CHAPTER 1-INTRODUCTION TO GEOMETRY
SECTION 1:
Points
- In the diagram below, three points are represented by three dots. This is how you would recognize
points.
Lines
- Made up of points and are straight
- Extend infinitely far in both directions
Line Segments
- Like lines, they are made up of points and are straight.
- However, a segment has a definite beginning and end.
- Named in terms of its two endpoints.
Rays
- Made up of points and are straight
- Begins at an endpoint and then extends infinitely far in only one direction
- When a ray is named, it must be named by the endpoint first so that it is clear where the ray begins
Angles
- An angle is made up of two rays with a common endpoint.
- The common endpoint is called the vertex of an angle.
- The rays are called the sides of the angle.
Triangles
- Have three segments as its sides.
- Have three angles as its sides.
- The triangle is the union of three segments
- The intersection of any two sides is a vertex of the triangle
SECTION 2:
Classifying Angles by Size
- An acute angle is an angle whose measure is greater than 0 and less than 90.
-A right angle is an angle whose measure is 90.
-An obtuse angle is an angle whose measure is greater than 90 and less than 180.
-A straight angle is an angle whose measure is 180.
Parts of a Degree
- Each degree of an angle is divided into 60 minutes (‘), and each minute of an angle is divided into 60
seconds (“)
*60’ = 1 degree
*60” = 1 minute
Thus,
87.5 degrees = 87 degrees and 30 minutes
60.4 degrees= 60 degrees and 24 seconds
90 degrees= 89 degrees and 60 seconds
Congruent Angles and Segments
Congruent angles: Angles that have the same measure
*angles A and B are congruent*
Congruent
-Often
angles and segments.
segments: Segments that have the same length
identical tick marks are used to indicate congruent
SECTION 3:
Collinear: Points that lie on the same line
Noncollinear: Points that do not lie on the same line
Betweenness of Points
-In order to say that a point is between two other points, all three other points must be collinear.
Triangle Inequality
- For any three points, there are only two possibilities:
1.)The points are collinear (one point is between the other two. Two of the distances add up to
the third.)
2.)The points are noncollinear (the three points determine a triangle.)
Assumptions from Diagrams
You Should Assume:
-Straight lines and Angles
-Collinearity of Points
-Betweenness of Points
-Relative positions of points
You Should NOT Assume
-Right angles
-Congruent segments
-Congruent angles
-Relative sizes of segments and angles
SECTION 4:
Theorem: a mathematical statement that can be proved.
-Theorem 1: If two angles are right angles, then they are congruent.
-Theorem 2: If two angles are straight angles, then they are congruent.
Two-column Proof: Proving a problem in two columns, a statement and a reason.
SECTION 5:
Midpoints and Bisectors of Segments
-A point that divides a segment into two congruent segments bisects the segment and the bisection point
is called the midpoint.
*only segments have midpoints, rays and lines don’t*
Trisection Points and Trisecting a Segment
-Two points that divide a segment into three congruent segments trisect the segment. The two points
where the segment is divided is called the trisection point.
*only segments have trisection points, rays and lines don’t*
Angle Bisectors
-A ray that divides an angle into two congruent angles bisects the angle and the dividing ray is called the
bisector.
Angle Trisectors
-Two rays that divide an angle into three congruent angles trisect the angle and the two diving rays are
called trisectors.
SECTION 6:
Counter
Example:
When
the
conclusion made in a proof is false.
Paragraph Proof: Written out proof instead of in a two-column form,
O
P Given: O=67.5º
Prove: O  P
P=67º30’
Proof: Since there are 60 minutes in 1 degree, 67º30’ equals 67.5º. Since O and P have the same measure,
they are congruent.
SECTION 7:
Postulate: is an unproved assumption.
Deductive Structure: is a system of thought in which conclusions are justified by means of previously assumed or
proved statements.
Every deductive structure contains:
1. Undefined terms
2. Assumptions known as postulates
3. Definitions
4. Theorems and other conclusions
*Example: 1. If the sidewalk is wet, then it rained last night.
-The sentence is called a conditional statement
-The “if” part is called a hypothesis
-The “then” part is called a conclusion
*Example: 2. If it rained last night, then the sidewalk is wet.
Converse Statement: when you write your conditional statement reversed, but still gives the same
concepts.
REMEMBER
-Definitions are always reversible
-Theorems and postulates are not always reversible
PRACTICE:
1. Write the converse of this statement
If the angle is 45 degrees, then the angle is acute.
2. What are the four elements found in a deductive structure?
ANSWERS TO SECTION 7:
1. If the angle is acute, then the angle is 45 degrees.
2. Undefined terms, assumptions known as postulates, definitions, theorems and other
conclusions.
SECTION 8:
Converse: the conditional statement reversed.
Inverse: is the conditional statement in the negative
Contrapositive: is the conditional statement reversed and negative
*Example: If you live in Georgia, then you live in Atlanta.
Converse- If you live in Atlanta, then you live in Georgia.
Inverse- If you don’t live in Georgia, then you don’t live in Atlanta.
