Geometry
Wasatch High School
2011-2012
Student Name _______________________
Teacher Name _______________________
Geometry
State Standards
Standard 1: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.
Objective 1: Use inductive and deductive reasoning to develop mathematical arguments.
a. Write conditional statements, converses, and inverses, and determine the truth value of these statements.
b. Formulate conjectures using inductive reasoning.
c. Prove a statement false by using a counterexample.
Objective 2: Analyze characteristics and properties of angles.
a. Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence.
b. Identify and determine relationships in adjacent, complementary, supplementary, or vertical angles and
linear pairs.
c. Classify angle pairs formed by two lines and a transversal.
d. Prove relationships in angle pairs.
e. Prove lines parallel or perpendicular using slope or angle relationships.
Objective 3: Analyze characteristics and properties of triangles.
a. Prove congruency and similarity of triangles using postulates and theorems.
b. Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the
Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the
Pythagorean Theorem.
c. Prove and apply theorems involving isosceles triangles.
d. Apply triangle inequality theorems.
e. Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides
of a triangle, and justify the concurrency theorems.
Objective 4: Analyze characteristics and properties of polygons and circles.
a. Use examples and counterexamples to classify subsets of quadrilaterals.
b. Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.
c. Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures,
perimeter, and area of regular polygons.
d. Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle,
and solve problems using their properties.
e. Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.
Objective 5: Perform basic geometric constructions, describing and justifying the procedures used.
a. Investigate geometric relationships using constructions.
b. Copy and bisect angles and segments.
c. Construct perpendicular and parallel lines.
d. Justify procedures used to construct geometric figures.
e. Discover and investigate conjectures about geometric properties using constructions.
Objective 6: Analyze characteristics and properties of three-dimensional figures.
a. Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).
b. Identify three-dimensional objects from different perspectives using nets, cross-sections, and twodimensional views.
c. Describe the symmetries of three-dimensional figures.
d. Describe relationships between the faces, edges, and vertices of polyhedra.
Standard 2: Students will use the language and operations of algebra to explore geometric
relationships with coordinate geometry.
Objective 1: Describe the properties and attributes of lines and line segments using coordinate geometry.
a. Verify the classifications of geometric figures using coordinate geometry to find lengths and slopes.
b. Find the distance between two given points and find the coordinates of the midpoint.
c. Write an equation of a line perpendicular or a line parallel to a line through a given point.
Objective 2: Describe spatial relationships using coordinate geometry.
a. Graph a circle given the equation in the form (x − h)2 + ( y − k)2 = r 2, and write the equation when given
the graph.
b. Determine whether points in a set are collinear.
2
Standard 3: Students will extend concepts of proportion and similarity to trigonometric ratios.
Objective 1: Use triangle relationships to solve problems.
a. Solve problems using the properties of special right triangles, e.g., 30°, 60°, 90° or 45°, 45°, 90°.
b. Identify the trigonometric relationships of sine, cosine, and tangent with the appropriate ratio of sides of
a right triangle.
c. Express trigonometric relationships using exact values and approximations.
Objective 2: Use the trigonometric ratios of sine, cosine, and tangent to represent and solve for missing
parts of triangles.
a. Find the angle measure in degrees when given the trigonometric ratio.
b. Find the trigonometric ratio given the angle measure in degrees, using a calculator.
c. Find unknown measures of right triangles using sine, cosine, and tangent functions and inverse
trigonometric functions.
Standard 4: Students will use algebraic, spatial, and logical reasoning to solve measurement
problems.
Objective 1: Find measurements of plane and solid figures.
a. Find linear and angle measures in real-world situations using appropriate tools or technology.
b. Develop surface area and volume formulas for polyhedra, cones, and cylinders.
c. Determine perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones,
and spheres when given the formulas.
d. Calculate or estimate the area of an irregular region.
e. Find the length of an arc and the area of a sector when given the angle measure and radius.
3
Homework (This list is subject to change).
Logic
Notebook Pages 6-9
Homework:
p.8 #17-23 odd, 27, 33, 35, 38. Explain your reasoning.
Homework:
p.27 #11,13,15,22,23,25,28,31
Points, Lines, Planes, Segments, Measurements
Homework:
Notebook Pages 10-13
p.16 #9-12,14,16,17,18,23, 24-29 all
p.59 #13-27 odds, 31
p.71 #15-23 odds, 25-33 odds
Angles Notebook Pages 13-20
Homework:
p. 93 #9-19 odds, 24,27,28
Homework:
p. 108 #11-23 odds, 27
Homework:
p.113 #8,9,10,14,16,23 p.120 #13,14,16,19,22,25,28
p.126 #9,11,13,14,15,17,21,24
Coordinate Geometry
Notebook Pages 20-25
Homework:
p.266 #13-19 odds, 27
Homework:
p.172 #11,15,18,25,28
Parallel Lines
p.80 #21-27 odds, 30, 36-40
p.178# 13,15,17,25
Notebook Pages 25-29
Homework:
p.178 #14,16,18,26-30
Homework:
p.152 #13-37 odds, 46
Homework:
p.166 #9-19, 25,26
Triangles
p.100 #11,17,23,27
Notebook Pages 30-37
Homework:
p.191 #9-21 odds, 26,27,32
Homework:
p.196 #9-19 odds, 22, 26-29
Homework:
p.231 #9,11,13,17
p.238 #9-15 odds
Homework:
p.286 #12,14,18,19
p.293 #9-21 odds, 22
Similar Triangle
p.353 #17-37 odds, 41,42
Homework:
p.359 #9-19 odds, 24, 25,30
Homework:
p.366 #7,8,9,11,16,20
Homework:
p.242 #7-11, 17,18
p.299 #9,11,13
Notebook Pages 37-40
Homework:
Congruent Triangles
p.249 #7,9,11,15
p.372 #15-21 odds, 24,28
Notebook Pages 41- 42
p.213 #12-15,17,19,22,23
p.218 #15,17,19-22,25-28
Trigonometry
Notebook Pages 49-55
Homework:
p.260 #17-35 odds,
Homework:
p.557 #7-13 odds,14,16
p.562 #7-13 odds, 16,17
Homework:
p.568 #9-19 odds, 20
p.576 #13-27 odds, 31, 36
4
Polygon and Polyhedron
Notebook Pages 49-55
Homework:
p.405 #13-33
p.437 #13-17 odds
Homework.
