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4.7 Use Isosceles and Equilateral Triangles
Warm Up
Lesson Presentation
Lesson Quiz
4.7
Warm-Up
Classify each triangle by its sides.
1. 2 cm, 2 cm, 2 cm
ANSWER
equilateral
2. 7 ft, 11 ft, 7 ft
ANSWER
isosceles
3. 9 m, 8 m, 10 m
ANSWER
scalene
4.7
Warm-Up
4. In ∆ABC, if m
ANSWER
A = 70º and m
B = 50º, what is m
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
4.7
In
Example 1
DEF, DE
DF . Name two congruent angles.
SOLUTION
DE
DF , so by the Base Angles Theorem,
E
F.
4.7
Guided Practice
Copy and complete each statement.
1.
If HG
ANSWER
2.
If
KHJ
ANSWER
HK , then
?
? .
HGK, HKG
KJH, then ?
KH, KJ
? .
4.7
Example 2
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
ANSWER
Divide each side by 3.
The measures of P, Q, and R are all 60°.
4.7
3.
Guided Practice
Find ST in the triangle at the right.
ANSWER
4.
5
Is it possible for an equilateral triangle to have an
angle measure other than 60°? Explain.
ANSWER
No; The Triangle Sum Theorem and the fact that the
triangle is equilateral guarantees the angles
measure 60° because all pairs of angles could be
considered base angles of an isosceles triangle.
4.7
Example 3
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
Therefore, y = 4.
KL .
4.7
Example 3
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.
4.7
Example 4
Lifeguard Tower
In the lifeguard tower, PS
and
QPS
PQR.
a.
QR
What congruence postulate
can you use to prove that
QPS
PQR?
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 
Postulate, QPS  PQR.
4.7
Example 4
Lifeguard Tower
In the lifeguard tower, PS
and
QPS
PQR.
b.
Explain why
QR
PQT is isosceles.
SOLUTION
b.
From part (a), you know that 1
2 because
corresp. parts of
are . By the Converse
of the Base Angles Theorem, PT QT , and
PQT is isosceles.
4.7
Example 4
Lifeguard Tower
In the lifeguard tower, PS
and
QPS
PQR.
c.
Show that
PTS
QR
QTR.
SOLUTION
c.
You know that PS
QR , and 3
4 because
corresp. parts of
are . Also, PTS
QTR
by the Vertical Angles Congruence Theorem. So,
PTS
QTR by the AAS Congruence Theorem.
4.7
5.
Guided Practice
Find the values of x and y in the diagram.
ANSWER
x = 60
y = 120
4.7
6.
Guided Practice
Use parts (b) and (c) in Example 4 and
the SSS Congruence Postulate to give a
different proof that PTS
QTR
ANSWER
By the Segment Addition Postulate QT + TS = QS and
PT + TR = PR. Since PT  QT from part (b) and TS  TR
from part (c), then QS  PR. PQ  PQ by the Reflexive
Property and it is given that PS  QR, therefore
QPS  PQR by the SSS Congruence Postulate.
4.7
Lesson Quiz
Find the value of x.
1.
ANSWER
8
4.7
Lesson Quiz
Find the value of x.
2.
ANSWER
3
4.7
Lesson Quiz
3. If the measure of vertex angle of an isosceles
triangle is 112°, what are the measures of the
base angles?
ANSWER
34°, 34°
4.7
Lesson Quiz
4. Find the perimeter of triangle.
ANSWER
66 cm