Contrapositive- If you don’t live in Atlanta, then you don’t live in Georgia.
Theorem: If a conditional statement is true, then the Contrapositive of the statement is also true.
Chain Rule: Each proof that you do involves a series of steps in a logical sequence.
*Example: If you study hard, then you will earn good grades. If you earn good grades, then your parents will be
happy. What can you conclude?
If you study hard, then your parents will be happy.
PRACTICE:
1. Write the converse, inverse, and contrapositive
If two angles are right angles, then they are congruent.
2. Draw a conclusion
If gremlins grow grapes(G), then elves eat earthworms(E).
If trolls don’t tell tales(T), then the wizards weave willows(W).
If trolls tell tales, then elves don’t eat earthworms
ANSWERS TO SECTION 8:
1. Converse- If they are congruent, then two angles are right angles.
Inverse- If two angles aren’t right angles, then they aren’t congruent.
Contrapositive- If they aren’t congruent, then they aren’t right angles.
2. G--- E
E--- ~T (T--- E is equivalent to E--- ~T)
~T--- W
G---W
IF GREMLINS GROW GRAPES, THEN WIZARDS WEAVE WILLOWS.
SECTION 9:
Two basic steps for Probability Problems:
1. Determine all the possibilities in a logical manner. Count them.
2. Determine the number of the possibilities that are “favorable.”
Probability Formula:
Number of winners / Number of possibilities
PRACTICE:
1. If you flip a coin once, what is the probability of getting heads? Tails? If you flip the coin twice, what is the
probability of getting both heads?
2. If you have a cent bag with 10, 5, 25, 1, what is the probability of getting greater than 10? Less than or equal
to 10?
ANSWERS TO SECTION 9:
1. P(H)- ½
P(T)- ½
P(H,H)- ¼
2.P(X>10)- ¼
P(X less than or equal to 10)- ¾
CHAPTER 2 REVIEW
2.1 Perpendicularity (pg. 61)
• Perpendicularity- lines, rays, or
segments that intersect at right angles
2.2 Complementary and Supplementary Angles (pg. 66)
• Complementary Angles- angle pairs that have a sum of 90º. They do not have to be adjacent and any
two angles that add up to 90º.
• Supplementary Angles- angle pairs that have a sum of 180º. They do not have to be adjacent and any
two angles that add up to 180º.
Complementary
Supplementary
2.3 Drawing Conclusions (pg. 72)
Procedure For Drawing Conclusions:
1. Memorize theorems, definitions, and
postulates.
2. Look for key words and symbols in the
given information.
3. Think of all the theorems, definitions,
and postulates that involve those keys.
4. Decide which theorem, definition, or postulate allows you to draw a conclusion.
5. Draw a conclusion, and give a reason to justify the conclusion. Be certain that you have not used the
reverse of the correct reason.
2.4 Congruent Supplements And Complements (Pg. 76)
• Theorem 4: If angles are supplementary to the same angle, then they are congruent.
• Theorem 5: If angles are supplementary to congruent angles, then they are congruent.
• Theorem 6: If angles are complementary to the same angle, then they are congruent.
• Theorem 7: If angles are complementary to congruent angles, then they are congruent.
2.5 Addition and Subtraction Properties (pg. 82)
• Theorem 8: If a segment is aded to two congruent segments, the sums are congruent.
• Theorem 9: If an angle is added to two congruent angles, the sums are congruent. (Addition Property)
• Theorem 10: If congruent segments are added to congruent segments, the sums are congruent.
(Addition Property)
• Theorem 11: If congruent angles are added to congruent angles, the sums are congruent. (Addition
Property)
• Theorem 12: If a segment or angle is subtracted from congruent segments or angles, the differences
are congruent. (Subtraction Property)
• Theorem 13: If congruent segments or angles are subtracted from congruent segments or angles, the
differences are congruent. (Subtraction Property)
2.6 Multiplication and Division Properties (pg. 89)
• Theorem 14: If segments (or angles) are congruent, their like multiples are congruent. ( multiplication
property).
• Theorem 15: If segments (or angles) are congruent, their like divisions are congruent. (division
property)
Using the multiplication and division properties in proofs:
1.
Look for a double use of the word midpoint or trisect or bisects in the given information.
2.
The multiplication property is used when the segments or angles in the conclusion are greater
than those in the given information.
3.
The division Property is used when the segments or angles in the conclusions are smaller than
those in the given information.
2.7 Transitive and Substitution Properties (pg. 95)
Theorem 16: If angles (or segments) are congruent to the same angle (or segment), they are congruent
to each other. (transitive property)
Theorem 17: If angles (or segments) are congruent to congruent angles (or segments), they are
congruent to each other. (transitive property)
Example:
Suppose that < A is congruent to < B and < A is congruent to < C. is < B congruent to < C? - yes
because it can be proven using the transitive property, if angles (or segments) are congruent to the same
angle (or segment), they are congruent to each other.
Substitution Property:
If you plug in a value in for a variable into an expression or equation, this allows you to substitute
quantities for each other into an expression as long as those quantities are equal.
