only needed to
make two creases.
Constructions, part 2:
Locating a Warehouse (Link:
Illustrative Mathematics)
You have been asked to place a warehouse so that it
is an equal distance from the three roads indicated
on the following map. Find this location and show
your work.
a) Show how to fold your paper to physically
construct this point as an intersection of two
creases.
b) Explain why the above construction works,
and in particular why you only needed to
make two creases.
c) Explain how this construction is different from
the construction you performed in “Placing a
Fire Hydrant”.
AF Geometry, Unit #3
Quadrilaterals:
In quadrilateral ABCD, pictured at right, ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷 and
̅̅̅̅ ≅ 𝐴𝐷
̅̅̅̅.
Is this a Parallelogram? (Link: 𝐵𝐶
Illustrative Mathematics)
From the given information, can we deduce that ABCD
is a parallelogram? Explain.
AF Geometry, Unit #3
SAMPLE PROBLEMS FROM STANDARDIZED TESTS
G-CO.10 Prove theorems about
triangles. Theorems include:
measures of interior angles of a
triangle sum to 180 degrees; base
angles of isosceles triangles are
congruent; the segment joining
midpoints of two sides of a triangle
is parallel to the third side and half
the length; the medians of a triangle
meet at a point.
SAT-aligned items by standard
AF Geometry, Unit #3
G-CO.10 Prove theorems about
triangles. Theorems include:
measures of interior angles of a
triangle sum to 180 degrees; base
angles of isosceles triangles are
congruent; the segment joining
midpoints of two sides of a triangle
is parallel to the third side and half
the length; the medians of a triangle
meet at a point.
AF Geometry, Unit #3
G-CO.10 Prove theorems about
triangles. Theorems include:
measures of interior angles of a
triangle sum to 180 degrees; base
angles of isosceles triangles are
congruent; the segment joining
midpoints of two sides of a triangle
is parallel to the third side and half
the length; the medians of a triangle
meet at a point.
G-CO.10 Prove theorems about
triangles. Theorems include:
measures of interior angles of a
triangle sum to 180 degrees; base
angles of isosceles triangles are
congruent; the segment joining
midpoints of two sides of a triangle
is parallel to the third side and half
the length; the medians of a triangle
meet at a point.
AF Geometry, Unit #3
G-CO.10 Prove theorems about
triangles. Theorems include:
measures of interior angles of a
triangle sum to 180 degrees; base
angles of isosceles triangles are
congruent; the segment joining
midpoints of two sides of a triangle
is parallel to the third side and half
the length; the medians of a triangle
meet at a point.
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
AF Geometry, Unit #3
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
**Note: these problems require the
use of the polygon interior angle
sum formula, not specifically
covered by G-CO.9.**
AF Geometry, Unit #3
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
**Note: these problems require the
use of the polygon interior angle
sum formula, not specifically
covered by G-CO.9.**
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
AF Geometry, Unit #3
G-CO.9 Prove theorems about
lines and angles. Theorems
include: vertical angles are
congruent; when a transversal
crosses parallel lines, alternate
interior angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
BOTH G-CO.9 AND G-CO.10
AF Geometry, Unit #3
OTHER EVIDENCE



In-class questioning (CFUs)
Daily homework (should be pulled from various sources, including the SAT where applicable, and should build to at least the
rigor level of the IA)
Small-group instruction (tutorial, pull out SGI, etc.) data
AF Geometry, Unit #3
Stage 3: Learning Plan
LEARNING PLAN
Lesson Content
MPSs
Aim/Exit Ticket
#
Standards
G-CO.9
3.01
MP.1 WWBAT use properties of simple angle pair relationships (vertical
(basic
angle
pairs –
vertical
angles,
linear
pairs)
angles, complementary / supplementary angles) to find missing
angle measures by writing and solving equations.
1. An angle measures 28o.
a. What is the measure of the angle’s supplement?
b. What is the measure of the angle’s complement?
Key Points, Resources and Notes


c. What is the measure of an angle that forms a linear pair
with the angle?
Use the figure at right to answer each question.
2. If m∠6 = 78o, then m∠7 = ________
3. If m∠8 = 94o, then m∠6 = ________
4. If m∠9 = 124o, then m∠8 =
________
5. Find x.


