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5.4 Isosceles and Equilateral Triangles
Geometry
What conjectures can you make about congruent
angles and sides?
Topic/Objective
Use properties of isosceles triangles.
 Use properties of equilateral triangles.

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Opposite Angles and Sides
E
EF is opposite D.
E is opposite side DF.
D
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F
Geometry 5.4 Isosceles, Equilateral Triangles
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Isosceles Triangles
Vertex Angle
Leg
Leg
Base
Angles
Base
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A construction.
Begin with an isosceles
triangle, ABC.
C
Draw the angle bisector
from the vertex angle.
The angle bisector
intersects the base at M.
ACM  BCM. Why?
SAS
A
M
B
A  B. Why?
CPCTC
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Geometry 5.4 Isosceles, Equilateral Triangles
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Theorem 5.6 Base Angles Theorem.
If two sides of a triangle are congruent,
then the angles opposite them are
congruent.
 (Easy form) The base angles of an
isosceles triangle are congruent.

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Visually:
This:
Means this:
The base angles of an isosceles triangle are congruent.
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Example Problem
Solve for x.
x + x + 52 = 180
52°
2x + 52 = 180
2x = 128
x°
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x°
x = 64
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Example 2
Solve for x and y.
42°
In an isosceles
triangle, base angles
are congruent.
y°
So y is…
x°
42°
Now use the triangle
angle sum theorem:
x + 42 + 42 = 180
x + 84 = 180
x = 96°
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Example 3.
You try it.
x = 65°
50°
Find x and y.
y°
y = 32.5°
x°
2y + 115 = 180
y° 50°
32.5°`
2y = 65
y = 32.5°
y°
115° 65°
x°
65°
x°
2x + 50 = 180
180 – 65 = 115
2x = 130
x = 65
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Example 4
Solve for x.
(2x)°
3x – 25 = 2x
x = 25
(3x – 25)°
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Theorem 5.7
Converse of the Base Angles Theorem.
 If two angles of a triangle are congruent,
then the sides opposite them are
congruent.

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Example 5
4x + 52
4(8) + 52 = 84
Solve for x,
then find the
length of the
legs.
2x + 68
Since base angles are
equal,
opposite sides are
equal.
4x + 52 = 2x + 68
2x + 52 = 68
2x = 16
x=8
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Example 6
Find the length of each side.
You do it.
40
4x – 2
5x
30
5x = 3x + 16
2x = 16
x=8
3x + 16
40
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Equilateral Triangles
Corollaries to Base Angles Theorem
If a triangle is equilateral, then it is also
equiangular.
 If a triangle is equiangular, then it is also
equilateral.

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Example 7
All sides are congruent.
Solve for x.
3x – 10 = x + 10
2x = 20
3x – 10
x + 10
x = 10
2x = x + 10
x = 10
2x
3x – 10 = 2x
x – 10 = 0
x = 10
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One last problem. Solve for x and y.
y°
x°
50°
Solution…
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Solution…
40°
x°
80°
y°
50°
?
50°
70°
70°
60°
60°
This triangle is
equilateral.
Each angle is?
60°
These angles form
straight angle. The
missing angle is?
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Summarize what you have learned today

The base angles of an isosceles
triangle are congruent.

Equilateral triangles
are Equiangular.
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