Chapter 4Chapter 4

Congruent TrianglesCongruent Triangles

4.14.1

Triangles and AnglesTriangles and Angles

Parts of TrianglesParts of Triangles

Vertex Vertex Points joining the sides of a trianglePoints joining the sides of a triangle

Adjacent SidesAdjacent Sides Sides that share a common vertexSides that share a common vertex

Classification by SidesClassification by Sides

EquilateralEquilateral 3 congruent sides3 congruent sides

IsoscelesIsosceles At least 2 congruent sidesAt least 2 congruent sides

ScaleneScalene No congruent sidesNo congruent sides

Classification by AnglesClassification by Angles

AcuteAcute 3 acute angles3 acute angles

EquiangularEquiangular 3 congruent angles3 congruent angles

RightRight 1 right angle1 right angle

ObtuseObtuse 1 obtuse angle1 obtuse angle

Parts of Isosceles Parts of Isosceles TrianglesTriangles

LegsLegs The sides that are congruent.The sides that are congruent.

BaseBase The non-congruent side.The non-congruent side.

Isosceles trianglesIsosceles triangles

Base angles are congruent.Base angles are congruent.

Base

Base angles

legsVertex angle

Parts of Right TrianglesParts of Right Triangles

HypotenuseHypotenuse The side that is opposite the right The side that is opposite the right

angle. It is always the longest side.angle. It is always the longest side. LegsLegs

The sides that form the right angleThe sides that form the right angle

Right TrianglesRight Triangles

leg

leghypotenuse

Interior AnglesInterior Angles

The angles on the inside of a The angles on the inside of a triangle.triangle.

Triangle Sum ConjectureTriangle Sum Conjecture

The sum of the measures of the The sum of the measures of the angles in every triangle is 180angles in every triangle is 180..

ExampleExample

Find the measure of each angle.

2x + 10

x x + 2

Exterior AnglesExterior Angles

The angles that are adjacent to the interior The angles that are adjacent to the interior anglesangles

The exterior angles always add to equal The exterior angles always add to equal 360°360°

DefinitionsDefinitions

Exterior AngleExterior Angle AdjacentAdjacent

Interior Interior

AngleAngle

Remote Interior AnglesRemote Interior Angles

Exterior Angles of a Exterior Angles of a TriangleTriangle

Use your straightedge to draw a Use your straightedge to draw a triangle.triangle.

Extend one side out as shownExtend one side out as shown

A

B

C

xa

b

c

Exterior Angles of a Exterior Angles of a TriangleTriangle

Trace angles a and b onto a Trace angles a and b onto a transparency so that they are transparency so that they are adjacent.adjacent.

How does this compare to angle x?How does this compare to angle x?

A

B

C

xa

b

cab

Triangle Exterior Angle Triangle Exterior Angle ConjectureConjecture

The measure of the exterior angle The measure of the exterior angle of a triangle is equal to the sum of of a triangle is equal to the sum of the measures of the remote the measures of the remote interior anglesinterior angles

A

B

C

x = a + ba

b

c

ExampleExample

Find the missing measuresFind the missing measures

80°

53°

ExampleExample

Find the missing measuresFind the missing measures

60° 120°

ExampleExample

Page 199 #37Page 199 #37

(2x – 8)°

x° 31°

4.24.2

Congruence and Congruence and TrianglesTriangles

TermsTerms

CongruentCongruent Figures that are exactly the same size and Figures that are exactly the same size and

shape are congruentshape are congruent Corresponding anglesCorresponding angles

The angles that are in corresponding The angles that are in corresponding positions are congruentpositions are congruent

Corresponding sidesCorresponding sides The sides that are in corresponding The sides that are in corresponding

positions are congruentpositions are congruent

Naming Congruent FiguresNaming Congruent Figures

When a congruence statement is When a congruence statement is made it is important to match up made it is important to match up corresponding parts.corresponding parts.

Third Angle TheoremThird Angle Theorem

If two angles in one triangle are If two angles in one triangle are equal to two angles in another equal to two angles in another triangle, then the third angles in triangle, then the third angles in each triangle are also equal. each triangle are also equal.