If two or more angles are complementary, their
measures add to 90o; if two or more angles are
supplementary, then their measures add to 180o..
o Emphasize vocabulary – for example, “a
linear pair consists of supplementary angles”
Always name the angle pair relationship before
you write an equation – and be specific! (ex. ∠𝑋
and ∠𝑌 are vertical angles, so they are
congruent. THEN 6x + 7 = 3x + 12, and solve. This
will prevent careless mistakes – for example,
having vertical angles that sum to 180o.
Show all work when solving equations!
Two or more angles that when combined make a
straight line must have measures that add up to
180o. Two angles that are across from each other
when two straight lines intersect are vertical
angles, and they are congruent. Make sure to
show non-example (below) and explain that
because the auxiliary lines are not straight, the
angles that are across from each other (∠𝐴 and
∠𝐶) aren’t vertical.
AF Geometry, Unit #3
3.02
G-CO.9
(basic
angle
pairs –
vertical
angles,
linear
pairs)
MP.1
WWBAT use properties of complex vertical angles and sets of
linear angles to find missing angle measures by writing and
solving equations.
1. In the figure below, AB is a straight line. What is the value
of y?
(A) 36
(B) 72
(C) 108
(D) 120
(E) 135
2. Find x and y.
3. Three lines intersect in a point, as shown in the figure
below. Which of the following pairs of angle measures is
NOT sufficient for determining all six angle measures?
(F) t and z
(G) t and y
(H) s and x
(I) r and t
(J) r and s
Same key points as above, with the addition of the
following:
 When given multiple sets of lines, it is easiest to
draw over the given lines to visually separate
and make it easier to identify vertical angles
 Name ALL pairs of angles you see first, and try to
write equations. Find one equation that has only
one variable, and solve it, then substitute if
necessary and keep going until you’ve found all
the variables.
AF Geometry, Unit #3
3.03
G-CO.9
(angle
pairs
formed by
parallel
lines and a
transversal)
MP.1
MP.7
WWBAT classify and state the relationships between angles
formed by two parallel lines and a transversal.
State the relationship between each pair of angles.
1. ∠A and ∠D are ________________.
2. ∠B and ∠F are ________________.
3. ∠D and ∠H are _______________.
4. Which of the following correctly describes the relationship
between ∠G and ∠B?
(A) They are congruent, because they are vertical angles
(B) They are supplementary, because they are alternate
exterior angles
(C) They are congruent, because they are alternate
interior angles
(D) None of the above
5. Which angle pair(s) in the diagram above are
supplementary?
(A) i only
i.
∠A and ∠B
(B) i and ii only
ii.
∠E and ∠G
(C) iii only
iii.
∠F and ∠D
(D) i, ii, and iii
6. In the diagram below, a||b.
If m∠4 = 65o, find m∠5 and m∠8. Explain your reasoning.

When two parallel lines are intersected by a
transversal, four types of angle pairs are formed
(in addition to vertical angles and linear pairs).
The name of the angle pair describes the
relationship:
o “corresponding” angles are located in the
same relative position in relation to the
transversal and parallel lines
o “alternate” angles are on opposite sides of
the transversal
o “same-side” or “consecutive” angles are on
the same side of the transversal
o “interior” angles are between the two
parallel lines
o “exterior” angles are on the outside
(exterior!) of the parallel lines
 Corresponding angles are congruent.
o Alternate interior and alternate exterior
angles are also congruent
o Same-side (consecutive) interior angles are
supplementary
o All three of the theorems above can be
derived by assuming corresponding angles
are congruent, and then using properties of
vertical angle congruence.
 If no angle pair relationship can be easily
determined, try to use properties of vertical
angles (or linear pairs) to identify other angles
in the diagram, so that you can determine a
relationship
 Before you read the multiple choice answer
choices, try to figure out what the correct
answer is – and write it down – then see which
answer choice matches. This will help you avoid
“distractors”!
AF Geometry, Unit #3
3.04
G-CO.9
(angle
pairs
formed by
parallel
lines and a
transversal)
MP.1
MP.6
MP.7
Given two parallel lines and a transversal, WWBAT use properties
of angle pair relationships to find missing angle measures by
writing and solving equations.
1. Find the value of x.
2. In the diagram below, m∠B = (6x – 8)o and m∠E = (8x + 34)o.
What is m∠F?
(A) 11o
(B)
21o
(C)
58o
(D) 122o
Same key points as 3.01 – 3.03, above, with the
addition of the following:
 When you need to calculate the values of
multiple variables, it’s helpful to first name ALL
pairs of angles you see first, and try to write
equations. Find one equation that has only one
variable, and solve it, then substitute if
necessary and keep going until you’ve found all
the variables.
 CASIO (including naming ALL of your steps) can
be especially helpful in multi-step problems,
because you will know exactly what you’ve
found.
o Always re-read the problem to check that
you’ve found the correct quantity! If you were
asked to find an angle measure, don’t just
find the value of a variable!
3. Find x and y.
3.05
G-CO.10
(interior
angles of a
triangle
add to
180o)
MP.1
WWBAT use the Triangle Angle Sum Theorem to find missing
angles of triangles by writing and solving equations.
1. Find the value of each variable.
a.
b.