p. 499 #15,17,21-29 odds, 32,33
Quadrilateral, Polygons Notebook Pages 56- 62
Homework:
p.411 #9-17 odds, 18
Homework:
p.320 #11-23 odds, 28,29 p.325 #7-12, 14,15,16
Homework:
p.330 #16-21, 23-37 odds, 39-44
Homework:
p.337 #10,13-21 odds, 30,32,33,34
Circles
Notebook Pages 62- 70
Homework:
p.457 #13-27,29,30
Homework:
p.466 #13-23 odds, 31,41 p.590 #9,11,12-14,19,21,27
Homework:
p.596 #9-14, 19,27
Homework:
p.616 #8-13
Homework:
p.621 #15-25 odds,26,27,34,36
Area and Volume
p.481 #11,13,17,19,22,26
p.604 #9-17 odds
Notebook Pages 71- 83
Homework:
p.416 #9,12,13,15,21
p.423 #11,12,14,15
Homework:
p.429 #7,8,9,11,12,15
p.486 #9,11,13,17
Homework:
p.487 #19-21, 24,25,27,29 p.482 #27
Homework:
p.508 #7,9,11,16,18,19
Homework:
p.509 #13-15,17
Homework:
p.520 #7-14 all
p.531 #7-13 (surface area only)
Homework:
p.513 #12,14,15,18,19,20
p.525 #9-17 odds, 22
p.531 #8,11,16,21
5
Utah State Standards
1.1.a Write conditional statements, converses, and inverses, and determine the truth
value of theses statements.
1.1.b Formulate conjectures using inductive reasoning.
1.1.c Prove a statement false by using a counterexample.
Conjecture
Inductive reasoningEx. Make a conjecture about “Pascal’s Triangle”. Explain your reasoning.
Ex. Make a conjecture about the next number based on the pattern. 1, 3, 6, 10, 15
Explain your reasoning.
Ex. Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240
Explain your reasoning.
Ex. Make a conjecture about the next number based on the pattern. 1,1,2,3,5,8,…
Explain your reasoning.
Ex. Make a conjecture about the next number based on the pattern. 1,4,9,16,…
Explain your reasoning.
6
True means always true.
It only takes one false example to show that a conjecture is not true.
Counterexample
Ex. Find a counterexample to the statement: All geometry students have blue eyes.
Example p.6 #4
Ex. Based on the table showing unemployment rates for various counties in Kansas, find
a counterexample for the following statement: The unemployment rate is highest in the
counties with the most people. Source: Labor Market Information Services-Kansas Dept. of Human Resources
County
Shawnee
Jefferson
Jackson
Douglas
Osage
Wabaunsee
Pottawatomie
Civilian Labor Force
90,254
9,937
8,915
55,730
10,182
3,575
11,025
Rate
3.1%
3.0%
2.8%
3.2%
4.0%
3.0%
2.1%
Homework: p.8 #17-23 odd, 27, 33, 35, 38. Explain your reasoning.
Worksheet p.44 Exercise Set 1.2 evens. Explain your reasoning.
Conditional Statement
If p, then q.
p
q
7
Ex. Write your own example of a conditional statement.
Ex. Identify the hypothesis and conclusion.
Ex. Put the given information in a conditional statement. Get $1500 cash back when you
buy a new car.
Converse statement
If
, then
.
Negation ~
Inverse statement
If
, then
.
Contrapositive statement
If
, then
.
Ex. Write the converse, inverse, and contrapositive of the statement:
If a shape is a square, then it is a rectangle.
Ex. Write the converse, inverse, and contrapositive of the statement and determine the
truth value for each. If false, give a counterexample.
If a person lives in Heber, then that person lives in Utah.
Truth value:
Converse:
Truth value:
8
Inverse:
Truth value:
Contrapositive:
Truth value:
Building a logical argument:
Ex.
If a person buys worms, then they can go fishing.
If a person can go fishing, then they can catch dinner.
New logical statement:
Ex.
If x=7, then 4x=28.
If 4x=28, then 20x=140.
New logical statement:
Homework:
p.27 #11,13,15,22,23,25,28,31
CDAS
9
Utah State Standards
1.2.a Use accepted geometric notation for lines, segments, congruence.
1.5.b Copy segments using constructions.
2.2.b Determine whether points in a set are collinear.
4.1.a Find linear measures in real-world situations using appropriate tools or
technology.
Reading p.77-79 Introducing Geometry
point*No shape or size
Diagram
Named by
Line*No thickness or width
Diagram
Named by
RayDiagram
Named by
Plane*No thickness
Diagram
Named by
CollinearCoplanar-
10
Practice. Model and study real life objects.
What term would we use to model the floor?
What term would we use to model the hand on a clock?
What term would we use to model the corner of the driveway where the concrete
meets the road?
Intersection-
2 planes
2 lines
a plane and a line
11
*Points, lines, and planes have no real measurement. In real life, we have things with a
certain shape and size. Things we can measure!
Line segment or SegmentDiagram
Named by
Practice p.15 #3-7
Length of a segment
Practice with rulers!
centimeter side
inches side
Practice measuring the following segments
1. Use cm.
2. Use inches.
3. Use inches.
Segment Addition
Draw a diagram of a point that lies between two others.
12
Practice. p.59 #4,6
Homework. p.16 #9-12,14,16,17,18,23, 24-29 all
p.59 #13-27 odds, 31
p.71 #15-23 odds, 25-33 odds
Utah State Standards
1.2.a Use accepted geometric notation for rays, angles.
1.5.b Copy and bisect angles using constructions.
1.5.d Justify procedures used to construct geometric figures.
4.1.a Find angle measures in real-world situations using appropriate tools or
technology.
Claudius __________ first used the unit of measure we think of as a ______________.
Rotation: once around ______, half a turn ______
1 = ________ of a turn around a circle.
Opposite raysDiagram
AngleDiagram
Label the following:
Sides of the angle
Vertex
Interior and exterior of an angle
13
Naming an angle
A
Practice.