Examples 1 Examples 1 (page 205)(page 205)

ΔΔLMN LMN ΔΔPQRPQR

105°ML

N45°

P

R

Q

What is the measure of:

P

M

R

N

Which side is congruent to

segment QR

Segment LN

Example 2Example 2

Given ABC PQR, find the values of x and y.

A

B CP

QR

85°

50°

(6y – 4)°

(10x + 5)°

4.34.3

Proving Triangles Proving Triangles Congruent SSS and SAS Congruent SSS and SAS

Warm-UpWarm-Up

Complete the following statementComplete the following statement

BIG BIG B

I

G

R

A

T

DefinitionsDefinitions

included angle included angle An angle that is between two given An angle that is between two given

sides.sides.

included side included side A side that is between two given A side that is between two given

angles.angles.

Example 1Example 1

Use the diagram. Use the diagram. Name the Name the included angle included angle between the pair between the pair of given sides.of given sides.

KP

J L

PLandKPLKandPKKLandJK

Triangle Congruence Triangle Congruence ShortcutShortcut

SSSSSS If the three sides of one triangle are If the three sides of one triangle are

congruent to the three sides of congruent to the three sides of another triangle, then the triangles another triangle, then the triangles are congruent.are congruent.

Triangle Congruence Triangle Congruence ShortcutsShortcuts

SASSAS If two sides and the included angle of If two sides and the included angle of

one triangle are congruent to two one triangle are congruent to two sides and the included angle of sides and the included angle of another triangle, then the triangles another triangle, then the triangles are congruent.are congruent.

Example 2Example 2

Complete the congruence Complete the congruence statement.statement.

Name the congruence shortcut Name the congruence shortcut used.used.S

T

U

V

WSTW

Example 3Example 3

Determine if the following are Determine if the following are congruent. congruent.

Name the congruence shortcut Name the congruence shortcut used.used.H

I

J

L

MN

HIJ LMN

Example 4Example 4

Complete the congruence Complete the congruence statement.statement.

Name the congruence shortcut Name the congruence shortcut used.used.

B

OX

C

A

R

XBO

Example 5Example 5

Complete the congruence Complete the congruence statement.statement.

Name the congruence shortcut Name the congruence shortcut used.used.SPQ

S

P

Q

T

4.44.4

Proving Triangles Proving Triangles Congruent ASA and AASCongruent ASA and AAS

Triangle Congruence Triangle Congruence ShortcutsShortcuts

ASAASA If two angles and the included side of If two angles and the included side of

one triangle are congruent to two one triangle are congruent to two angles and the included side of angles and the included side of another triangle, then the triangles another triangle, then the triangles are congruent.are congruent.

Triangle Congruence Triangle Congruence ShortcutsShortcuts

SAASAA If two angles and a non-included side If two angles and a non-included side

of one triangle are congruent to the of one triangle are congruent to the corresponding angles and side of corresponding angles and side of another triangle, then the two another triangle, then the two triangles are congruent.triangles are congruent.

Example 1Example 1

Complete the congruence Complete the congruence statement.statement.

Name the congruence shortcut Name the congruence shortcut used.used. Q U

ADQUA

Example 2Example 2

Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used.Name the congruence shortcut used.

MM R R N N

QQ P P

RMQ

Example 3Example 3

Determine if the following are Determine if the following are congruent. congruent.

Name the congruence shortcut Name the congruence shortcut used.used.

A

B

C

F

E

DABC FED

4.64.6

Isosceles, Equilateral, Isosceles, Equilateral, and Right Trianglesand Right Triangles

Warm-Up 1Warm-Up 1

Find the measure of each angle.

90°

90°

30°

60°a

b

Warm-Up 2Warm-Up 2

Find the measure of each angle.

110

15090

The base angles of an isosceles The base angles of an isosceles triangle are congruent.triangle are congruent.

If a triangle has at least two If a triangle has at least two congruent angles, then it is an congruent angles, then it is an isosceles triangle.isosceles triangle.

If the sides are congruent then the If the sides are congruent then the base angles are congruent.base angles are congruent.

