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Objectives
Apply properties of isosceles and equilateral
triangles.
• legs of an isosceles triangle
• vertex angle
• base angles
Recall that an isosceles triangle has at least two
congruent sides. The congruent sides are called the
legs. The vertex angle is the angle formed by the
legs. The side opposite the vertex angle is called the
base, and the base angles are the two angles that
have the base as a side.
3 is the vertex angle.
1 and 2 are the base angles.
I DO: Finding the Measure of an Angle
Find mG.
mJ = mG Isosc. ∆ Thm.
(x + 44) = 3x
44 = 2x
Substitute the
given values.
Simplify x from
both sides.
Divide both
sides by 2.
Thus mG = 22° + 44° = 66°.
x = 22
I DO: Finding the Measure of an Angle
Find mF.
mF = mD = x°
Isosc. ∆ Thm.
mF + mD + mA = 180 ∆ Sum Thm.
Substitute the
x + x + 22 = 180 given values.
Simplify and subtract
2x = 158 22 from both sides.
x = 79 Divide both
sides by 2.
Thus mF = 79°
You Do! Example 2A
Find mH.
mH = mG = x°
Isosc. ∆ Thm.
mH + mG + mF = 180 ∆ Sum Thm.
Substitute the
x + x + 48 = 180 given values.
Simplify and subtract
2x = 132 48 from both sides.
x = 66 Divide both
sides by 2.
Thus mH = 66°
Find mN.
You Do! Example 2B
mP = mN Isosc. ∆ Thm.
(8y – 16) = 6y
2y = 16
y = 8
Substitute the
given values.
Subtract 6y and
add 16 to both
sides.
Divide both
sides by 2.
Thus mN = 6(8) = 48°.
I Do: Using Properties of Equilateral Triangles
Find the value of x.
∆LKM is equilateral.
Equilateral ∆  equiangular ∆
(2x + 32) = 60
2x = 28
x = 14
The measure of each  of an
equiangular ∆ is 60°.
Subtract 32 both sides.
Divide both sides by 2.
I DO: Using Properties of Equilateral Triangles
Find the value of y.
∆NPO is equiangular.
Equiangular ∆  equilateral ∆
5y – 6 = 4y + 12
y = 18
Definition of
equilateral ∆.
Subtract 4y and add 6 to
both sides.
You Do! Example 3
Find the value of JL.
∆JKL is equiangular.
Equiangular ∆  equilateral ∆
4t – 8 = 2t + 1
2t = 9
t = 4.5
Definition of
equilateral ∆.
Subtract 4y and add 6 to
both sides.
Divide both sides by 2.
Thus JL = 2(4.5) + 1 = 10.
Find Missing Measures
A. Find mR.
Since QP = QR, QP  QR. By the
Isosceles Triangle Theorem, base
angles P and R are congruent, so
mP = mR . Use the Triangle Sum
Theorem to write and solve an equation
to find mR.
Triangle Sum Theorem
mQ = 60, mP = mR
Answer:
mR = 60
Simplify.
Subtract 60 from each side.
Divide each side by 2.
A. Find mT.
A. 30°
B. 45°
C. 60°
D. 65°
B. Find TS.
A. 1.5
B. 3.5
C. 4
D. 7
Find the value of each variable.
A. x = 20, y = 8
B. x = 20, y = 7
C. x = 30, y = 8
D. x = 30, y = 7
Exit Slip (Pick one)
Find each angle measure.
1. mR 28° and mP 124°
Find each value.
3. x
5. x
20
4. y
26°
6