geo_fl_ch04_07
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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