Name the vertex of 2 .
4
3
Name the sides of 4 .
Write another name for BDC .
B
C
2
1 D
We measure an angle in units of ____________.
Notation for the measure of an angle
A __________________ is a tool to help us measure angles.
Practice. Measure the following angles using a protractor.
Most computer programs will also measure angles for us!
14
Classifying Angles by their measure
Acute angles
Diagram
Right Angles
Diagram
Obtuse Angles
Diagram
Straight Angles
Diagram
Practice. Classify each angle as right, acute, or obtuse.
WXY
Z
W
WXZ
Y
X
HW: p. 93 #9-19 odds, 24,27,28
p.100 #11,17,23,27
15
Utah State Standards
1.2.a Use accepted geometric notation for rays, angles.
1.5.b Copy and bisect angles using constructions.
1.5.d Justify procedures used to construct geometric figures.
4.1.a Find angle measures in real-world situations using appropriate tools or
technology.
Angle Addition
Draw an angle with vertex Q and a point H in its interior. Draw a ray from Q through
H.
Practice. P.108 #7,8
Congruent AnglesDiagram
Notation
Ex. Finding Angle Measure with algebra.
Given: ABC  DBF
mABC  6 x  2
C
A
mDBF  8 x  14
Find x
6x+2
F
B
8x -14
D
16
Angle bisectorDiagram
T
S
R
An angle bisector creates two ___________ angles.
Practice.
QPand QR are opposite rays. QT bisects RQS
If mRQT  6 x  5 and mSQT  7 x  2 ,
Find mRQT .
Q
P
Homework Problems p. 108 #11-23 odds, 27
Utah State Standards
1.2.b Identify and determine relationships in adjacent, complementary,
supplementary, or vertical angles and linear pairs.
1.5.c Construct perpendicular lines
Special Pairs of Angles (two angles)
Adjacent Angles-
Diagram
*Things to watch out for…
Shared interior
No common vertex
17
Linear PairDiagram
Supplementary Angles2 Diagrams
Ex. Supplement of an angle
1. What is the supplement of a 120 degree angle?
2. How could you use algebra to write the “supplement of an angle”?
3. Find the measures of two supplementary angles if the measure of one angle is
6 less than five times the measure of the other angle.
Complementary Angles2 Diagrams
Theorem- If the non-common sides of two adjacent angles form a right angle, then the
angles are ___________________ angles.
Ex. Complements of an Angle
1. What is the complement of a 40 degree angle?
2. How could you use algebra to write the “complement of an angle”?
18
3. Find the measures of two complementary angles if the difference of the measures
of the two angles is 12.
Vertical Angles
Diagram
*Must be formed by two intersecting lines!
Vertical angles are _____________.
Perpendicular LinesPerpendicular lines intersect to form ________ right angles.
Diagram
Practice.
Find the measure of angle 1.
22 
1
19
L
Find x so that KO  HM
M
K
N
3x+6
J
I
9x
O
H
*Never assume lines are perpendicular!
HW: p.113 #8,9,10,14,16,23 p.120 #13,14,16,19,22,25,28
p.126 #9,11,13,14,15,17,21,24
CDAS
Utah State Standards
2.1.b Find the distance between two given points and find the coordinates of the
midpoint.
1.5.b Bisect segments using constructions.
4.2.b Solve problems using the distance formula.
Coordinate System
Who is Rene Descartes?
Points on the system
Quadrants
20
Practice. Drawing geometric figures on the coordinate system.
QR on a coordinate plane contains Q(-2,4) and R(4, -4). Add point T so
that T is collinear with these points.
Practice Square Roots p.552
21
Investigation. Distance in Coordinate Geometry
Segment length on the coordinate system
We need the coordinates of the endpoints.
The Distance between any two points ( x1 , y1 ) and ( x2 , y2 ) :
Ex. Using the distance formula to find the length of a segment
1. Find the length of EF given the coordinates E (-4, 1) and F(3, -1)
2. Find the length of GH given the coordinates G (2, 8) and H (-2, 2)
Perimeter of a closed figure
Ex. Finding perimeter on a coordinate Plane
*Use the distance formula
Find the perimeter of triangle PQR if P(-5,1), Q(-1,4), and R(-6,-8).
22
CongruentSymbol
Notation - _____________________ means AB = CD.
Diagram-
Midpoint of a Segment-
Practice. p.66 #7
Investigation. Midpoint Conjecture
Finding the midpoint on a coordinate system
Ex. Finding the midpoint of a segment.
1. Find the coordinates of M, the midpoint of PR for P (-1, 2) and R (6, 1).
2. Find the coordinates of D if E is the midpoint of DF . E (-6,4) and F(-5, -3).
Segment Bisector-
HW: p.266 #13-19 odds, 27
p.80 #21-27 odds, 30, 36-40
23
Utah State Standards
1.2.e Prove lines parallel or perpendicular using slope or angle relationships
2.1.c Write an equation of a line perpendicular or a line parallel to a line through a
given point.
Investigation Equations of Lines
The equation of a line on a coordinate plane describes where it is and how it slants up or
down.
Slope
Practice. p.171 #4-6
Positive Slope
Negative Slope
Slope = 0
Slope undefined
Slope-intercept form of a line.
y = mx + b
m is the ___________. b is the _______________.
Practice. p. 177 #6, 3, 7
Ex. Writing the equation of a line on a coordinate plane.
1. Write the equation of a line given the slope = 2, and the y-intercept is (0, 6)
24
2. Write the equation of a line given the slope =3, and the point (-1, 4) that lies
on the line.
3. Write the equation of a line given two points on the line (-2, 6) and (1, 5).
HW: p.172 #11,15,18,25,28
p.178# 13,15,17,25
CDAS
Utah State Standards
1.2.e Prove lines parallel or perpendicular using slope or angle relationships
2.1.c Write an equation of a line perpendicular or a line parallel to a line through a
given point.
Parallel Lines-
Symbol
Postulate- Two lines are parallel if and only if their slopes are _____________________.
Perpendicular Lines-
Symbol
Postulate- Two lines are perpendicular if and only if their slopes are ________________.