Isosceles trianglesIsosceles triangles

Example 1Example 1

35°

x

Example 2Example 2

15°

b

a

Example 3Example 3

Find each missing measure

63°

10 cmm n

p

Equilateral TrianglesEquilateral Triangles

If a triangle is equilateral, then it is If a triangle is equilateral, then it is equiangular.equiangular.

If a triangle is equiangular, then it If a triangle is equiangular, then it is equilateral.is equilateral.

Hypotenuse-Leg (HL)Hypotenuse-Leg (HL)

If the hypotenuse and the leg of a If the hypotenuse and the leg of a right triangle are congruent to the right triangle are congruent to the hypotenuse and leg of a second hypotenuse and leg of a second right triangle, then the two right triangle, then the two triangles are congruent.triangles are congruent.

Example 4Example 4

Find the value of xFind the value of x

12 in

2x in

Example 5Example 5

Find the value of x and y.Find the value of x and y.

yx

Example 6Example 6

Find the value of x and y.Find the value of x and y.

75°

x°

y°

x°

Chapter 8Chapter 8

SimilaritySimilarity

8.18.1

Ratio and ProportionRatio and Proportion

RatiosRatios

The ratio of a to b can be written The ratio of a to b can be written asas a/ba/b a : ba : b

The denominator cannot be zeroThe denominator cannot be zero

Simplifying RatiosSimplifying Ratios

Ratios should be expressed in simplified Ratios should be expressed in simplified formform 6:8 = 3:46:8 = 3:4

Before reducing, make sure that the Before reducing, make sure that the units are the same.units are the same. 12in : 3 ft12in : 3 ft

12in : 36 in12in : 36 in

1: 31: 3

Examples (page 461)Examples (page 461)

Simplify each ratioSimplify each ratio10.10. 16 students16 students

24 students24 students

12.12. 22 feet22 feet

52 feet52 feet

18.18. 60 cm60 cm

1 m1 m

Examples (page 461)Examples (page 461)

Simplify each ratioSimplify each ratio20. 20. 2 mi2 mi

3000 ft3000 ft

24. 24. 20 oz.20 oz.

4 lb4 lb

There are 5280 ft in 1 mi.There are 5280 ft in 1 mi.

There are 16 oz in 1 lb.There are 16 oz in 1 lb.

Examples (page 461)Examples (page 461)

Find the width to length ratioFind the width to length ratio14.14.

16.16.

20 mm

16 mm

2 ft

12 in.

Using Ratios Example 1Using Ratios Example 1

The perimeter of the isosceles The perimeter of the isosceles triangle shown is 56 in. The ratio triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of LM : MN is 5:4. Find the length of the sides and the base of the of the sides and the base of the triangle.triangle.

N

L

M

Using Ratios Example 2Using Ratios Example 2

The measures of the angles in a The measures of the angles in a triangle are in the extended ratio triangle are in the extended ratio 3:4:8. Find the measures of the 3:4:8. Find the measures of the angles angles

3x

4x

8x

Using Ratios Example 3Using Ratios Example 3

The ratios of the side lengths of The ratios of the side lengths of ΔΔQRS to the corresponding side QRS to the corresponding side lengths of lengths of ΔΔVTU are 3:2. Find the VTU are 3:2. Find the unknown lengths.unknown lengths.

S

T

R

Q

V

U

2 cm

18 cm

ProportionsProportions

ProportionProportion Ratio = RatioRatio = Ratio Fraction = FractionFraction = Fraction

Solving ProportionsSolving Proportions Cross multiplyCross multiply Let the means equal the extremesLet the means equal the extremes

,a c

If thenb d

Properties of ProportionsProperties of Proportions

Cross Product PropertyCross Product Property

Reciprocal PropertyReciprocal Property

,a c b d

If thenb d a c

ad bc

Solving Proportions Solving Proportions Example 1Example 1

9 6

14 x

Solving Proportions Solving Proportions Example 2Example 2

5

4 10

s s

Solving Proportions Solving Proportions Example 3Example 3

A photo of a A photo of a building has the building has the measurements measurements shown. The shown. The actual building is actual building is 26 ¼ ft wide. 26 ¼ ft wide. How tall is it?How tall is it?