Begin by identifying one of the following:
o a triangle with only one missing angle
o a linear pair, set of vertical angles, etc.
Use one of the above relationships to find the
measure of one missing angle (or write and
solve one equation), then repeat until you have
found all desired quantities.
 Always name the angle pair relationship first –
and be specific! (ex angles of a triangle add to
180o, so (5x + 3) + (7x – 1) + (12x – 100) = 180,
AF Geometry, Unit #3
2.
3.06
G-CO.10
(interior
angles of a
triangle
add to
180o)
MP.1
Use the diagram below to find the value of 𝑤 + 𝑡.
(A)
37
(B)
62
(C)
99
(D)
118
(E)
155
WWBAT use the Triangle Angle Sum Theorems to solve SAT-like
problems involving finding missing angles of triangles by writing
and solving equations with variable expressions and ratios.
1. In ΔPQR, m∠P = (4x)o, m∠Q = (9x – 50)o, and m∠R = 35o. Is ∠Q
acute, right, or obtuse? Show your work or explain how you
found your answer.
2. Find the measure of the smallest
angle of the triangle shown in the
diagram, at right.
Show your work or explain how you
found your answer. (NOTE: the
diagram is not drawn to scale).
3. In a certain triangle, the measures of the angles are in the
ratio 2 : 3 : 4. By how many degrees does the measure of the
largest angle of the triangle exceed the measure of the
smallest angle?
(A) 20o
(B) 30o
(C) 40o
(D) 50o
then solve)
Once you’ve found a value of a variable, you can
find other quantities – by substituting the
variable you can find angle measures (a good
time to check that you’ve done the algebra
correctly!), and then you can classify the
polygon based on its angles.
 CASIO (including naming ALL of your steps) can
be especially helpful in multi-step problems,
because you will know exactly what you’ve
found.
o Always re-read the problem to check that
you’ve found the correct quantity! If you were
asked to find an angle measure, don’t just
find the value of a variable!
Same key points as above.

NOTE: this lesson explicitly drives at success on SATstyle problems involving angles in triangles. The SAT
frequently assesses scholar knowledge of angle pair
relationships, and these problems often have
relatively low degrees of difficulty.
AF Geometry, Unit #3
3.07
G-CO.10
(interior
angles of a
triangle
add to
180o)
MP.1
MP.6
MP.7
G-CO.10
(interior
angles of a
triangle
add to
180o)
**NOTE: this
lesson focuses
on extending
the rule of 180
to polygons, to
derive the
properties of
their exterior
WWBAT calculate the sum of the interior angles of polygons;
WWBAT calculate the measure of one interior angle of a regular
polygon.
1. Find the sum of the measures of the interior angles of a
hexagon.
**NOTE: this
lesson focuses
on extending
the rule of 180
to polygons, to
derive the
formula for the
sum of their
interior
angles**
3.08
(E) 60o
WWBAT use inductive reasoning to make generalizations about
the sum of the interior angles of polygons;
2. A certain regular convex polygon has an interior angle sum
of 2160o. How many sides does the polygon have?
3. Find the measure of one of the interior angles of the stop
sign, shown below.
MP.1
MP.7
4. The interior angles of a regular n-gon each have a measure of
165o. Find n.
(A) 3
(B) 15
(C) 24
(D) 177.8
WWBAT use inductive reasoning to make generalizations about
the measures of, and sum of the measures of, exterior angles of
polygons;
WWBAT calculate the measures of interior and exterior angles of
regular and irregular polygons using the Polygon Angle Sum
Theorems.
1. Draw a polygon in the space below. Label one interior angle
and one exterior angle of the polygon.
2. What is the sum of the measures of the exterior angles of a
convex heptagon?
3. If all interior angles of the polygon below are congruent,

The Polygon Interior Angle Sum Theorem can
be derived from splitting polygons into nonoverlapping triangles. Any n-gon can be split
into (n – 2) non-overlapping triangles, and
therefore the sum of the interior angles of any
n-gon is 𝑆 = (𝑛 − 2) ∙ 180.
o For a regular n-gon, the measure of each
(𝑛−2)∙180
interior angle is 𝐸 =
.
𝑛
o For irregular n-gons, the interior angle sum
can be calculated, and then the measures of
the known angles can be used to calculate
the measure of the unknown angle.
 When you think you’ve solved the problem, it
can help to re-substitute your answer to check
that it gives you the correct angle sum (or
individual angle measure) to make sure you
didn’t make an algebraic or arithmetic error!