2.8 Vertical Angles (pg.100)
Know Opposite Rays
• Opposite Rays: Two collinear rays that have a common endpoint and extend in different directions.
ex.)
--->
--->
• AC and AB are opposite rays
• Non opposite rays do not share a common endpoint
--->
ex.) PT and RS are not opposite, since they do not have a common endpoint.
• Vertical angles: if the rays forming the two sides of one and the rays forming the sides of the other are
opposite rays.
Vertical angles are congrue
CHAPTER 4 STUDY GUIDE REVIEW
- When proving 2 or more triangle you use detour proofs to help solve the problem.
- After you prove the triangle congruent always use CPCTC
- By locating a midpoint on a straight line you add the two values then divide by 2
ex)
______________________________
2
?
14
2+14=16/2=8
so ?=8
- The Midpoint Formula(Coordinate Plane)= (x1+x2/2),(y1+y2/2)
so the midpoint is equal to (Slope of X, Slope of Y)
-Sometimes in geometry there are problems that do not give a diagram so you should set up the problem by
drawing a diagram
ex) Set up a proof for the following: The medians of a triangle are cong. if the triangle is equilateral.( Look at
Pg 177 in the textbook to see if you answered correctly)
Setup: Given- Triangle XYZ is equilateral. Lines PZ, RY and QX are medians.
Prove- Line PZ is congruent to line RY which is cong. to line QX
-If two right angles are both supp. and cong. then they are right angles. This is because supp angles add up to
180 and 180 divided by 2= 90( Right Angle)
-The distance between two objects is the length of the shortest path between them which is a straight line
-Equidistant means two lines are equal distance from a given point
- A perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment
- If 2 points are each equidistant from the endpoints of a segment then the 2 pts. determine the perp. bisector
of the segment.( 2 pts eqd. then perp bisector)
- If a point is on the perp bisector of a segment then it is eqd. from the endpoints of that segment( pt. on perp
bisector is equidistant)
- A plane is a surface in which 2 points are connected by a line
- Coplanar= lie in same plane
- Noncoplanar= do not lie on same plane
- Transversal= a line that intersects two coplanar lines in two distinct pts.
exterior= outside figure/plane
interior= inside figure/plane
Angle Pairs Determined by Parallel Lines
- Corresponding = formed by 2 lines and a transversal and they are on the same side but one is interior and
the other is exterior
- Alt Interior = formed by 2 lines and a transversal and are opposite angles in the interior of the figure
- Alt Exterior = formed by 2 lines and a transversal and are opposite angles in the exterior of the figure
-Parallel Lines = 2 coplanar lines that do not intersect
-The slope is found by the formula y2-y1 divided by x2-x1
ex) Find the slope of the segment joining (-2,3) and (6,5)
so it would be 5-3 / 6-(-2) which is 2/8 or ¼
Slope Types
-vertical lines have no slope because they are undefined which is when division is by a zero
-zero slope is a horizontal line
-positive slope is a rising line and a negative slope is a falling line
- If two nonvertical lines are parallel then their slopes are equal
- perpendicular lines intersect each other
- to find a perp. slope take the reciprocal
ex) the slope is ⅖ so the slope perp to it would be -5/2
CHAPTER 5 STUDY GUIDE
5.1 Indirect Proof
Procedure
1.
2.
3.
4.
List the possibilities for the conclusion
Assume that the negation of the desired conclusion is correct
Write a chain of reasons until you reach an impossibility. This will be a contradiction of either (a)
given information or (b) a theorem, definition, or other known fact
State the remaining possibility as the desired conclusion.
5.2 Proving that Lines are Parallel
Theorem 30- The measure of an exterior angle of a triangle is greater than the measure of either remote
interior angle.
Theorem 31- If two lines are cut by a transversal such that two alternate interior angles are congruent, the
lines are parallel.
Theorem 32- If two lines are cut by a transversal such that two alternate exterior angles are congruent, the
lines are parallel.
Theorem 33- If two lines are cut by a transversal such that two corresponding angles are congruent, the lines
are parallel.
Theorem 34- If two lines are cut by a transversal such that two interior angles on the same side of the
transversal are supplementary, the lines are parallel.
Theorem 35- If two lines are cut by a transversal such that two exterior angles on the same side of the
transversal are supplementary, the lines are parallel.
Theorem 36- If two coplanar lines are perpendicular to a third line, they are parallel.
5.3 Congruent Angles Associated with Parallel Lines
postulate- Through a point not on a line there is exactly one parallel to the given line.
Theorem 37- If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent.
Theorem 38- If two parallel lines are cut by a transversal, then any pair of the angles formed are either
congruent or supplementary.
Theorem 39- If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent.
Theorem 40- If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent.
Theorem 41- If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the
transversal are supplementary.
Theorem 42- If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the
transversal are supplementary.
