Chapter 4

Congruent Triangles

Chapter Objectives

• Classification of Triangles by Sides• Classification of Triangles by Angles• Exterior Angle Theorem• Triangle Sum Theorem• Adjacent Sides and Angles• Parts of Specific Triangles• 5 Congruence Theorems for Triangles

Lesson 4.1

Triangles and Angles

Lesson 4.1 Objectives

• Identify the parts of a triangle• Classify triangles according to their sides• Classify triangles according to their angles• Calculate angle measures in triangles

Classification of Triangles by Sides

Name Equilateral Isosceles Scalene

Looks Like

Characteristics 3 congruent sides At least 2 congruent sides

No Congruent Sides

Classification of Triangles by Angles

Name Acute Equiangular Right Obtuse

Looks Like

Characteristics 3 acute angles

3 congruent angles

1 right angles

1 obtuse angle

Example 1

• You must classify the triangle as specific as you possibly can.

• That means you must name– Classification according to angles– Classification according to sides

• In that order!

• Example

Obtuse isosceles

Vertex

• The vertex of a triangle is any point at which two sides are joined.– It is a corner of a triangle.– There are 3 in every triangle

Adjacent Sides and Adjacent Angles

• Adjacent sides are those sides that intersect at a common vertex of a polygon.– These are said to be adjacent

to an angle.

• Adjacent angles are those angles that are right next to each other as you move inside a polygon.– These are said to be adjacent

to a specific side.

Special Parts in a Right Triangle

• Right triangles have special names that go with it parts.

• For instance:– The two sides that form the right angle are called the

legs of the right triangle.– The side opposite the right angle is called the

hypotenuse.• The hypotenuse is always the longest side of a right triangle.

legs

hypotenuse

Special Parts of an Isosceles Triangle

• An isosceles triangle has only two congruent sides– Those two congruent sides are called legs.– The third side is called the base.

legs

base

More Parts of Triangles

• If you were to extend the sides you will see that more angles would be formed.

• So we need to keep them separate– The three original angles are called interior angles because they

are inside the triangle.– The three new angles are called exterior angles because they lie

outside the triangle.

Example 2

Classify the following triangles by their sides and their angles.

Scalene

ObtuseScalene

RightIsosceles

Acute

Theorem 4.1:Triangle Sum Theorem

• The sum of the measures of the interior angles of a triangle is 180o.

A

B

C

mA + mB + mC = 180o

Example 3

Solve for x and then classify the triangle based on its angles.

3x + 2x + 55 = 180 Triangle Sum Theorem

5x + 55 = 180 Simplify

5x = 125 SPOE

x = 25 DPOE

75

50

Acute

Theorem 4.2:Exterior Angle Theorem

• The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

m A +m B = m C

A

B

C

Example 4

Solve for x

x = 50 + 70 Exterior Angle Theorem

x = 120 Simplify

Corollary to theTriangle Sum Theorem

• A corollary to a theorem is a statement that can be proved easily using the original theorem itself.– This is treated just like a theorem or a postulate in proofs.

• The acute angles in a right triangle are complementary.

C

A

BmA + mB = 90o

Homework 4.1

• In Class– 1-9

• p199-201

• In HW– 10-26, 31-39, 41-47, 49, 50, 52-68

• Due Tomorrow

Lesson 4.2

Congruence and Triangles

Lesson 4.2 Objectives

• Identify congruent figures and their corresponding parts.

• Prove two triangles are congruent.• Apply the properties of congruence to

triangles.

Congruent Triangles

• When two triangles are congruent, then – Corresponding angles are congruent.– Corresponding sides are congruent.

• Corresponding, remember, means that objects are in the same location.– So you must verify that when the triangles are drawn

in the same way, what pieces match up?

Naming Congruent Parts• Be sure to pay attention to the proper notation when

naming parts. ABC DEF

• This is called a congruence statement.

A

B

C

D

E

F A D B E C F

and

AB DEBC EFAC DF

Theorem 4.3:Third Angles Theorem

• If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Prove Triangles are Congruent

• In order to prove that two triangles are congruent, we must– Show that ALL corresponding angles are

congruent, and– Show that ALL corresponding sides are congruent.

• We must show all 6 are congruent!

Example 5Complete the following statements.a) Segment EF ___________

a) segment OPb) P ________

b) Fc) G ________

c) Qd) mO = ________

d) 110o

e) QO = ________e) 7 km

f) GFE __________f) QPO

• Yes, the order is important!

Theorem 4.4:Properties of Congruent Triangles

• Reflexive Property of Congruent Triangles ABC ABC

• Reflexive Property of

• Symmetric Property of Congruent Triangles– If ABC DEF, then DEF ABC.

• Symmetric Property of

• Transitive Property of Congruent Triangles– If ABC DEF and DEF JKL, then

ABC JKL.• Transitive Property of

Homework 4.2

• None!

Lesson 4.3

Proving Triangles are Congruent:SSS&

SAS

Lesson 4.3 Objectives

• Prove triangles are congruent using the SSS Congruence Postulate

• Prove triangles are congruent using the SAS Congruence Postulate

Postulate 19:Side-Side-Side Congruence Postulate

• If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

Postulate 20:Side-Angle-Side Congruence Postulate

• If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Which One Do I Use?• Remember there are 6 parts to every triangle.