An exterior angle of a polygon is formed by
extending one side of the polygon out. This line
forms an exterior angle with the adjacent side.
The exterior angle forms a linear pair with the
adjacent interior angle.
 The sum of the measures of the exterior angles
of any convex n-gon is 360o.
o For a regular n-gon, the measure of each
360
exterior angle is 𝐸 = .
𝑛
o For irregular n-gons, there are two ways to
calculate an exterior angle: (1) find the
measure of the adjacent interior angle, then
use properties of linear pairs, or (2) writing
an equation in which all exterior angles sum
AF Geometry, Unit #3
then x =
angles**
(A)
(B)
(C)
(D)
(E)

60
65
72
80
84
to 360o.
A focus should be made this day on stressing the
relationship between interior and exterior
angles of polygons (rather than memorizing that
exterior angles always sum to 360o); this aligns
to the way exterior angle questions are tested on
the SAT.
4. Use the diagram below to find the value of x.
(NOTE: the diagram is not drawn to scale).
3.09
Given angle measurements of triangles, WWBAT rank the sides
using triangle inequality relationships;
Given side lengths of triangles, WWBAT rank the angles using
triangle inequality relationships.
1.
2.


In any triangle, the longest side of the triangle is
always located across from the largest angle and
the shortest side of the triangle is always located
across from the smallest angle.
When asked to solve a problem in which a
diagram is given, it is helpful to first sketch a
diagram, in order to identify relationships.
AF Geometry, Unit #3
Use ΔGHJ, below, to answer questions 3 and 4.
3. Which is the longest
side of the triangle?
Explain.
4.
3.10
G-CO.10
(base
angles
theorem)
MP.1
MP.6
MP.7
Which is the shortest
side of the triangle?
Explain.
Given isosceles triangles, WWBAT use the base angles theorem
and rule of 180 to find missing angle measures.
1. In ΔPHA, below, ̅̅̅̅
𝐻𝑃 ≅ ̅̅̅̅
𝐻𝐴.
If 𝑚∠𝑃 = (3x – 10)o and 𝑚∠𝐴 = (2x + 6)o, find x.



2. Use the diagram below to find the values of h and k.


In an isosceles triangle, two sides are congruent.
These sides are referred to as the “legs”.
o The angles located opposite the legs are
called the “base angles”. The base angles of
an isosceles triangle are congruent.
The third (non-congruent) side of an isosceles
triangle is called the “base” of the triangle.
o The third (non-congruent) angle of an
isosceles triangle called the “vertex angle”.
o The vertex angle is located opposite the
base.
Make annotations on the triangles to show where
the base angles are located – for example, draw
arrows across from the congruent sides. Misidentification of the base angles will lead to
incorrect equations.
Always be sure of (and name!) an angle pair
relationship before you write an equation – and
be specific! (ex. ∠𝑋 and ∠𝑌 are base angles, so
they are congruent. THEN 6x + 7 = 3x + 12, and
solve. This, in addition to your annotations, will
prevent careless mistakes – for example, setting
the vertex angle equal to one of the base angles.
Show all work when solving equations!
AF Geometry, Unit #3
3.11
G-CO.10
(triangle
perp.
bisectors)
MP.1
WWBAT describe the properties of perpendicular bisectors of
triangles;
Given perpendicular bisectors of triangles, WWBAT use their
properties to find missing angle and segment lengths.
The perpendicular bisectors of ΔABC meet at point P, as
shown below. Use the diagram to answer questions 1 and 2.
1. Which of the following is the
proper name for P?
(A) Incenter
(B) Circumcenter
(C) Orthocenter
(D) Centroid


