Warm-Up Exercises
Classify each triangle by its sides.
1. 2 cm, 2 cm, 2 cm
equilateral
4. In ∆ABC, if m
2. 7 ft, 11 ft, 7 ft
isosceles
3. 9 m, 8 m, 10 m
scalene
A = 70º and m B = 50º, what is m
ANSWER
C?
60º
5. In ∆DEF, if m D = m E and m F = 26º, What are
the measure of D and E
ANSWER
77º, 77º
Warm-Up Exercises
Target
Proving triangles
congruent.
GOAL:
4.8 Use theorems about isosceles
and equilateral triangles.
Warm-Up Exercises
Vocabulary
vertex angle
legs
• isosceles triangle –
base
base angles
vertex angle is always opposite the base
base angles are always opposite the legs
Warm-Up Exercises
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.
Warm-Up1Exercises
EXAMPLE
Apply the Base Angles Theorem
In
DEF, DE
DF . Name two congruent angles.
SOLUTION
DE
DF , so by the Base Angles Theorem,
E
F.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Copy and complete the statement.
1.
If HG
HK , then
?
? .
SOLUTION
HGK
2.
If
KHJ
HKG
KJH, then ?
? .
KJH, then , KH
KJ
SOLUTION
If
KHJ
Warm-Up2Exercises
EXAMPLE
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
o
m
P = 60
Divide each side by 3.
ANSWER
The measures of
P,
Q, and
R are all 60° .
Warm-Up
Exercises
GUIDED
PRACTICE
3.
for Example 2
Find ST in the triangle at the right.
SOLUTION
STU is equilateral, then it is
equiangular
ANSWER
Thus ST = 5
( Base angle theorem )
Warm-Up
Exercises
GUIDED
PRACTICE
4.
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.
Warm-Up3Exercises
EXAMPLE
Use isosceles and equilateral triangles
ALGEBRA
Find the values of x and y
in the diagram.
SOLUTION
STEP 1
STEP 2
Find the value of y. Because
KLN is
equiangular, it is also equilateral and KN
Therefore, y = 4.
KL .
Find the value of x. Because LNM
LMN,
LN
LM and
LMN is isosceles. You also
know that LN = 4 because
KLN is equilateral.
Definition of congruent segments
LN = LM
4=x+1
Substitute 4 for LN and x + 1 for LM.
3=x
Subtract 1 from each side.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
Lifeguard Tower
In the lifeguard tower, PS
and
QPS
PQR.
QR
a.
What congruence postulate
can you use to prove that
QPS
PQR?
b.
Explain why
isosceles.
c.
Show that
PQT is
PTS
QTR.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
SOLUTION
a.
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.
b.
From part (a), you know that 1
2 because
corresponding parts of congruent triangles are
congruent. By the Converse of the Base Angles
PQT is isosceles.
Theorem, PT QT , and
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
c.
You know that PS
QR , and 3
4 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.
Warm-Up
Exercises
GUIDED
PRACTICE
5.
for Examples 3 and 4
Find the values of x and y in the diagram.
ANSWER
y° = 120°
x° = 60°