If we multiplied the two slopes together we get _______.
1
Ex. A line has a slope of 3, A second line has a slope of  .
3
25
1. Write the equation of a line perpendicular to the line described by
y = -2x + 5 , given a point on the line (-3, 4).
Check by graphing:
2. Write the equation of a line parallel to the line described by y  4 x  7 , given
a point on the line (-6, 1).
Check by graphing:
HW: p.178 #14,16,18,26-30
Utah State Standards
1.2.c Classify angle pairs formed by two lines and a transversal.
________________ lines are coplanar lines that do not intersect.
Segments and rays on those lines are also ___________.
Notation:
Diagram with symbols:
Symbol for not parallel to
26
_____________planes never intersect.
Example and Diagram:
________________ lines do not intersect and are NOT coplanar (nor parallel)
Segments and rays on those lines are also __________.
Diagram
Practice problems p. 144 #4-11
A line that intersects two or more lines in a plane at different points is called a
__________________.
Ex.
Special angles formed by transversals
We will say line p is our transversal.
Angles on the same side of the transversal
are called ______________.
Angles on opposite sides of the transversal
are called _____________.
The lines crossed are q and r
The space between lines q and r is known
as ________________.
The space outside lines q and r is know as
________________.
Consecutive Interior Angles:
Sometimes called Same-Side Interior Angles
Alternate Interior Angles:
Alternate Exterior Angles:
Corresponding Angles: in the same relative position.
27
Investigation. Special Angle Relationships
Practice Problems p. 152 #4-7
**Special angles formed by a transversal are even more special when the lines crossed
are ___________________.
Postulate: If two parallel lines are cut by a transversal, then corresponding angles are
_________________.
Diagram
Ex. angle measures of special angle pairs.
kl
k
m11  51
Find m16
10
12
15
14
l
16
11
13
17
j
Theorem- If two parallel lines are cut by a ______________, then each pair of alternate
interior angles is _______________.
Theorem- If two parallel lines are cut by a _______________, then each pair of
consecutive interior (same-side interior) angles is ____________________.
Theorem- If two parallel lines are cut by a _______________, then each pair of alternate
exterior angles is ____________________.
28
Practice. p.153 #38
HW: p.152 #13-37 odds, 46
Utah State Standards
1.2.e Prove lines parallel using angle relationships
If we are not on the coordinate system, how do we prove that lines are parallel?
Recall: slopes of parallel lines are ____________.
Ways to prove lines are parallel
1. Show corresponding angles are ___________.
2. Show alternate exterior angles are ____________.
3. Show consecutive interior angles are ______________.
4. Show alternate interior angles are ____________.
5. Show that two lines are _____________ to the same line.
*We must use one of these 5 rules to prove lines are parallel!!!!
Practice. p. 165 #4,5
HW: p.166 #9-19 (Give a reason with vocabulary words), 25,26
CDAS
29
Utah State Standards
1.3 Analyze characteristics and properties of triangles
1.3.c. Prove and apply theorems involving isosceles triangles.
2.1.a Verify the classifications of geometric figures using coordinate geometry to
find the lengths and slopes.
TriangleDiagram
Sides:
Vertices:
Angles:
Investigation Triangle Classification
Classifying Triangles by Angles
Acute Triangle
Diagram
*special example: equiangular triangle
Obtuse Triangle
Diagram
Right Triangle
Diagram
Practice problems: classification by angle p. 190 #4-6
30
Classifying Triangles by Sides
Scalene Triangle
Diagram
Isosceles Triangle
Diagram
Vertex angleBase angles-
Practice. p.190 #4-6 by sides
Practice. 190 #7
Homework: p.191 #9-21 odds, 26,27,32
Investigation. Angles of a Triangle p.193 of textbook.
31
Angle Sum Theorem- The sum of the measures of the angles of a triangle is _________.
Practice Problems:
1. Find x
2. Find the missing angle measures.
x
43
79
85
x
y
74
35
Corollary: The acute angles of a right triangle are __________________.
z
Investigation. Discovering Properties of Isosceles Triangles
Isosceles Triangle TheoremDiagram and Abbreviation
Converse of Isosceles Triangle TheoremDiagram and Abbreviation
*Look at an equilateral triangle…
Corollary- A triangle is equilateral if and only if it is ___________________.
Corollary- Each angle measure of an equilateral triangle measures ________.
Diagram
32
Exterior Angle
Diagram
Remote Interior Angles
Exterior Angle Theorem- the measure of an exterior angle of a triangle is equal to the
sum of _______________________________________________________________.
Practice.
Find the measure of each numbered angle.
3
2
50
1
120
4
78
HW: p.196 #9-19 odds, 22, 26-29
5
56
p.249 #7,9,11,15
Utah State Standards
1.3.e Identify medians, altitudes, and angle bisectors of a triangle, and the
perpendicular bisectors of the sides of a triangle, and justify the concurrency
theorems.
1.5.a Investigate geometric relationships using constructions
1.5.e Discover and investigate conjectures about geometric properties using
constructions.
Concurrent LinesDiagram
Point of Concurrency
33
Sketchpad Investigation. Segments of a Triangle.
Perpendicular Bisector of a Segment
Are the 3 perpendicular bisectors of a triangle concurrent lines?
CircumcenterEquidistant from-
Angle bisector
Are the 3 angle bisectors of a triangle concurrent lines?
Incenter-
Equidistant from-
Medians of a triangleDiagram
Are the 3 medians concurrent lines?
CentroidLocated-
34
Practice. p.231 #6,7
AltitudeDiagram
Are the 3 altitudes of a triangle concurrent lines?
Orthocenter-
HW: p.231 #9,11,13,17
p.238 #9-15 odds
p.242 #7-11, 17,18
Utah State Standards
1.3.d Apply triangle inequality theorems.
Investigation. Triangle Inequalities.
Inequality-
Exterior Angle Inequality Theorem
Diagram
Ex. p.286 #5
35
Angle-Side Inequality Relationships
Diagram
Practice.
Determine the relationship between the measures of the given angles.
