Transcript lg_ch04_08 Use Isosceles and Equilateral Triangles _teacher

4.8 Use Isosceles and Equilateral
Triangles
• You will use theorems
about isosceles and
equilateral triangles.
• Essential Question:
How are the sides and
angles of a triangle
related if there are two
or more congruent sides
or angles?
You will learn how to answer this
question by learning the Base
Angles Theorem and its converse.
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
HKG
?
? .
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
Copy and complete the statement.
2.
If
KHJ
KJH, then ?
? .
SOLUTION
If
KHJ
KJH, then , KH
KJ
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.
Can an equilateral triangle have an
angle of 61?
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 its 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.
Explain how you could find
m ∠ M.
Warm-Up3Exercises
EXAMPLE
Use isosceles and equilateral triangles
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.
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
c.
Show that
PQT is isosceles.
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
corresp. parts of
are . By the Converse
of the Base Angles Theorem, PT QT , and
PQT is isosceles.
Warm-Up4Exercises
EXAMPLE
Solve a multi-step problem
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.
Warm-Up
Exercises
GUIDED
PRACTICE
5.
for Examples 3 and 4
Find the values of x and y in the diagram.
SOLUTION
y° = 120°
x° = 60°
Warm-Up
Exercises
GUIDED
PRACTICE
6.
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
Warm-Up
Exercises
GUIDED
PRACTICE
Since PT
for Examples 3 and 4
QT from part (b) and
from part (c) then,
TS
TR
QS
PR
PQ
PQ
Reflexive Property and
PS
QR
Given
ANSWER
Therefore
Postulate
QPS
PQR . By SSS Congruence
Daily
Homework
Quiz
Warm-Up
Exercises
Find the value of x.
1.
ANSWER
8
Daily
Homework
Quiz
Warm-Up
Exercises
Find the value of x.
2.
ANSWER
3
Daily
Homework
Quiz
Warm-Up
Exercises
3. If the measure of vertex angle of an isosceles
triangle is 112°, what are the measures of the
base angles?
ANSWER
34°, 34°
Daily
Homework
Quiz
Warm-Up
Exercises
4. Find the perimeter of triangle.
ANSWER
66 cm
• Essential Question:
How are the sides and
angles of a triangle
related if there are two
• Angles opposite congruent sides
or more congruent sides
of a triangle are congruent and
or angles?
• You will use theorems
about isosceles and
equilateral triangles.
conversely.
• If a triangle is equilateral, then
it is equiangular and conversely.
If two sides of a triangle are
congruent, then the angles
opposite them are congruent. The
converse is also true.