2. If AP = 5 and AB = 8, what is BP?
Justify your answer.
̅̅̅̅̅ , 𝑇𝐶
̅̅̅̅ , and 𝐶𝑉
̅̅̅̅ are the
3. In the diagram below, 𝑊𝐶
̅̅̅̅
perpendicular bisectors of sides 𝑋𝑌, ̅̅̅̅
𝑍𝑋, and ̅̅̅̅
𝑌𝑍 ,
̅̅̅̅
respectively. Complete the paragraph proof to show that 𝐶𝑋
̅̅̅̅.
≅ 𝐶𝑍
Since ______is the perpendicular
X
̅̅̅̅, point T is the
bisector of 𝑋𝑍
______________ of ̅̅̅̅
𝑋𝑍.
W
C
T
Y
V
Then, by ____________________,
̅̅̅̅ ≅ ______. Additionally, ̅̅̅̅
𝑋𝑇
𝑇𝐶 ≅
______ by ___________________.
Z
̅̅̅̅ angles
Since ______ is the perpendicular bisector of 𝑋𝑍
________ and _______ are right angles, and therefore are
congruent by _________. Therefore, ΔXTC ≅ __________ by
_____________________. Since corresponding parts of
________________________________ are congruent, ̅̅̅̅
𝐶𝑋 ≅ ̅̅̅̅
𝐶𝑍.

When three or more lines, rays, or segments
intersect at the same point, they are called
concurrent. The point of intersection is called the
point of concurrency.
Any point that lies on the perpendicular bisector
of a segment is equidistant from the endpoints of
that segment.
o The three perpendicular bisectors of a
triangle intersect at a point, called the
circumcenter, which is equidistant from the
vertices of the triangle (the endpoints of the
sides).
o The above theorem can be proven using
properties of congruent triangles (see exit
ticket, question #3).
o The Pythagorean Theorem can be used to
find missing segment lengths, since the
triangles formed by a perpendicular bisector
and the line from a vertex to the circumcenter
will always be a right triangle.
The circumcenter of a triangle can be used to
construct a circle passing through all three
vertices of the triangle (the circumscribed circle
of the triangle).
o The circumcenter of a triangle may lie inside
the triangle (if the triangle is acute), on one
side of the triangle (if the triangle is right), or
outside the triangle (if the triangle is obtuse).
o For a right triangle, the circumcenter will
always lie on the triangle’s hypotenuse.
AF Geometry, Unit #3
3.12
G-CO.10
(triangle
angle
bisectors)
MP.1
WWBAT describe the properties of angle bisectors of triangles;
Given angle bisectors of triangles, WWBAT use their properties to
find missing angle and segment lengths.
̅̅̅̅ are the angle bisectors of ΔABC, shown below.
̅̅̅̅, 𝑯𝑷
̅̅̅̅̅, and 𝑮𝑷
𝑭𝑷
Use the diagram to answer questions 1 and 2.
1. Which of the following is the
proper name for P?
(A) Incenter
(B) Circumcenter
(C) Orthocenter
(D) Centroid
2. If PY = 2 and HP = 3, what is HY?
Justify your answer.