RSU , SUR
TSV , STV
R
5.2
6.6
S
5.3
T
4.8
3.6
4.4
RSV , RUV
V
U
5.1
Triangle Inequality TheoremDiagram
Example. Determine whether the given measures can be the lengths of the sides of a
triangle.
1. 2, 4, 5
2. 6, 8, 14
36
3. In PQR , PQ=7.2 and QR=5.2. Which measure cannot be PR?
a. 7
b. 9
c. 11
d. 13
Hinge Theorem (SAS Inequality Theorem)
Diagram
HW: p.286 #12,14,18,19
p.293 #9-21 odds, 22
p.299 #9,11,13
CDAS
Utah State Standards
1.2.a Use accepted geometric notation for similarity.
1.3.a Prove congruency and similarity of triangles using postulates and theorems.
RatioCan be expressed as
or
or said:
The denominator cannot = 0.
Ex. The total number of students who participate in sports programs at Wasatch
High School is 440. The total number of students in the school is 1100. Find the
athlete-to-student ratio to the nearest tenth.
Proportion-
*Ratios must have the same units!
37
Ex.
2 10

3 15
To Solve a Proportion3 x
Ex. 
5 75
Ex.
3 x  5  13

4
2
Ex. A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale
model is made with a length of 16 inches. Find the width of the model.
HW: p.353 #17-37 odds, 41,42
Utah State Standards
1.2.a Use accepted geometric notation for similarity.
1.3.a Prove congruency and similarity of triangles using postulates and theorems.
Similar Polygons-
m AB = 2.86 cm
JF = 1.43 cm
m BC = 6.02 cm
FG = 1.01 cm
m DA = 2.01 cm
m ED = 3.10 cm
Ex.
GH = 1.55 cm
B
HI = 2.24 cm
m CE = 4.48 cm
IJ = 3.01 cm
A
J
F
G
D
I
C
H
E
38
Symbol for Similar
Notation
Corresponding Angles
Corresponding Sides
Practice. p.359 #4,5
Scale Factor-
*Depends on the order of comparison
Practice. p.359 #6,7
Ex.
ABCDE ~ RSTUV
Find the scale factor of polygon ABCDE to RSTUV.
Find x and y.
S
B
6
C
A
D
E
x
4
R
T
U
V
3
y+1
8
HW: p.359 #9-19 odds, 24, 25,30
39
Investigation Shortcuts.
Utah State Standards
1.2.a Use accepted geometric notation for similarity.
1.3.a Prove congruency and similarity of triangles using postulates and theorems.
In similar triangles, we have three sets of congruent angles and three sets of sides in
proportion. But there are shortcuts to prove similarity.
AA ~ (angle angle similarity)Diagram
SSS ~ (side side side similarity)Diagram
SAS~ (side angle side similarity)Diagram
Practice. p.365 #3,4,6
p.372 #6,7
HW: p.366 #7,8,9,11,16,20
p.372 #15-21 odds, 24,28
CDAS
40
Utah State Standards
1.3.a Prove congruence and similarity of triangles using postulates and theorems.
Congruent TrianglesDiagram
Notation
Corresponding parts*CPCTC-
Practice. p.205 #7-9
Reading and Investigation. Shortcuts.
Ways to Prove Triangles are Congruent
1. All three pairs of corresponding angles are ___________ and all three pairs of
corresponding sides are ____________.
2. SSS
3. SAS
4. ASA
5. AAS
*6. HL
SSS- Side-Side-Side CongruenceDiagram
41
SAS- Side-Included Angle-Side CongruenceDiagram
*must be the included anglePractice. p.213 #5,6
ASA- Angle-Included Side-Angle Congruence
Diagram
AAS- Angle-Angle-Side Congruence
Diagram
Practice.p.218 #6-9
HL- Hypotenuse and Leg Congruence
*For right triangles only!!!
HypotenuseDiagram
HW: p.213 #12-15,17,19,22,23
p.218 #15,17,19-22,25-28
CDAS
42
Utah State Standards
1.3.b Prove the Pythagorean Theorem in multiple ways, find missing sides of right
triangles using the Pythagorean Theorem, and determine whether a triangle is a
right triangle using the converse of the Pythagorean Theorem
Right triangle review
Diagram
Investigation. Pythagorean Theorem.
Pythagorean TheoremFormula
Ex. Finding a missing side of a right triangle.
Ex. p.259#11
Diagram
Ex. 2 Find x.
3
6
x
Converse of the Pythagorean TheoremTo test if you have a right triangle, set up the Pythagorean Theorem formula
using the longest side given as the hypotenuse.
If it works (makes a true equation) and if the numbers are whole numbers, then
the triangle is a right triangle! And the set of the lengths of the sides is called a
__________________________
*The value that you put alone (the hypotenuse) is always the longest side.
43
Ex. problems. Are the given sides, the sides of a right triangle?
8, 15, 16
20, 48, 5
Common Triples.
HW: p.260 #17-35 odds
Utah State Standards
3.1.a Solve problems using the properties of special right triangles
Take a look at this triangle.
Why is it isosceles?
45
c
a
45
Let’s look at if a= 3, what are the
lengths of the other two sides.
b
Let’s look at if a= 7, what are the
lengths of the other two sides.
Let’s look at if a= x, what are the
lengths of the other two sides.
Theorem- In a 45-45-90 triangle, the length of the hypotenuse is ________ times the
length of a leg.
This theorem is a time-saver! (you don’t have to do the Pythagorean theorem every time)
Diagram
44
Ex. Find a missing side of a 45-45-90
1. Each leg of a 45-45-90 has a length of 8. What is the length of the
hypotenuse?
2. Find the length of a leg of a 45-45-90 triangle, if the length of the hypotenuse
is 6.
Take a look at this triangle.
60
It’s equiangular and equilateral!
We’re going to chop it in half by
drawing an altitude.
60
60
We’ll look at a side length of 8.
Let’s look at a side length of 10.
Theorem- In a 30-60-90 triangle, the length of the hypotenuse is _________ times the
length of the shorter leg, and the length of the longer leg is _________ times the length of
the shorter leg.
Another time-saver!
Diagram
45
Ex. Practice problems 30-60-90 triangles.