Theorem 43- In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
Theorem 44- If two lines are parallel to a third line, they are parallel to each other. (transitive property)
5.4 Four Sided Polygons
polygons:
Convex
polygons: a polygon where each interior angle has a measure less than 180
Quadrilaterals:
Parallelogram: a quadrilateral in which both pairs of opposite sides are parallel
Rectangle: a parallelogram in which at least one angle is right
Rhombus: a parallelogram in which at least two consecutive sides are congruent
kite: two isosceles triangles that share a common imaginary base
square: a parallelogram that is both a rectangle and a rhombus
trapezoid: a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases of the
trapezoid.
isosceles trapezoid: a trapezoid in which nonparallel sides are congruent
Properties of Parallelograms
1. Opposite sides are congruent
2. Opposite sides are parallel
3. Opposite angles are congruent
4. Adjacent angles are supplementary
5. Diagonals bisect each other
Properties of a Rectangle
1. All properties of parallelogram apply
2. All angles are 90 degrees
3. Diagonals are congruent
Properties of a Kite
1. Two disjoint pairs of consecutive sides are congruent by definition
2. Diagonals are perpendicular
3. One diagonal is the perpendicular bisector of the other
4. One diagonal bisects a pair of opposite angles
5. One pair of opposite angles are congruent
Properties of a Rhombus
1. All of the properties of parallelogram apply by definition
2. Two consecutive sides are congruent by definition
3. All sides are congruent
4. The diagonals bisect angles
5. The diagonals are perpendicular bisectors of each other
6. The diagonals divide the rhombus into four congruent triangles
Properties of Squares
1. All the properties of a rectangle apply by definition
2. All properties of a rhombus apply by definition
3. Diagonals form four isosceles triangles
Properties of an Isosceles Trapezoid
1. The legs are congruent by definition
2. The bases are parallel by definition
3. The lower base angles are congruent
4. The upper base angles are congruent
5. Diagonals are congruent
6. Any lower base angle is supplementary to any upper base angle
Proving that a quadrilateral is a Parallelogram
1. If both pairs of opposite sides of a quadrilateral are perpendicular then it is a parallelogram
2. If both pairs of opposite sides of a quadrilateral are congruent then it’s a parallelogram
3. if one pair of opposite sides of a quadrilateral are both parallel and congruent then it’s a parallelogram
4. If the diagonals of a quadrilateral bisect each other then it is a parallelogram
5. If both pairs of opposite angles of a quadrilateral are congruent then it’s a parallelogram
Proving that a quadrilateral is a Rectangle
1. If a parallelogram contains at least one right angle then it is a rectangle
2. If the diagonals of a parallelogram are congruent the parallelogram is a rectangle
3. If all four angles of a quadrilateral are right angles then it is a rectangle
Proving that a quadrilateral is a Kite
1. If two disjoint pairs of consecutive sides of a quadrilateral are congruent then it is a kite
2. If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal then it’s a kite
Proving that a quadrilateral is a Rhombus
1. If a parallelogram contains a pair of consecutive sides that are congruent then it’s a rhombus
2. If either diagonal of a parallelogram bisects two angles of the parallelogram then it’s a rhombus
3. If the diagonals of a quadrilateral are perpendicular bisectors of each other then the quadrilateral is a
rhombus
Proving that a quadrilateral is a square
1. If a quadrilateral is both a rectangle and a rhombus then it is a square
Proving that a Trapezoid is Isosceles
1. If the nonparallel sides of a trapezoid are congruent then it is isosceles
2. If the lower or the upper base angles of a trapezoid are congruent then it is isosceles
3. If the diagonals of a trapezoid are congruent then it is isosceles
CHAPTER 6 STUDY GUIDE
Plane – A flat surface where if any two points on the surface are connected by a line, all points of the line are also on
the surface. Example:
The point of intersection of a line and a plane is the foot of the
FOOT
If three points are collinear, they do not determine a plane.
Points A, D, B (right) do not determine a plane.
If three points are not collinear, they create a plane.
Points C, E, A (right) determine a plane.
Theorem – A line and a point not on the line determine a plane.
line.
Theorem – Two intersecting lines determine a plane.
Theorem – Two parallel lines determine a plane.
If a line intersects a plane not containing
it, the intersection is at exactly one point.
If two planes intersect, their intersection is exactly
one line (in this example, Line ADB).
A line is perpendicular to a plane if it is perpendicular to every one of the lines in the plane that pass through its foot.
Example:
Theorem – If a line is perpendicular to two distinct lines that lie in a plane and that lie in a plane and pass through its
foot, then it is perpendicular to the plane.
See Page 278 for Practice Problems
A line and a plane are parallel if they do not intersect.
Two lines are parallel if they do not intersect.
Lines that are not on the same plane are not parallel.
A
In other words, lines that are not coplanar are skew
lines. Lines A and B are Skew Lines.
B
There are no skew planes; planes are either intersecting or parallel.
Theorem – If a plane intersects two parallel planes, the
lines of intersection are parallel. Lines AB and P are
parallel.