When three or more lines, rays, or segments
intersect at the same point, they are called
concurrent. The point of intersection is called the
point of concurrency.
Any point that lies on the bisector of an angle is
equidistant from the two sides of the angle.
o The three angle bisectors of a triangle
intersect at a point, called the incenter, which
is equidistant from the sides of the triangle.
The incenter of a triangle can be used to
construct a circle that is tangent to the three sides
of the triangle (the inscribed circle of the
triangle).
o The incenter of a triangle always lies in the
interior of the triangle.
3. In the diagram below, P is the point of concurrency of the
angle bisectors of ΔABC. Which of the following statements
are true?
̅̅̅̅ ≅ ̅̅̅̅
(A) 𝐴𝑃
𝐶𝑃
(B) ∠𝑆𝐵𝑃 ≅ ∠𝑄𝐵𝑃
(C) ̅̅̅̅
𝑅𝑃 ≅ ̅̅̅̅
𝑃𝑄
(D) Both (A) and (B)
(E) Both (B) and (C)
3.13
G-C.3
(construct
inscribed
& circumscribed
circles)
MP.5
MP.6
MP.7
Given a triangle, WWBAT construct the inscribed or
circumscribed circles of the triangle by constructing the incenter
or circumcenter of the triangle.
1. Adriana performed a construction, resulting in the diagram
below. Circle the word that correctly fills in each blank to
describe the figure Adriana created.
Point L is the (circumcenter,
incenter) of ΔABC. The point
was found by constructing the
(perpendicular bisectors,
angle bisectors) of the
triangle. Point L is equidistant
from the (sides, vertices) of
the triangle.
Same as key points for 3.11 and 3.12, above, with the
addition of the following:
 Points of concurrency are useful when creating
geometric constructions.
o The incenter of a triangle can be used to
construct a circle inscribed inside the
triangle.
o The circumcenter of a triangle can be used to
construct a circle around the triangle (that is,
a circle containing the vertices of the
triangle). This is called a circumscribed
circle.
 It can be helpful to first identify the type of
construction you are creating: ask yourself
AF Geometry, Unit #3
2. Construct the circumcircle of the triangle shown below.
whether you:
1) Want the circle to touch the sides (and thus,
want the center to be equidistant from the
sides.
2) Want the circle to touch the vertices (and
thus, want the center to be equidistant from
the vertices).
This thought process will allow you to determine
whether to construct the perpendicular bisectors
or angle bisectors of the triangle.
NOTE: an excellent vocabulary review activity can be
found here, and may be useful for reviewing
properties of incenters and circumcenters. Mastery of
these concepts is crucial to success with circle
constructions.
3.14
G-CO.10
(triangle
medians)
MP.1
WWBAT describe the properties of medians of triangles;
Given medians of triangles, WWBAT use their properties to find
segment measures by writing and solving equations.
Fill in the blanks:
1. A median of a triangle is a ____________ whose endpoints
are a _____________ and the ____________ of the opposite
side.
2. The centroid of a triangle divides each median into two
segments whose lengths are in the ratio _____ : ______.
Use the diagram below, to answer questions 3 – 7.
Assume G is the centroid of ΔABC, CG = 13, AD = 15, FG = 8,
̅̅̅̅.
and ̅̅̅̅
𝑨𝑫 ⊥ 𝑪𝑩
3. Find the length of ̅̅̅̅
𝐴𝐺 .
4. Find the length of ̅̅̅̅
𝐺𝐷 .
5. Find the length of ̅̅̅̅
𝐶𝐷 .
6. Find the length of ̅̅̅̅
𝐺𝐸 .
7. Find the length of ̅̅̅̅
𝐺𝐵 .
NOTE: make sure to give scholars enough space to
complete the construction!
 When three or more lines, rays, or segments
intersect at the same point, they are called
concurrent. The point of intersection is called the
point of concurrency.
 A median of a triangle is a segment from a vertex
to the midpoint of the opposite side. A median
bisects one side of the triangle.
o The three medians of a triangle are
concurrent. The point of concurrency, called
the centroid, always lies in the interior of the
triangle.
o The centroid is two-thirds the distance from
each vertex to the midpoint of the opposite
side (or, stated another way, the centroid
divides a median into two segments whose
lengths are in the ratio 2:1, with the longer
segment nearest the vertex).
NOTE: this may also be a good day for vocabulary
review, and questions may be added to the exit ticket
about incenters/circumcenters. Emphasis should be
placed on the properties of each type of special
segment, and on properties of the concurrency point.
AF Geometry, Unit #3
3.15
G-CO.11
(identify
basic
properties
of quadrilaterals)
WWBAT describe properties of parallelograms.

Given a parallelogram, WWBAT use its properties to find missing
angle and segment measures.
Each figure shown below is a parallelogram.
Find the indicated quantity. Justify your answers for full
credit.
1. Find 𝑚∠𝑅.
2. Given RT = 19.8, find RP.