1. Find AC.
B
60
14
A
2. Find QR.
4 3
C
P
3
Q
30
R
HW: p.557 #7-13 odds,14,16
p.562 #7-13 odds, 16,17
Utah State Standards
3.1.b Identify the trigonometric relationships of sine, cosine, and tangent with the
appropriate ratio of sides of a right triangle.
3.1.c Express trigonometric relationships using exact values and approximations.
3.2.a Find the angle measure in degrees when given the trigonometric ratio.
3.2.b Find the trigonometric ratio given the angle measure in degrees, using a
calculator.
3.2.c Find unknown measures of right triangles using sine, cosine, and tangent
functions and inverse trigonometric functions.
4.2.c Solve problems involving trigonometric ratios.
Investigation. Trig ratios.
TrigonometryTrigonometric ratio*related to the acute angles of a right triangle (NOT the right angle).
46
Diagram:
Sine
Cosine
Tangent
SOH-CAH-TOA
Practice Problems.
N
Find sinL
cosL
tanL
17
sinN
cosN
tanN
8
L
M
15
*we’ll find a fraction and decimal for each.
If we know the degree of our angles, we can find the ratio of sides on our calculator.
*we can find the trig ratios!
Practice.
Find the sin67.
Find cos89.
Find tan56.
Find tan11.
47
Using trig ratios to solve problems.
Practice. p.568 #5
p.575#9
Solving for an angle measure, knowing the ratio of sides.
We set up trig equations just as in the previous example. The missing piece is
now the angle!
We use the inverse of sine, cosine, and tangent to help us solve these equations.
sinA = x
to find angle A
A = sin-1(x) “A is the inverse sine of x”
cosA= x
to find angle A
A = cos-1(x) “A is the inverse cosine of x”
tanA= x
to find angle A
A = tan-1 (x) “A is the inverse tangent of x”
On the calculator
N
ex.
to find angle L
8
L
M
15
Ex.
to find angle N
N
17
8
M
L
48
Angle of ElevationDiagram
Angle of DepressionDiagram
Practice p. 568 #6
p.575 #10
HW: p.568 #9-19 odds, 20
p.576 #13-27 odds, 31, 36
CDAS
Utah State Standards
1.4 Analyze characteristics and properties of polygons.
1.4.c Derive, justify, and use formulas for the perimeter and lines of symmetry of
regular polygons.
Polygon*sides that have a common endpoint are _________________.
*each side intersects exactly two other sides, but only at their ______________.
49
Naming a polygon
Sketches of some polygon examples and names
Concave vs. Convex polygons
Classifying polygons by the number of sides.
Number Polygon Name
of Sides
3
4
5
6
7
8
9
10
12
N
Regular polygon- A convex polygon with all __________ congruent and all
___________ congruent.
Diagram
Practice. Listen and Talk worksheet
Practice p. 404 #2,4-7,9,10
Diagonals of a polygon
50
Perimeter of a polygonPractice p.405 #8
Symmetry
Reflection symmetry
Need a mirror line
Practice. Determine how many lines of symmetry a square has.
Determine how many lines of symmetry a regular pentagon has.
p.436 #8,9
Rotation symmetry
Can a figure have both reflection and rotation symmetry? Draw an example.
HW: p.405 #13-33
p.437 #13-17 odds
CDAS
51
Utah State Standards
1.6.a Identify and classify prisms, pyramids, cylinders and cones based on the shape
of their base(s).
1.6.b Identify three-dimensional objects from different perspectives using nets,
cross-sections, and two-dimensional views.
1.6.c Describe the symmetries of three-dimensional figures.
1.6.d Describe the relationships between the faces, edges, and vertices of polyhedra.
Orthogonal drawings
Ohio Dept. of Education
Practice drawings with cube designs.
Build 3D Figures from nets.
52
Polyhedron
Face
Edge
Vertex
Examples:
Prism
Naming a prism
Examples
Regular Prism
Pyramid
Naming a pyramid
Examples
Euler’s Theorem:
53
Regular polyhedron
Platonic Solids
Names and descriptions
Solid Non-polyhedrons
Cylinder
Sketch
Cone
Sketch
Sphere
Sketch
54
Cross-sections of Solids
The circle on the right is a cross-section of the cylinder on the left.
The triangle on the right is a cross-section of the cube on the left.
Homework. p. 499 #15,17,21-29 odds, 32,33
CDAS
55
Utah State Standards
1.4.c Derive, justify and use formulas for the number of diagonals, lines of
symmetry, angle measures of regular polygons.
Investigation. Interior angles of polygons.
Interior Angle Sum Theorem-
Each interior angle
Ex. Finding the sum of the interior angles of a polygon.
In a pentagon, what is the sum of the interior angles?
Ex. Finding the measure of EACH interior angle of a polygon.
In a regular hexagon, what is the measure of each interior angle?
Ex. Finding the number of sides, given the sum of the measure.
Given that the sum of the measure of interior angles of a polygon is 900,
how many sides does the polygon have?
Exterior angles of a polygonDiagram
Relationship between an exterior and interior angle
Investigation. Exterior angles of a polygon.
Exterior Angle Sum TheoremDiagram
56
Ex. Find the measure of each exterior angle of a regular nonagon.
Ex. Find the measure of an interior angle and an exterior angle of a regular 18-gon.
HW: p.411 #9-17 odds, 18
Utah State Standards
1.4.a Use examples and counterexamples to classify subsets of quadrilaterals.
1.4.b Prove properties of quadrilaterals using triangle congruence relationships,
postulates, and theorems.
Quadrilateral
Consecutive sides
Non-consecutive sides
ParallelogramDiagram
Symbol
Investigation. What Does It Take to Make a Parallelogram?
Properties of Parallelograms
Theorem- Opposite sides of a parallelogram are __________________.
Theorem- Opposite angles of a parallelogram are __________________.
Theorem- Consecutive angles in a parallelogram are __________________.
Theorem- If a parallelogram has 1 right angle, then _________________________.
57
Theorem- The diagonals of a parallelogram _________________________________.
Practice. p.319 #2,4-8
The ONLY ways to prove a quadrilateral is a parallelogram.
Definition- Both pairs of opposite sides are _________________.