Parallelism of Lines and Planes
1. If two planes are perpendicular to the same line, they are parallel to each other.
2. If a line is perpendicular to one of two parallel planes, it is perpendicular to the other plane as well.
3. If two planes are parallel to the same plane, they are parallel to each other.
4. If two lines are perpendicular to the same plane, they are parallel to each other.
5. If a plane is perpendicular to one of two parallel lines, it is perpendicular to the other line as well.
See Page 284 for Practice Problems
CH. 7 STUDY GUIDE - POLYGONS
7.1 Triangle Application Theorems
Theorem 50: The sum of the measures of the three angles of a triangle are equal to 180
A
B
C
Proof: According to the parallel postulate, there exists exactly one line through point A parallel to BC,
so the figure below can be drawn.
Because of the straight angle, we know that <1 + <2 + <3 which should be at the top equal to 180. For
this drawing, since A=A and B = B, we can substitute through alternate interior lines so they are equal
to 180. A + C + B= 180
Exterior Angle: Is an angle that is adjacent to and supplementary to and interior angle of a polygon
Theorem 51: The measure of an exterior angle of a triangle is equal to the sum of the measures of the
remote interior angle.
For example the measurement of Z (exterior) is equal to the measurements of XY (Interior)
Theorem 52: A segment joining the midpoints of two sides of a triangle is parallel to the third side and
its length is one half the length of the third side.
7.2 Two Proof Oriented Theorems
Theorem 53: If two angles of one triangle are congruent to two angles of a second triangle, Then the
third angles are congruent.
A
B
F
C
D
E
Given <A = <F
<B = <D
Conclusion: <C=<E
Proof: since the sum of the angles in each triangle is 180, the sums may be equal. Then apply the
subtraction property and we see the measurements are equal making the two missing angles
congruent.
The triangles do not need to be congruent to apply this theorem.
Theorem 54: If there exists a correspondence between the vertices of two triangles such that two
angles and a nonincluded side of one are congruent to the corresponding parts of another, then the
triangles are congruent. (AAS)
Given < Q is congruent to Angle R
Angle Q is congruent to Angle R
Prove Angle X is congruent to angle T
<QMX is congruent to <RMT Vertial <’s
<X is Congruent to <T
7.3 Formulas Involving Polygons
No. of sides Polygon
Triangle-3
Quadrilateral-4
Octagon-8
Nonagon-9
Pentegon-5
Decagon-10
Hexagon-6
Dodecagon-12
Heptagon-7
Pentadecagon-15
n-gon- n
Given
No Choice theorem
What is the Sum of the measures of a five angle figure?
A five figured angle produces three triangles
3 (180) = 540
This is proven from Thrm. 55
Theorem 55- The Sum of S i of the angles of a polygon with n sides is given by the formula S i = (n2)180.
Theorem 56 If one exterior angles is taken at each vertex, the sum Se of the exterior angles of a
polygon is given by the formula Se = 360
6(180) =1080 according to theorem 55
1080-720=360
Theorem 57 The number d of diagonals that can be drawn in a polygon of n sides is given by the
formula
𝒅=
𝒏(𝒏 − 𝟑)
𝟐
Example= 5(5-3)
2
=5
What is the name of a polygon if the sum of the measures =1080?
Si – (n-2)180
1080= (n -2)180
1080 = 180 n – 360
1440 = 180n
8=n
Octagon
7.4 Regular Polygon
Square, Equilateral triangle, Regular pentagon and Regular hexagon
Theorem 58: The measure E of each exterior angle of an equiangular polygon of n sides is given by the
formula
E= 360/N
If each exterior angle of a polygon is 18 degree, how many sides does the polygon have?
18 = 360/N
18N = 360
N = 20
CHAPTER 8
Ratio and Proportion
Ratio- a quotient of two numbers
*4 ways to write a ratio
5/3 5:3 5 to 3 5 ÷ 3
A ratio is given in its lowest form for example 15/6 is 5/2
Slope= Rise/ run
Rise- height difference between two points on a grid
Run- Horizontal difference between two points on a grid
Proportion- An equation showing that
-2 ways to write
a/b=c/d
or
A= 1,
B=2,
a:b=c:d
C=3,
D=4
slope examples
(1,2) (3,1)
1-2 = -1
3-1= 2
Coordinates
= -1/2 or -0.5
In a ratio A and D are extremes
In a ratio B and C are means
*If you multiply the means together they are equal to the products of the
extremes
*If the means in a ratio are both equal then
they are called a Geometric mean or Mean
Proportional
Arithmetic mean- average of two numbers
ex) 3 and 27
3+27= 30/2= 15
Geometric Meanex) 3:x=x:27
x^2=81
x=+- 9
Methods of Proving Triangles Similar
Postulates:
-If a correspondence of the vertices of two triangles exist so that the 3
angles of that triangle are the same as the other, than they are similar
(AAA)
-If a correspondence of the vertices of two triangles exist so that at least 2
angles of that triangle are the same as the other, then they are similar
(AA)
-If the two triangles sides are rationally proportionate then they are similar
(SSS~)
-If theres a correspondence between the two triangles and the ratios of 2
sides are proportionate and the angles are congruent then the triangles are
similar (SAS~)
8.4 Congruences And Proportions In Similar Triangles
We can use the definition of similar polygons to prove that:
1. Corresponding sides of the triangles are proportional (The ratios of
the measures of corresponding sides are equal.)