– Identify which parts of the triangle do you know (100% sure) are congruent.

– Rotate around the triangle keeping one thing in mind.• Cannot rotate so that 2 parts in a row are missed!• That means as you rotate by counting angle, then side, then angle,

then side, then angle, and then side you cannot miss two pieces in a row!

– You can skip 1, but not 2!!– Be sure the pattern that you find fits the same pattern in

the same way from the other triangle.• If it fits, they are congruent.

Example 6Decide whether or not the congruence statement is true.Explain your reasoning.

Reflexive Property of Congruence

The statement is true because ofSSS Congruence

Reflexive Property of Congruence

The statement is not true because the vertices areout of order.

The statement is not true because the vertices areout of order.Because the segment is shared between

two triangles, and yet it is the same segment

Example 7

Decide whether or not there is enough information to conclude SAS Congruence.

Reflexive Property of Congruence

Yes!

Yes!

No

Homework 4.3

• In Class– 1-5

• p216-218

• HW– 6-20

• Due Tomorrow

Lesson 4.4

Proving Triangles are Congruent:ASA

&AAS

Lesson 4.4 Objectives

• Prove that triangles are congruent using the ASA Congruence Postulate

• Prove that triangles are congruent using the AAS Congruence Theorem

Postulate 21:Angle-Side-Angle Congruence

• If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

Theorem 4.5:Angle-Angle-Side Congruence

• If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.

Example 8

Complete the proof

Given

Given

Reflexive POC

SSS Congruence

Homework 4.4

• In Class– 1-7

• p223-227

• HW– 8-18 evens

• Due Tomorrow

Lesson 4.5

Using Congruent Triangles

Lesson 4.5 Objectives

• Observe that corresponding parts of congruent triangles are congruent

Showing Triangles are Congruent

• You only have 4 shortcuts right now to show that two triangles are congruent to each other.

1. SSS Congruence2. SAS Congruence3. ASA Congruence4. AAS Congruence

• Otherwise you need to show all 6 parts of a triangle have matching congruent parts to another triangle.

• If you can use one of the above 4 shortcuts to show triangle congruency, then we can assume that all corresponding parts of the triangles are congruent as well.

Surveying

MNP MKL– Given

• Segment NM Segment KM– Definition of a midpoint

LMK PMN– Vertical Angles Theorem

KLM NPM– ASA Congruence

• Segment LK Segment PN– Corresponding Parts of Congruent Triangles

Example 9Tell which triangles you show to be congruent in order to prove the

statement is true. What postulate or theorem would help you show the triangles are congruent.

Show:STV UTV

Reflexive Property of Congruence

STV UTV

SSS Congruence

Corresponding Parts of Congruent Triangles

Show:Segment XY Segment ZW

Reflexive Property of Congruence

Alternate Interior Angles Theorem (Parallel Lines)

WXZ YZX

ASA Congruence

Corresponding Parts of Congruent Triangles

Homework 4.5

• In Class– 1-3

• p232-235

• HW– 4-18, 25-36

• Due Tomorrow

Lesson 4.6

Isosceles,Equilateral,

andRight Triangles

Lesson 4.6 Objectives

• Use properties of isosceles and equilateral triangles.

• Identify more properties based on the definitions of isosceles and equilateral triangles.

• Use properties of right triangles.

Isosceles Triangle Theorems

• Theorem 4.6: Base Angles Theorem– If two sides of a triangle

are congruent, then the angles opposite them are congruent.

• Theorem 4.7: Converse of Base Angles Theorem– If two angles of a

triangle are congruent, then the sides opposite them are congruent.

Example 10

Solve for x

Theorem 4.7

4x + 3 = 15

4x = 12

x = 3

Theorem 4.6

7x + 5 = x + 47

6x + 5 = 47

6x = 42

x = 7

Equilateral Triangles

• Corollary to Theorem 4.6– If a triangle is

equilateral, then it is equiangular.

• Corollary to Theorem 4.7– If a triangle is

equiangular, then it is equilateral.

Example 11Solve for x

Corollary to Theorem 4.6

In order for a triangle to be equiangular, all angles must equal…

5x = 60

x = 12

Corollary to Theorem 4.6

It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest!

2x + 3 = 4x - 5

3 = 2x - 5

8 = 2x

x = 4

Theorem 4.8:Hypotenuse-Leg Congruence Theorem

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

• HL

Example 12

Determine if enough information is given to conclude the triangles are congruent using HL Congruence

Reflexive Property of Congruence

Yes they are congruent!

Reflexive Property of Congruence

Neither triangle is a right triangle, so…

Not congruent

Homework 4.6

• In Class– 1-7

• p239-242

• HW– 8-28 even, 33, 34

• Due Tomorrow

Lesson 4.7

TrianglesAnd

Coordinate Proof

Lesson 4.7 Objectives

• Place geometric figures in a coordinate plane.• Use the Distance Formula to verify congruent

triangles.

Coordinate Proof• A coordinate proof involves placing geometric figures in a

coordinate plane.• Then you employ the following tools to prove concepts from

your picture– Distance Formula

– Midpoint Formula(x2 – x1)2 + (y2 – y1)2

(x1 + x2)

(y1 + y2)( ),2 2

Homework 4.7

• WS