WARM-UPGive reasons for each step in this proof.

B

DA

E

C

Given: C is the midpoint of AC, C is the midpoint of DBProve: AB is congruent to ED

Statements Reasons

C is the midpoint of AC, C is the midpoint of DB

AC is congruent to CE

DC is congruent to CB

<ACB is congruent to <DCE

Triangle ACB is congruent to triangle ECD

AB is congruent to ED

Vertical angles are congruent

SAS

CPCTC

Midpoint Theorem

Given

Midpoint Theorem

CHAPTER 4SECTION 5

Proving Triangles Congruent: AAS

VOCABULARYAngle-Angle-Side Theorem (AAS)- If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent.

VOCABULARYIsosceles Triangle Theorem- If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Theorem 4-7- If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollary 4-3- A triangle is equilateral if and only if it is equiangular.

Corollary 4-4- Each angle of an equilateral triangle measures 60 degrees.

Example 1) In isosceles triangle DEF, <D is the vertex angle. If m<E = 2x + 40, and m<F = 3x + 22, find the measure of each angle of the triangle.

FE

D

Since it is isosceles, the base angles are congruent.

m<E = m<F2x + 40 = 3x + 2240 = x + 2218 = x

Plug 18 in for x in either equation.

m<E = 2x + 40m<E = 2(18) + 40m<E = 36 + 40m<E = 76 = m<F

Now all the angles in a triangle add up to 180.

180 = m<E + m<F + m<D180 = 76 + 76 + m<D180 = 152 + m<D28 = m<D

Example 2) In isosceles triangle ISO with base SO, m<S = 5x - 18 and m<O = 2x + 21. Find the measure of each angle of the triangle.

OS

I

Since it is isosceles, the base angles are congruent.

m<O = m<S2x + 21 = 5x - 1821 = 3x – 1839 = 3x13 = x

Plug 13 in for x in either equation.

m<O = 2x + 21m<O = 2(13) + 21m<O = 26 + 21m<O = 47 = m<S

Now all the angles in a triangle add up to 180.

180 = m<O + m<S + m<I180 = 47 + 47 + m<I180 = 94 + m<I86 = m<I

Example 3) Given: AB congruent to EB and <DEC congruent to <BProve: Triangle ABE is equilateral.

C

A

B

Statements Reasons

AB congruent to EB and <DEC congruent to <B

<A congruent to <AEB

<AEB is congruent to <DEC

<A congruent to <AEB congruent to <B

Given

Isosceles Triangle Theorem

Transitive Property of Equality

Triangle AEB is equiangular Definition of equiangular triangles

D

E

Vertical angles are congruent

Triangle ABE is equilateral If a Triangle is equiangular, it is equilateral.

Example 4) Find the value of x.

All the angles in a triangle add up to 180.

180 = 60 + y + y180 = 60 + 2y120 = 2y60 = y

So the top triangle is an equilateral triangle. So all the sides equal 2x + 5.

The bottom triangle is an isosceles triangle.

2x + 5 = 3x – 135 = x – 1318 = x

602x +

5

3x - 13

y y

Example 5) Find the value of x.

This is an isosceles triangle so two of the angles are equal to 3x + 8. All the angles in a triangle add up to 180.

180 = 2x + 20 + 3x + 8 + 3x + 8180 = 8x + 36144 = 8x18 = x

2x + 20

3x + 8

Example 6) In triangle ABD, AB is congruent to BD, m<A is 12 less than 3 times a number, and m<D is 13 more than twice the same number. Find m<B.

Start by drawing a diagram.

Now since AB is congruent to BD we knowTriangle ABD is an isosceles triangle.

m<A = m<D3x – 12 = 2x + 13x – 12 = 13x = 25

Plug 25 in for x in either equation.

m<A = 3x – 12m<A = 3(25) – 12m<A = 75 – 12m<A = 63 = m<D

B

DA

All the angles in a triangle add up to 180.

180 = m<A + m<B + m<D180 = 63 + m<B + 63180 = 126 + m<B54 = m<B

Example 7)Given: AD congruent to AE and <ACD congruent to <ABEProve: Triangle ABC is isosceles.

E

Statements Reasons

AD congruent to AE and <ACD congruent to <ABE

<A is congruent to <A

AC is congruent to AB

Triangle ACD is congruent to triangle ABE

Given

AAS

CPCTC

Congruence of angles is reflexive

C

A

B

D

Definition of isosceles triangleTriangle ABC is isosceles.

Refer to the figure.

12

11

10

9

8

7

6

54

32

1

TSR

CF D

Example 8) Name the included side for <1 and <4.FD

Example 9) Name the included side for <7 and <8.DS

Example 10) Name a nonincluded side for <5 and <6.FD, FR, RS, ST, CT, or DC

Example11) Name a nonincluded side for <9 and <10.DS, CT, ST, SR, RF, or FD

Refer to the figure.

12

11

10

9

8

7

6

54

32

1

TSR

CF D

Example 12) CT is included between which two angles?<10 and <11

Example 13) In triangle FDR, name a pair of angles so that FR is not included.<2 and <4 or <1 and <4

Example 14) If <1 is congruent to <6, <4 is congruent to <3, and FR is congruent to DS, then triangle FDR is congruent to triangle ______ by ______.SRD; AAS

Refer to the figure.

12

11

10

9

8

7

6

54

32

1

TSR

CF D

Example 15) If <5 is congruent to <7, <6 is congruent to <8, and DS is congruent to DS, then triangle TDS is congruent to triangle ______ by ______.RDS; ASA

Example 16) If <4 is congruent to <9, what sides would need to be congruent to show triangle FDR is congruent to triangle CDT?FD congruent to CD and DR congruent to DT