2. Corresponding angles of the triangles are congruent.
EX. 1
GIVEN: Triangle ABC is similar to triangle DEF
PROVE: Angle A is congruent to angle D
Statements:1. Triangle ABC is similar to triangle DEF
2 Angle A is congruent to angle D
Reasons: 1. Given
2. Corresponding angles of similar triangles are congruent
EX. 2
GIVEN: Triangle ABC is similar to triangle DEF
PROVE: AB = AC
DE
DF
Statements: 1. Triangle ABC is congruent to triangle DEF
2. AB = AC
DE DF
Reasons: 1. Given
2. Corresponding sides of similar triangles are proportional.
EX. 3 (Ready? This one is kind of a long problem.)
8.5 Three Theorems Involving Proportions
Theorems:
1. If a line is parallel to one side of a triangle and intersects the other two
sides, it divides those two sides proportionally. (Side Splitter Theorem)
2. If three or more parallel lines are intersected by two transversals, the
parallel lines divide the transversals proportionally.
3. If a ray bisects an angle of a triangle, it divides the opposite side into
segments that are proportional to the adjacent sides (Angle Bisector
Theorem).
EXAMPLES:
First
is
the
side-splitter
thereom:
side-splitter: a line in a triangle that is parallel to one side and intersects the two other sides
Proportion:
Next is the theorem with parallel lines:
If three or more parallel lines are intersected by two transversals, the parallel
lines divide the transversals proportionally.
Proportion:
The last thereom is the angle bisector thereom:
If a ray bisects an angle of a triangle, it divides the opposite side into segments
that are proportional to the adjacent sides.
EX. 1 Side Splitter Theorem
If BX = 2 cm, XA = 3 cm, BY = 3 cm, then find the length of YC.
Solution:
According to Side Splitter Theorem
BX/XA = BY/YC
..............(1)
Plug in the values of BX, XA, BY in the equation (1)
=>
2/3=3/YC
2 * YC = 3 * 3 cm
YC = 9/2 cm
YC = 4.5 cm
The value of YC is 4.5 cm.
EX. 2
Find the value of x .
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
So write a proportion.
Substitute the values.
Use the cross product.
Divide each side by 8.
Therefore, the value of x is 7.5
EX. 3 Angle Bisector Theorem
By the Angle Bisector Theorem,
Proof:
Extend
to meet
at point E.
CHAPTER 10: CIRCLES
10.1 The Circle
Definitions:
Circle- set of all points in a plane that are a given distance from a given point in the plane
Center (of a circle)- the given point on a circle
Radius- a segment that joins the center to a point on the circle is also called a radius
Concentric- two or more coplanar circles with the same center
Two circles are congruent if they have congruent radii
Interior- a point that is inside of a circle (its distance from the center is less than the radius)
Exterior- point outside a circle (its distance from the center is greater than the radius)
A point is on a circle if its distance from the center is equal to the radius
Chords- points on a circle that can be connected by segments; a segment joining any two points on a circle
Diameter- a chord that passes through the center of the circle
The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the
chord
Theorems:
1. If a radius is perpendicular to a chord, then it bisects the chord (insert picture)
2. If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord
3. The perpendicular bisector of a chord passes through the center of the circle
Example:
Given: Triangle XYZ is isosceles (XY≅XZ); circle Y & circle Z; BC||YZ;
Prove: circle R ≅ circle S
X
B
C
Y
Z
Z
*See problem #3 on p. 442 for Statements and Reasons
10.2 Congruent Chords
Theorems:
1.
2.
If two chords of a circle are equidistant from the center, then they are congruent
If two chords of a circle are congruent, then they are equidistant from the center of the circle

P
*The two lines protruding from point P to the two chords are congruent because the
two chords are equidistant
10.3 Arcs of a Circle
Definitions:
Arc- consists of two points on a circle and all points on the circle needed to connect the points by a single path
The center of an arc is the center of the circle of which the arc is a part
Central angle- an angle whose vertex is at the center of a circle
Minor arc- an arc whose points are on or between the sides of a central angle
Major arc- an arc whose points are on or outside of a central angle
Semicircle- an arc whose endpoints are the endpoints of a diameter
The measure of a minor arc/semicircle is the same as the measure of the central angle that intercepts the arc
The measure of a major arc is 360 minus the measure of the minor arc with the same endpoints
Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles
Theorems:
1. If two central angles of a circle are congruent, then their intercepted arcs are congruent
2. If two arcs of a circle are congruent, then the corresponding central angles are congruent
3. If two central angles of a circle are congruent, then the corresponding chords are congruent
4. If two chords are congruent, then the corresponding central angles are congruent
5. If two arcs are congruent, then the corresponding chords are congruent
6. If two chords of a circle are congruent, then the corresponding arcs are congruent
Example:
115°

If the measure of the minor arc angle is 115°, then the major arc angle is 245° because
360-115=245
10.4 Secants and Tangents
Definitions:
Secant- a line that intersects a circle at exactly two points (every secant contains a chord of the circle)
Tangent- a line that intersects a circle at exactly one point (this point is called the point of tangency or point of
contact)
Tangent Segment- the part of a tangent line between the point of contact and a point outside the circle
Secant segment- the part of a secant line that joins a point outside the circle to the farther intersection point of the
secant and the circle
External part- the part of a secant line (of a secant segment) that joins the outside point to the nearer intersection
point
Internally tangent- when one of two tangent circles lies inside the other
Externally tangent- if each of two tangent circles lies outside the other
Common tangent- a line tangent to two circles (internal if: it lies between the circles; external if: it is not between
the circles)
Postulates:
A tangent line is perp. to the radius drawn to the point of contact
If a line is perp. to a radius at its outer endpoint, then it’s tangent to the circle
Theorem:
1.
