Holt Geometry

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Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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• Warm Up
• 1. Find each angle measure.
• True or False. If false explain.
• 2. Every equilateral triangle is isosceles.
• 3. Every isosceles triangle is equilateral.

60 60 60
True
False an isosceles triangle can have only two
congruent sides.
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Objectives
Prove theorems about isosceles and equilateral
triangles. Apply properties of isosceles and
equilateral triangles.
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Vocabulary
legs of an isosceles triangle vertex
angle base base angles
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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.
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Example 1 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 given values.
x x 22 180
Simplify and subtract 22 from both sides.
2x 158
Divide both sides by 2.
x 79?
Thus m?F 79
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Example 2 Finding the Measure of an Angle
Find m?G.
m?J m?G
Isosc. ? Thm.
Substitute the given values.
(x 44)? 3x?
Simplify x from both sides.
44 2x
Divide both sides by 2.
x 22?
Thus m?G 22 44 66.
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TEACH! Example 1
Find m?H.
m?H m?G x
Isosc. ? Thm.
m?H m?G m?F 180
? Sum Thm.
Substitute the given values.
x x 48 180
Simplify and subtract 48 from both sides.
2x 132
Divide both sides by 2.
x 66?
Thus m?H 66
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TEACH! Example 2
Find m?N.
m?P m?N
Isosc. ? Thm.
Substitute the given values.
(8y 16)? 6y?
Subtract 6y and add 16 to both sides.
2y 16
Divide both sides by 2.
y 8?
Thus m?N 6(8) 48.
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The following corollary and its converse show the
connection between equilateral triangles and
equiangular triangles.
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Ex. 3A Using Properties of Equilateral Triangles
Find the value of x.
?LKM is equilateral.
Equilateral ? ? equiangular ?
The measure of each ? of an equiangular ? is 60.
(2x 32)? 60?
Subtract 32 both sides.
2x 28
Divide both sides by 2.
x 14
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Ex. 3B Using Properties of Equilateral Triangles
Find the value of y.
?NPO is equiangular.
Equiangular ? ? equilateral ?
Definition of equilateral ?.
5y 6 4y 12
Subtract 4y and add 6 to both sides.
y 18
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TEACH! Example 3
Find the value of JL.
?JKL is equiangular.
Equiangular ? ? equilateral ?
Definition of equilateral ?.
4t 8 2t 1
Subtract 4y and add 6 to both sides.
2t 9
t 4.5
Divide both sides by 2.
Thus JL 2(4.5) 1 10.
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Example 4 Using Coordinate Proof
Prove that the segment joining the midpoints of
two sides of an isosceles triangle is half the
base.
Given In isosceles ?ABC, X is the mdpt. of AB,
and Y is the mdpt. of BC.
Prove XY AC.
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Example 4 Continued
Proof Draw a diagram and place the coordinates
as shown.
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TEACH! Example 4
The coordinates of isosceles ?ABC are A(0, 2b),
B(-2a, 0), and C(2a, 0). X is the midpoint of AB,
Y is the midpoint of AC, and Z(0, 0), . Prove
?XYZ is isosceles.
Proof Draw a diagram and place the coordinates
as shown.
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Check It Out! Example 4 Continued
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Lesson Quiz Part I
Find each angle measure. 1. m?R 2. m?P Find
each value. 3. x 4. y 5. x
28
124
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20
26
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Lesson Quiz Part II
6. The vertex angle of an isosceles triangle
measures (a 15), and one of the base angles
measures 7a. Find a and each angle measure.
a 11 26 77 77