Theorem- Both pairs of opposite sides are __________________.
Theorem- Both pairs of opposite angles are __________________.
Theorem- The diagonals __________________________________.
Theorem- One pair of sides is both _______________ and ________________.
Practice. p.325 #3,4
HW: p.320 #11-23 odds, 28,29
p.325 #7-12, 14,15,16
58
Utah State Standards
1.4.a Use examples and counterexamples to classify subsets of quadrilaterals
1.4.b Prove properties of quadrilaterals using triangle congruence relationships,
postulates, and theorems.
RectangleDiagram
Investigation. What Does It Take to Make a Rectangle?
Theorem- If a parallelogram is a rectangle, then the diagonals are
__________________.
*It also has all the same properties of a parallelogram!
Ex. Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x.
S
T
R
U
Ex. Quadrilateral LMNP is a rectangle. Find x and y.
N
M
6y+2
L
5x+8
3x+2
P
Theorem- If the diagonals of a parallelogram are congruent, then the parallelogram is
_______________________.
59
RhombusPlural “rhombi”
Diagram
Investigation. What Does It Take to Make a Rhombus?
*All properties of a parallelogram still apply!
Theorem- The diagonals of a rhombus are _________________
Theorem- If the diagonals of parallelogram are perpendicular, then that parallelogram is
a ___________________.
Theorem- Each diagonal of a rhombus bisects a pair of _________________________.
Ex. Use rhombus LMNP and the given information to find each value.
M
1. Find mPNL if mMLP  64 .
L
Q1
N
P
SquareDiagram
*Has all properties of a parallelogram, rectangle, and rhombus.
KiteDiagram
HW: p.330 #16-21, 23-37 odds, 39-44
60
Utah State Standards
1.4.a Use examples and counterexamples to classify subsets of quadrilaterals
2.1.a Verify the classifications of geometric figures using coordinate geometry to
find lengths and slopes.
TrapezoidDiagram
Bases
Base angles
Legs
Isosceles TrapezoidDiagram
Theorem- Both pairs of base angles are __________________ in an isosceles trapezoid.
Theorem- The diagonals of an isosceles trapezoid are __________________.
Median of a trapezoidSometimes called midsegment
Diagram
Theorem- The median of any trapezoid is ______________ to the bases, and its measure
is equal to _________________ the __________ of the measures of the bases.
Formula
Ex. DEFG is an isosceles trapezoid with median MN .
1. Find DG if EF = 20 and MN = 30.
E
3
F
4
M
N
D 1
2
G
61
2. Find m1, m2, m3, m4 if m1  3x  5, m3  6 x  5 .
HW: p.337 #10,13-21 odds, 30,32,33,34
Utah State Standards
1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed
angle, and tangent of a circle, and solve problems using their properties
CircleCenter
Radius
Radii
Naming a circleChord
Diameter
*made of collinear radii
Diameter =
Practice Problems.
D
1. Name the circle.
F
2. Name a radius of the circle.
C
A
3. Name a chord of the circle.
4. Name a diameter of the circle.
B
E
5. If AE=5, find the DE.
62
Concentric circles-
What is π?
Circumference
Formula
Practice Problems. Leave π in your answers. Then use a calculator for a decimal
approximation..
1. Find C if r = 7.
2. Find C if d = 12.5
3. Find d and r to the nearest hundredth if C=136.9
HW: p.457 #13-27,29,30
p.481 #11,13,17,19,22,26
Utah State Standards
1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed
angle, and tangent of a circle, and solve problems using their properties
1.4.e Show the relationship between intercepted arcs and inscribed or central angles,
and find their measures.
Central angle
Diagram
Sum of central angles (with no interior points in common)
63
Practice Problem
1. Find x.
A
D
25x
2. Find mAOD .
O
3x
2x
C
E
B
Arc
*Measure of an arc =
Different than the length of an arc.
Minor Arc
Major Arc
Semicircle
Arc addition postulate
*What is the sum of all non-overlapping arcs on a circle?
C
Practice Problems.
1. Find mBE
D
2. Find mCBE
B
50
A
F
3. Find ACE
E
64
Investigation. Intercepted Arcs.
Inscribed Angles
Diagram
Intercepted arc
Measure of inscribed angle =
Practice Problems.
A
mAB=140, mBC=100,mAD=mDC. Find the
measure of the numbered angles.
B
5
1
2
O
4
3
D
C
Inscribed angle intercepting a semicircle-
HW: p.466 #13-23 odds, 31,41
p.590 #9,11,12-14,19,21,27
65
Utah State Standards
1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed
angle, and tangent of a circle, and solve problems using their properties
1.4.e Show the relationship between intercepted arcs and inscribed or central angles,
and find their measures.
Tangent
Point of tangency
Diagram
If a line is tangent to a circle, then it is _______________ to the radius drawn to the
point of tangency.
Diagram
If two segments from the same exterior point are tangent to the same circle, then they are
_____________________.
Practice. p.595 #6
Secant
Diagram
66
Measures of angles formed by secants and tangents
2 secants/chords intersecting in the interior of the circle
Diagram
Secant and a tangent
Diagram
2 secants
Diagram
2 tangents
Diagram
Practice. p.603 #4,5
67
Practice.
R
Find mRPS if mPT=114 and mTS=136
P
Q
S
114
T
136
HW: p.596 #9-14, 19,27
p.604 #9-17 odds
Utah State Standards
1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed
angle, and tangent of a circle, and solve problems using their properties
1.4.e Show the relationship between intercepted arcs and inscribed or central angles,
and find their measures.
Segments in a circle
Two chords intersecting in the circle
Diagram
Two secant segments
Diagram
68
Secant and a tangent segment
Diagram
Practice.
E
8
Find x.
H
10
x
F
I
24
G
HW: p.616 #8-13
69
Utah State Standards
2.2.a Graph a circle given the equation in the form ( x  h) 2  ( y  k ) 2  r 2 .
Equation of a circle
Center
Radius
Practice Problems.