3.
Find 𝑚∠𝐹.
4. Given 𝑚∠𝑌𝑋𝑉 = 52o and
𝑚∠𝑋𝑊𝑉 = 60o, find 𝑚∠𝑉𝑋𝑊.


Parallelograms are quadrilaterals in which both
pairs of opposite sides are parallel. Several
important theorems follow from this fact:
o If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
o If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
o If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
o If a quadrilateral is a parallelogram, then its
diagonals bisect each other.
Always name the relationship between the sides
or angles (justification!) before you write an
equation – and be specific! (ex. opposite sides of
a parallelogram are congruent, so 6x + 7 = 42;
then solve. This will help to prevent careless
mistakes.
Show all work when solving equations!
For a challenge problem, consider something
like the following:
In the diagram below, GFED is a parallelogram.
Find 𝑚∠𝐹.
5. In the diagram below, EFGH is a parallelogram whose
diagonals intersect at U. If UE = 24, HU = 2y + 2, UG = 10x – 6,
and UF = 18, find the values of x and y.
(A) x = 3, y = 8
(B) x = 8, y = 3
(C) x = 2.4, y = 11
(D) x = 11, y = 2.4
AF Geometry, Unit #3
3.16
G-CO.11
(prove
theorems
about
parallelograms)
MP.3
Given a parallelogram, WWBAT prove statements about it by
applying theorems about parallelograms and properties of
congruent triangles.
For questions 1 – 5, below, decide whether there is enough
information to classify the quadrilateral as a parallelogram. If
not, state any additional information you would need.
1. Opposite sides are parallel
2. Two pairs of consecutive sides are congruent.
3. Diagonals are congruent.
4. Consecutive angles are supplementary.
5. All four sides are congruent.
For question #6, below, write a proof. You may choose to
create a two-column, flow, or paragraph proof.
Given: ΔAXB ≅ ΔCXD
Prove: ABCD is a parallelogram
3.17
G-CO.11
(classify
quadrilaterals)
MP.3
MP.6
WWBAT classify quadrilaterals as parallelograms, rectangles,
rhombi, and/or squares by applying properties of their sides and
angles.
1. Can quadrilateral TWZV, below, be classified as a rectangle?
Explain.
Same key points as above, with the addition of:
 Proving any of the five properties of
parallelograms is sufficient to prove that a
quadrilateral is a parallelogram.
 Once it has been proven that a quadrilateral is a
parallelogram, other information can be proven,
using several different techniques:
1) Other properties of parallelograms, to prove
sides parallel/congruent, or to prove
statements about angles.
2) Properties of congruent triangles, which
follow from a parallelogram divided in half
by one or more diagonals.