1 7/8 in

2.75 in

8.28.2

Problem solving in Problem solving in Geometry with Geometry with

ProportionsProportions

Properties of ProportionsProperties of Proportions

,a c a b

If thenb d c d

,a c a b c d

If thenb d b d

Example 1Example 1

Tell whether the statement is true Tell whether the statement is true or falseor false A.A.

B. B.

15 3,

10 2

s sIf then

t t

3 5 3 5,

x yIf thenx y x y

Example 2Example 2

In the diagramIn the diagram

Find the length of LQ.Find the length of LQ.

MQ LQ

MN LP

P

M6

N

Q

1513

L 5

Geometric MeanGeometric Mean

Geometric MeanGeometric Mean The geometric mean between two The geometric mean between two

numbers a and b is the positive numbers a and b is the positive number x such that number x such that

a x

x b

Example 3Example 3

Find the geometric mean between Find the geometric mean between 35 and 175.35 and 175.

Example 4Example 4

You are building a scale model of You are building a scale model of your uncle’s fishing boat. The boat your uncle’s fishing boat. The boat is 62 ft long and 23 ft wide. The is 62 ft long and 23 ft wide. The model will be 14 in. long. How model will be 14 in. long. How wide should it be?wide should it be?

8.38.3

Similar PolygonsSimilar Polygons

Similar PolygonsSimilar Polygons

Polygons are similar if and only if Polygons are similar if and only if

the corresponding angles are the corresponding angles are congruent congruent

and and the corresponding sides are the corresponding sides are

proportional.proportional.

Similar figures are Similar figures are dilations dilations of each of each other. (They are reduced or other. (They are reduced or enlarged by a scale factor.)enlarged by a scale factor.)

The symbol for similar is The symbol for similar is

Example 1Example 1

Determine if the sides of the polygon are proportional.

12 m 8 m

8 m

6 m6 m

Example 2

Determine if the sides of the polygon are proportional.

15 m5 m

9 m

12 m

3 m

4 m

Example 3Example 3Find the missing measurements.

HAPIE NWYRS

HA

P

IE N

W

Y

RS

6

5 418 24

21AP =

EI =

SN =

YR =

Example 4Example 4

D

Find the missing measurements.

QUAD SIML

A

UQM

I

SL

20

25125º

8

95º65º

QD =

MI =

mD =

mU =

mA =

12

8.4/8.58.4/8.5

Similar TrianglesSimilar Triangles

Similar TrianglesSimilar Triangles

  To be similar, corresponding sides To be similar, corresponding sides must be proportional and must be proportional and corresponding angles are corresponding angles are congruent.congruent.

  

Similarity ShortcutsSimilarity Shortcuts

AA Similarity ShortcutAA Similarity Shortcut

If two angles in one triangle are If two angles in one triangle are congruentcongruent to two angles in to two angles in another triangle, then the triangles another triangle, then the triangles are similar.are similar.

Similarity ShortcutsSimilarity Shortcuts

SSS Similarity ShortcutSSS Similarity Shortcut

If three sides in one triangle are If three sides in one triangle are proportionalproportional to the three sides in to the three sides in another triangle, then the triangles another triangle, then the triangles are similar.are similar.

Similarity ShortcutsSimilarity Shortcuts

SAS Similarity ShortcutSAS Similarity Shortcut

If two sides of one triangle are If two sides of one triangle are proportionalproportional to two sides of to two sides of another triangle and another triangle and

their included angles are their included angles are congruentcongruent, then the triangles are , then the triangles are similar. similar.

Similarity ShortcutsSimilarity Shortcuts

We have three shortcuts:We have three shortcuts:

AAAA

SASSAS

SSSSSS

Example 1Example 1

4g

7

69

10.5

Example 2Example 2

h

32

24

50

k

30

Example 3Example 3

42m36

24

1. 1. A flagpole 4 meters tall casts a 6 A flagpole 4 meters tall casts a 6 meter shadow. At the same time of meter shadow. At the same time of day, a nearby building casts a 24 meter day, a nearby building casts a 24 meter shadow. How tall is the building?shadow. How tall is the building?