If two tangent segments are drawn to a circle from an external point, then those segments are congruent
(Two-Tangent theorem)
Example:
 R
*Radius R is perp. to the tangent line; thus, it forms two right angles
10.5 Angles Related to a Circle
Definitions:
Inscribed angle- an angle whose vertex is on a circle and whose sides are determined by two chords
Tangent-chord angle- an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord
that intersect at the tangent’s point of contact
Chord-chord angle- angle formed by 2 chords that intersect inside a circle but not at the center
Secant-secant angle- angle whose vertex is outside a circle and whose sides are determined by two secants
Secant-tangent angle- angle whose vertex is outside a circle; sides are determined by a secant and a tangent
Tangent-Tangent angle- angle whose vertex is outside of a circle; sides determined by 2 tangents
Theorems:
The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its
intercepted arc
The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord
angle and its vertical angle
The measure of a secant-secant, secant-tangent, or tangent-tangent angle (vertex outside a circle) is one-half the
difference of the measures of the intercepted arcs
Example:
E
D
F
*The measure of DE is 80°, so the measure of DEF is 40° because it’s half of 80
10.6 More Angle-Arc Theorems
Theorems:
1. If two inscribed/tangent-chord angles intercept the same arc (or congruent arcs), then they’re congruent
2. An angle inscribed in a semicircle is a right angle
3. The sum of the measures of a tangent-tangent angle and its minor arc is 180
Example:
130°
P
*If the measure of the arc is 130° then angle P is 50° because 180-130=50
10.7 Inscribed/Circumscribed Polygons
Definitions:
A polygon is inscribed in a circle if all of its vertices lie on a circle
A polygon is circumscribed about a circle if each of its sides is tangent to the circle
The center of a circle circumscribed about a polygon is the circumcenter of the polygon
The center of a circle inscribed in a polygon is the incenter of the polygon
Theorems:
If a quad. is inscribed in a circle, its opposite angles are supplementary
If a parallelogram is inscribed in a circle, it must be a rectangle
*Because the shape inscribed in the circle is a parallelogram, it is a rectangle as well.
10. 8 The Power Theorems
Theorems:
1.
If 2 chords of a circle intersect inside the circle, then the product of the measures of the segments of one
chord is equal to the product of the measures of the segments of the other chord (Chord-Chord Power
Theorem)
2. If a tangent segment and a secant segments are drawn from an external point to a circle, the square of the
measure of the tangent segment is equal to the product of the measures of the entire secant segment and its
external part (Tangent-Secant power theorem)
3. If two secant segments are drawn from an external point to a circle, then the product of the measures of one
secant segments and its external part is equal to the product of the measures of the other secant segments &
its external part (Secant-Secant power theorem)
Example:
Given: Chords VN and LS intersect at point E inside circle O.
Prove: EV× EN = EL× SE
10.9 Circumference and Arc Length
V
S
E
L
 O
Definitions:
Circumference- the perimeter of a circle (C=πd)
Theorems:
1.
The length of an arc is equal to the circumference times the fractional part of the circle determined by the
arc Length of PQ= (mPQ/360)πd
Example: Find the length of arc AB with an arc measure of 30° and a radius of 12cm Answer: you should get 2π cm
Example: Find the radius of a circle whose circumference is 70π Answer: you should get 35π
N
CHAPTER 12: SURFACE ARE AND VOLUME
Bases- parallel and congruent faces
Lateral edges- parallel edges joining the vertices of the bases
Lateral faces- faces of the prism that are not bases
Lateral surface area- sum of the area of the lateral faces
Total surface area- sum of the prism’s lateral area and the sum of the two
bases
lateral edge
bas
ee
Lateral face
Prisms
To Find Total Area:
Triangular
Prism
Rectangular
Prism
-
Find the area of both of the bases and of each of the
lateral faces and add them all up together
To Find Lateral Area:
-
Find the area of all the lateral faces and then add them
up all together.