A rectangle is a parallelogram in which the
adjacent sides are perpendicular to each other –
that is, a parallelogram in which all four angles
are right angles.
A parallelogram is a rectangle if and only if its
diagonals are congruent. This follows from the
SSS congruence postulate and the fact that
consecutive angles of a parallelogram are
supplementary (by using CPCTC to show that the
two consecutive angles must be congruent, we
can show that they must each equal 90o).
A parallelogram is a rhombus if all four of its
sides are congruent.
A parallelogram is a square if it is both a
AF Geometry, Unit #3
rectangle and a rhombus; that is, if the adjacent
sides are perpendicular to each other and if all
four sides are congruent.
2. Nathan and Mercedes are trying to classify a quadrilateral.
Nathan says, “this quadrilateral is a rhombus!”. Mercedes
says, “this quadrilateral is a rectangle!”. Could Nathan and
Mercedes both be correct? Explain.
Circle the figures for which each statement below is true
(NOTE: for some questions, more than one answer may be circled).
3. It is equiangular.
(A) Parallelogram
(B) Rectangle
(C) Rhombus
(D) Square
4. Opposite sides are congruent.
(A) Parallelogram
(B) Rectangle
(C) Rhombus
(D) Square
5. Adjacent angles are not necessarily congruent.
(A) Parallelogram
(B) Rectangle
(C) Rhombus
(D) Square
AF Geometry, Unit #3
COMMON MISCONCEPTIONS
What misconceptions will prevent scholars from reaching mastery?
Misconception
Clarification
Confusion when multiple lines intersect to
Have scholars re-draw the diagram with only two lines (or use a highlighter to
form complex vertical angles – which set is emphasize pairs of lines) to end up with pairs of vertical angles (with each VA being
congruent?
the sum of the smaller angles that make it up).
Thinking that same-side interior angles are Same-side interior angles are supplementary. This fact is derived from congruent
congruent (since all other sets of angle
corresponding angles – the exterior corresponding angle forms a linear pair with
pairs formed by a transversal and two
its’ adjacent interior angle, so the interior corresponding angle must also be
parallel lines are congruent).
supplementary to this angle.
Confusion with multiple variable problems
– what can I solve for first?
Postulates/theorems vs. converses – why
are there two? (example: corresponding
angles postulate, and corresponding
angles converse)
Answering an incorrect question,
especially when using theorems about
angle pairs (for example, only solving for x
when the problem asks for the measure of
the largest angle)
NOTE: this misconception provides a good opportunity for a confounding activity,
where scholars could potentially practice constructions and verify. angle measures –
if time, this would be good for a flex day.
Name ALL pairs of angles you see first, and try to write equations. Find one equation
that has only one variable, and solve it, then substitute if necessary and keep going
until you’ve found all the variables.
This is usually a grit issue, and one of the first places that scholars struggle to find an
“entry point” to a complex problem.
Re-emphasize the difference between a conditional statement and its converse:
hypothesis and conclusion are switched. The “if” statement is based on what you
know, and the “then” statement is what you can prove – so the postulate/theorem
and its converse are used to prove different things (specifically, the
postulate/theorem is used to prove that angles are congruent, while the converse is
used to prove that lines are parallel).
This misconception is actually two separate misconceptions that produce the same
result. Either:
1) The scholar is not carefully reading the problem statement, resulting in them
calculating the wrong value, OR
2) The scholar is trying to rush to complete the problem (this is especially true of
multiple-choice questions, where intermediary quantities are often present as
distractors).
Emphasize to scholars the importance of careful reading and understanding of the
AF Geometry, Unit #3
Setting incorrect angles equal to each other
in isosceles triangles – especially when the
triangles are rotated so that the “base” is
not the “bottom” segment.
question, both before you begin solving AND after you have begun calculations.
Answering the wrong question results in the wrong answer!
The base angles of an isosceles triangle are always located across from the
triangle’s legs (the two congruent sides). It is important to carefully identify the legs
of the triangle (and, if necessary, annotate to indicate the base angles) before
attempting to set up or solve any equations.
Errors when identifying or constructing
special segments of triangles – specifically,
perpendicular bisector vs. altitude vs.
median.
NOTE: this misconception provides a good opportunity for a confounding activity,
where scholars could potentially practice constructions and verify. angle measures –
if time, this would be good for a flex day.
The perpendicular bisector of the side of a triangle does not necessarily pass
through the opposite vertex – it passes through the midpoint of the side, and is
perpendicular to that side (only for isosceles and equilateral triangles will this
segment also pass through a vertex).
The altitude of a triangle does not necessarily bisect the opposite side. As above,
this will occur only for isosceles and equilateral triangles.
The median of a triangle is not necessarily perpendicular to the opposite side.
Again, this will occur only for isosceles and equilateral triangles.
Why do we only need two bisectors to find
a point of concurrency? (for both
perpendicular bisectors and angle
bisectors)
Difficulties using incenter to construct
inscribed circle (or circumcenter to
construct circumscribed circle)
When performing constructions on triangles, it is important to understand the
properties of the segment you are creating – is it perpendicular to a side? Does it
pass through a vertex? Through a midpoint? Asking yourself these questions will
allow you to perform the proper construction correctly.
When constructing incenters and circumcenters of triangles, we are looking for a
point that is equidistant from three non-collinear points. In both cases, the three
constructed lines (either the perpendicular bisectors or angle bisectors) will be
concurrent. However, since the intersection of any two lines is a point, only two of
the three segments must be constructed to identify this point.
Once you have located the desired point of intersection (the incenter or
circumcenter), think about the type of circle you are trying to draw and the
properties of the point:
 When constructing the inscribed circle, remember that the incenter is
equidistant from the sides of the triangle. Use a compass to measure the
AF Geometry, Unit #3
smallest distance between the incenter and one of the sides, then use this
radius to construct the circle.
 When constructing the circumscribed circle, remember that the circumcenter
is equidistant from the vertices of the triangle. Use a compass to measure the
distance between the circumcenter and one of the vertices, then use this
radius to construct the circle.
Unit 3 introduces to a large number of vocabulary words, and there will be more to
come in later units. It may be a good idea to stop and review the new words before
the students become overwhelmed – have them make flashcards, or play a
vocabulary game in class/on homework to review properties.
Vocabulary overload!
ANTICIPATED LESSON CALENDAR
Monday
Tuesday
10/21/13
IA 1 review
10/28/13
Lesson #4
11/4/13
Lesson #8
10/22/13
IA 1 review
10/29/13
Lesson #5
11/5/13
Quiz 3.1: angle pair
relationships
11/12/13
Lesson #13
11/19/13
Lesson #17
11/11/13
Lesson #12
11/18/13
No school – AF PD day
Wednesday
10/16/13
Lesson #1
10/23/13
IA 1 administration
10/30/13
Lesson #6
11/6/13
Lesson #9
Thursday
10/17/13
Lesson #2
10/24/13
IA 1 administration
10/31/13
Lesson #7
11/7/13
Lesson #10
Friday
10/18/13
Lesson #3
10/25/13
IA 1 administration
11/1/13
No school – data day #1
11/8/13
Lesson #11
11/13/13
Lesson #14
11/20/13
Flex day (or unit 3
review)
11/14/13
Lesson #15
11/21/13
Unit 3 exam
11/15/13
Lesson #16
11/22/13
Begin unit 4