1. Write an equation of a circle with center at (-2,4) and a radius of 2.
2. Write an equation of a circle with center at the origin and diameter of 6.
3. Graph the circle with equation (x+2)2 +(y-3)2=16.
HW: p.621 #15-25 odds,26,27,34,36
CDAS
70
Utah State Standards
1.4.c Derive, justify, and use formulas for the area of regular polygons.
4.1.d Calculate or estimate the area of an irregular region.
What is area?
Practice. p.416 #4,5,7
Area of a rectangle
Ex. What is the area of rectangle ABCD?
7
A
B
4
D
C
Investigation. Area of a Polygon
Area of a parallelogram
Ex. Find the area of each parallelogram.
1.
4
13
30
2.
10
60
71
Area of a triangle
Ex. Find the area of the triangle.
3.4
7.3
Area of a trapezoid
Ex. Find the area of the trapezoid.
16
14
12
24
Area of a rhombus
Ex. Find the area of the rhombus.
E
F
20
17
17
H
20
G
72
Area of a square
*a square is a rectangle, parallelogram, and a rhombus.
Ex. Find the area of the square.
5
What happens when you have a very strange shape like…?
HW: p.416 #9,12,13,15,21
p.423 #11,12,14,15
Utah State Standards
1.4.c Derive, justify, and use formulas for the area of regular polygons.
4.1.d Calculate or estimate the area of an irregular region.
Area of a regular polygon
Apothem
Central angle
73
Ex. Find the area of each regular polygon.
1. A pentagon with side length of 10 and apothem 6.9.
2. A hexagon with perimeter of 72 and apothem 6 3 .
Area of a circle
*derived from the polygon formula
Ex. Find the area for each problem.
1. A circle with radius of 8.
2. A caterer has a 48-inch diameter table that is 34 inches tall. She
wants a tablecloth that will touch the floor. Find the area of the
tablecloth in square inches.
Irregular Figures
Ex.
16
32
HW: p.429 #7,8,9,11,12,15
p.486 #9,11,13,17
74
Utah State Standards
4.1.e Find the length of an arc and the area of a sector when given the angle measure
and radius.
4.2.d Solve problems involving geometric probability.
Sector
Area of a Sector
Practice Problem. Find the area of the sector with central angle of 80 degrees in
a circle with radius 6.
Length of an Arc
*different than the measure of an arc.
Practice Problem. Find the length of an arc with central angle of 80 degrees in a
circle with radius 10.
Geometric Probability and Area
Comparing the area of part with the area of the whole figure.
P=
75
Practice Problems.
1. Find the probability that a point chosen at random lies in a sector of a circle
with central angle measure of 60 degrees. The radius of the circle is 9.
2. Find the probability that a point chosen at random lies in the shaded region.
Investigation. Geometric Probability.
HW: p.487 #19-21, 24,25,27,29
p.482 #27
Utah State Standards
1.6.b Identify three-dimensional objects from different perspectives using nets,
cross-sections, and two-dimensional views.
4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.
4.1.c Determine the perimeter, area, surface area, lateral area, and volume for
prisms, cylinders, pyramids, cones, and spheres when given the formulas.
Surface Area
Lateral Area
76
Prisms
*Named by their ________________.
Lateral Area of Prisms
Sketch
Shape of lateral faces of prism
Surface Area of Prism = Lateral Area + 2(Area of 1 Base)
Practice. p.508 #3,4,6
HW: p.508 #7,9,11,16,18,19
77
Utah State Standards
4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.
4.1.c Determine the perimeter, area, surface area, lateral area, and volume for
prisms, cylinders, pyramids, cones, and spheres when given the formulas.
Lateral Area of a Cylinder
Sketch
Base Area of a Cylinder
Surface Area of a Cylinder = Lateral Area + 2(Area of 2 base)
Practice. Find the surface area of a cylinder with a height of 18 feet and a base
radius of 14 feet.
Practice. Find the radius of the base of a right cylinder if the surface area is
528 square feet and the height is 10 feet.
HW: p.509 #13-15,17
78
Lateral Area of Regular Pyramids
*Named by their _______________.
Sketch
Slant height l
Shape of lateral faces of a pyramid
Surface Area of Regular Pyramids = Lateral Area + Area of 1 Base
Practice. Find the surface area of the regular pyramid to the nearest tenth.
6m
4m
8m
8m
Lateral Area of a Cone
Sketch
Surface Area of a Cone = Lateral Area + Area of 1 Base (Circle)
79
Practice. Find the surface area of a cone with height of 3.2cm and base radius of 1.4 cm.
Draw a sketch.
Surface Area of a Sphere
Practice. Find the surface area of a sphere with radius of 41 cm.
Practice. Find the surface area of a ball with a circumference of 24 inches to determine
how much leather is need to make the ball.
HW: p.520 #7-14 all
p.531 #7-13 (surface area only)
80
Utah State Standards
4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.
4.1.c Determine the perimeter, area, surface area, lateral area, and volume for
prisms, cylinders, pyramids, cones, and spheres when given the formulas.
Volume
Introduce project. There’s No Place Like Home
Volume of a Prism
Height
Area of the Base
Practice. p. 513 #8
Practice. The weight of water is 0.036 pounds times the volume of water in cubic inches.
How many pounds of water would fit into a rectangular child’s pool that is 12 inches
deep, 3 feet wide, and 4 feet long?
Volume of a Cylinder
81
Practice. Find the volume of a cylinder with height of 1.8cm and base radius of 2.2cm, to
the nearest tenth.
Investigation. Pyramids and Prisms. Cones and Cylinders.
Volume of a Pyramid
Ex. A student has a solid clock that is in the shape of a square pyramid. The clock has a
base side of 3 inches and a height of 7 inches. Find the volume of the clock.
Sketch
Volume of a Cone
Ex. Find the volume of a cone with a height of 12 feet and a base radius of 5 feet.
Volume of a Sphere
Practice. Find the volume of a sphere with radius of 15 cm.
82
Practice. Find the volume of a sphere with a Circumference of 25 cm.
Practice. Compare the volumes of a sphere and a cylinder with the same radius and
height as the radius of the sphere.
Sketch
HW: p.513 #12,14,15,18,19,20
p.525 #9-17 odds, 22
p.531 #8,11,16,21
CDAS
83