To Find the Volume:
-
𝑉𝑝𝑟𝑖𝑠𝑚 =𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
*𝑏𝑎𝑠𝑒being the area of the base of the prism
Pentagonal Prism
Hexagonal
Prism
Cones
To Find Lateral Area:
- 𝐿. 𝐴.𝑐𝑜𝑛𝑒=𝑙𝜋𝑟
*𝑙 being the slant height
*𝑟 being the radius
𝑙
To Find Total Area:
ℎ
- 𝑇. 𝐴.𝑐𝑜𝑛𝑒=𝑙𝜋𝑟+𝜋𝑟 2
*𝑙 being the slant height
*𝑟 being the radius
To Find the Volume:
𝑟
- 𝑉𝑐𝑜𝑛𝑒=1𝜋𝑟 2ℎ
3
*ℎ being the height
*𝑟 being the radius
Spheres
To Find Total Area:
- 𝑇. 𝐴.𝑠𝑝ℎ𝑒𝑟𝑒=4𝜋𝑟 2
*𝑟 being the radius
𝑟
To Find the Volume:
- 𝑉𝑠𝑝ℎ𝑒𝑟𝑒=4𝜋𝑟 3
3
*𝑟 being the radius
Cylinders
To Find Total Area:
To Find Lateral Area:
- 𝑇. 𝐴.𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝐶ℎ ×2𝜋𝑟 2
*𝑟 being the radius
*ℎ being the height
*𝐶 being the Circumference
ℎ
To Find the Volume:
- 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝐵ℎ=𝜋𝑟 2ℎ
𝑟
*ℎ being the height
*𝐵 being the area of the base
*𝑟 being the radius
- 𝐿. 𝐴.𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=2𝜋𝑟ℎ
*𝑟 being the radius
*ℎ being the height
Pyramids
Slant height- height of one of the lateral faces
A
Slant
Height
B
a
altitude
Lateral
Edge
To Find Surface Area:
E
-
Find the area of the base and of each of the lateral
faces and add them all up together
To Find the Volume:
F
C
altitude- line drawn from the vertex of the
pyramid to the center of the base; is
perpendicular to the base. If the length of the
altitude is given and the slant height is not, you
can still use the Pythagorean Theorem (𝑎2 +
𝑏 2 = 𝑐 2 ) to find the slant height.
- 𝑉𝑝𝑦𝑟𝑎𝑚𝑖𝑑=1𝑏𝑎𝑠𝑒 ×ℎ𝑒𝑖𝑔ℎ𝑡
D
3
*𝑏𝑎𝑠𝑒being the area of the base of the prism
b
a
1.
Find the total area of a triangular prism with the given dimensions
a. l = 9, a = 4, b = 6, c = 7
b. l = 10, a = 6, b = 5, c = 8
c. l = 15, a = 3, b = 4, c = 5
l
2. Find the Lateral Are, Total Area, and Volume of a cone with the following dimensions.
a. l= 21, h = 15, r = 10
b. l= 18, h = 12, r = 5
l
c. l= 9, h = 6, r = 3
h
r
3. Find the Total Area and Vol. of a sphere with the following dimensions.
𝑟
a. r = 16
b. r = 25
c. r = 8
4. Find the Total Area, Lateral Area, and Volume of a cylinder with the following dimensions.
a. h = 25, r = 6
𝑟
b. h = 18, r = 13
c. h = 5, r = 6
ℎ
5. Find the Surface Area and Volume of a pyramid with the following dimensions.
a. l= 10, b = 20, h = 15
b. l= 16, b = 9, h = 13
c. l= 28, b = 8, h = 22
h
l
b
GEOMETRY STUDY GUIDE
TRIG AND STATS
1: Trigonometry
SOH~ CAH~ TOA
AKA: Sine= opposite/hypotenuse
Cosine= adjacent/hypotenuse
Tangent= opposite/adjacent
This can be used to find: -Missing sides
-To help find area of triangles
-And basically any equations/diagrams involving awesome stuff like...
-To use trigonometry you insert what information you know into a
calculator using the sin, cos, or tan, buttons.
For example: using the first diagram of the study guide - the angle A is
50 degrees and the hypotenuse is 7... find the lengths of opposite and
adjacent as well as the area of the triangle
1) sin(50)= opp/7
2) multiply both sides by 7
3) (7)sin(50)= opp
4) calculator work
5) 5.3623= opp
OR
1)cos(50)=adj/7
steps 2 and 3 same
2) looks like-- (7)cos(50)= adj
3) calculator work
4) 4.49951= adj
SO
use ONE of the following ways then ‘pathagorize’ it!
* a2+b2=c2 ( the 2’s show it being squared)
4.499 squared+ b2= 7 squared
-> 20.24+ b2= 49
-> 28.76= b2
(square root it)
-> 5.3628= b
then you have====
adj=4.499
hyp= 7
opp=5.3628 ( notice its the same as the way we did it in the trig you
can do it either way!)
sooo..
the area= ½ * b* h
½ * 4.499 * 5.3628= 12.06
AREA= 12
yes that was one problem good luck to us all.....
2: Statistics
meAn - aka the Average
** add up the numbers and divide by total amount of values
Median - the Middle of the list of numbers
** must be in numerical order
mOde -the number that Occurs most Often
range - the difference between the highest and lowest value
3:Standard Deviation
**the square root of variance.. now Variance is basically when you find
the difference of each value from the mean, then square the
differences, add them up, and divide by the total amount of values,
that my friend is the Variance. Now if you’re still with me you take
that value (the Variance) and square root it to find the Standard
Deviation!!