1. Classify each triangle by its sides.
1. 2 cm, 2 cm, 2 cm
2. 7 ft, 11 ft, 7 ft
3. 9 m, 8 m, 10 m
equilateral
isosceles
scalene
4. In ∆ABC, if m A = 70º and m B = 50º, what is m C?
ANSWER
60º
5. In ∆DEF, if m D = m E and m F = 26º, What are the measure of D and E
ANSWER
77º, 77º

2. Proving triangles congruent.
Target
Proving triangles congruent.
GOAL:
4.8 Use theorems about isosceles and equilateral triangles.

3. Vocabulary
vertex angle
legs
isosceles triangle –
base
base angles
vertex angle is always opposite the base
base angles are always opposite the legs

4. Vocabulary
Base Angles Theorem 4.7 –
If two sides of a triangle are congruent, then the angles opposite them are congruent.
Corollary –
If a triangle is equilateral, then it is equiangular.
Base Angles Converse Theorem 4.8 –
If two angles of a triangle are congruent, then the sides opposite them are congruent.
Corollary –
If a triangle is equiangular, then it is equilateral.

5. EXAMPLE 1
Apply the Base Angles Theorem
In DEF, DE DF . Name two congruent angles.
SOLUTION
DE DF , so by the Base Angles Theorem, E F.

6. GUIDED PRACTICE
for Example 1
Copy and complete the statement.
If HG HK , then ? ? .
SOLUTION
HGK HKG
If KHJ KJH, then ? ? .
SOLUTION
If KHJ KJH, then , KH KJ

7. Find measures in a triangle
EXAMPLE 2
Find measures in a triangle
Find the measures of P, Q, and R.
The diagram shows that PQR is equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQR is equiangular. So, m P = m Q = m R.
3(m P)
= 180 o
Triangle Sum Theorem
m P
= 60 o
Divide each side by 3.
The measures of P, Q, and R are all 60° .
ANSWER

8. GUIDED PRACTICE
for Example 2
Find ST in the triangle at the right.
SOLUTION
STU is equilateral, then it is equiangular
Thus ST = 5
( Base angle theorem )
ANSWER

9. GUIDED PRACTICE
for Example 2
Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain.
SOLUTION
No; it is not possible for an equilateral triangle to have angle measure other then 60°. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60° because all pairs of angles could be considered base of an isosceles triangle.

10. Use isosceles and equilateral triangles
EXAMPLE 3
Use isosceles and equilateral triangles
ALGEBRA
Find the values of x and y in the diagram.
SOLUTION
STEP 1
Find the value of y. Because KLN is equiangular, it is also equilateral and KN KL . Therefore, y = 4.
STEP 2
Find the value of x. Because LNM LMN, LN LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.
LN = LM
Definition of congruent segments
4 = x + 1
Substitute 4 for LN and x + 1 for LM.
3 = x
Subtract 1 from each side.

11. EXAMPLE 4
Solve a multi-step problem
Lifeguard Tower
In the lifeguard tower, PS QR and QPS PQR.
QPS PQR?
What congruence postulate can you use to prove that
Explain why PQT is isosceles.
Show that PTS QTR.

12. EXAMPLE 4
Solve a multi-step problem
SOLUTION
Draw and label QPS and PQR so that they do not overlap. You can see that PQ QP , PS QR , and QPS PQR. So, by the SAS Congruence Postulate,
QPS PQR.
From part (a), you know that because corresponding parts of congruent triangles are congruent. By the Converse of the Base Angles Theorem, PT QT , and
PQT is isosceles.

13. EXAMPLE 4
Solve a multi-step problem
You know that PS QR , and because corresponding parts of congruent triangles are congruent. Also, PTS QTR by the Vertical Angles Congruence Theorem. So,
PTS QTR by the AAS Congruence Theorem.

14. GUIDED PRACTICE
for Examples 3 and 4
Find the values of x and y in the diagram.
ANSWER
y° = 120°
x° = 60°

15. GUIDED PRACTICE
for Examples 3 and 4
Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that
PTS QTR
SOLUTION
QPS PQR. Can be shown by segment addition postulate i.e a.
QT + TS = QS and PT + TR = PR

16. GUIDED PRACTICE
for Examples 3 and 4
Since PT QT from part (b) and
TS TR
from part (c) then,
QS PR
PQ PQ
Reflexive Property and
PS QR
Given
Therefore QPS PQR . By SSS Congruence Postulate
ANSWER