4m

6m

24m

2. 2. Five foot tall Melody casts an 84 inch Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot same time of day, his shadow is 1 foot shorter than hers?shorter than hers?

3. 3. A 10 meter rope from the top of a A 10 meter rope from the top of a flagpole reaches to the end of the flagpole reaches to the end of the flagpole’s 6 meter shadow. How tall is flagpole’s 6 meter shadow. How tall is the nearby football goalpost if, at the the nearby football goalpost if, at the same moment, it has a shadow of 4 same moment, it has a shadow of 4 meters?meters?

10m

6m

4m

4. 4. Private eye Samantha Diamond places Private eye Samantha Diamond places a mirror on the ground between herself a mirror on the ground between herself and an apartment building and stands and an apartment building and stands so that when she looks into the mirror, so that when she looks into the mirror, she sees into a window. The mirror is she sees into a window. The mirror is 1.22 meters from her feet and 7.32 1.22 meters from her feet and 7.32 meters from the base of the building. meters from the base of the building. Sam’s eye is 1.82 meters above the Sam’s eye is 1.82 meters above the ground. How high is the window?ground. How high is the window?

1.22 7.32

1.82

8.68.6

Proportions and Similar Proportions and Similar TrianglesTriangles

ProportionsProportions

Using similar triangles missing Using similar triangles missing sides can be found by setting up sides can be found by setting up proportions.proportions.

TheoremTheorem

Triangle Proportionality TheoremTriangle Proportionality Theorem If a line parallel to one side of a If a line parallel to one side of a

triangle intersects the other two triangle intersects the other two sides, then it divides the two sides sides, then it divides the two sides proportionally.proportionally.Q

S

T

R

U

|| , .RT RU

If TU QS thenTQ US

TheoremTheorem

Converse of the Triangle Converse of the Triangle Proportionality TheoremProportionality Theorem If a line divides two sides of a triangle If a line divides two sides of a triangle

proportionally, then it is parallel to proportionally, then it is parallel to the third side.the third side.Q

S

T

R

U

, || .RT RU

If thenTU QSTQ US

Example 1Example 1

In the diagram, segment UY is In the diagram, segment UY is parallel to segment VX, UV = 3, parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is UW = 18 and XW = 16. What is the length of segment YX?the length of segment YX?

U

Y

V

W

X

Example 2Example 2

Given the diagram, determine Given the diagram, determine whether segment PQ is parallel to whether segment PQ is parallel to segment TR.segment TR.

P

9

T

S

26

Q

9.75

24

R

TheoremTheorem

If three parallel lines intersect two If three parallel lines intersect two transversals, then they divide the transversals, then they divide the transversals proportionally.transversals proportionally.

TheoremTheorem

If a ray bisects an angle of a If a ray bisects an angle of a triangle, then it divides the triangle, then it divides the opposite side into segments whose opposite side into segments whose lengths are proportional to the lengths are proportional to the lengths of the other two sides.lengths of the other two sides.

Example 3Example 3

In the diagram, In the diagram, 1 1 2 2 3, AB 3, AB =6, BC=9, EF=8. What is x?=6, BC=9, EF=8. What is x?

A

ED

x

9

B6

F8

3

2

C

1

Example 4Example 4

In the diagram, In the diagram, LKM LKM MKN. MKN. Use the given side lengths to find Use the given side lengths to find the length of segment MN.the length of segment MN.

3

K

ML

15

17

N

5. Juanita, who is 1.82 meters tall, 5. Juanita, who is 1.82 meters tall, wants to find the height of a tree in wants to find the height of a tree in her backyard. From the tree’s base, her backyard. From the tree’s base, she walks 12.20 meters along the she walks 12.20 meters along the tree’s shadow to a position where the tree’s shadow to a position where the end of her shadow exactly overlaps end of her shadow exactly overlaps the end of the tree’s shadow. She is the end of the tree’s shadow. She is now 6.10 meters from the end of the now 6.10 meters from the end of the shadows. How tall is the tree?shadows. How tall is the tree?

1.8212.206.10