Download Geometry 5.4 Isosceles, Equilateral and Right Triangles

Geometry
5.4 Isosceles, Equilateral and
Right Triangles
Essential Question
What conjectures can you make about the
side lengths and angle measures of an
isosceles triangle?
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Goals
Use properties of isosceles triangles.
 Use properties of equilateral triangles.
 Use properties of right triangles.

January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Opposite Angles and Sides
E
EF is opposite D.
E is opposite side DF.
D
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F
5.4 Isosceles, Equilateral, and Right Triangles
Isosceles Triangles
Vertex Angle
Leg
Leg
Base
Angles
Base
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5.4 Isosceles, Equilateral, and Right Triangles
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
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Theorem 5.6
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.
This is known as the Base Angles Theorem.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Visually:
This:
Means this:
The base angles of an isosceles triangle are congruent.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 1
Solve for x.
x + x + 52 = 180
52°
2x + 52 = 180
2x = 128
x°
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x°
x = 64
5.4 Isosceles, Equilateral, and Right Triangles
Example 2
42°
x°
y°
Solve for x and y.
In an isosceles
triangle, base angles
are congruent.
So y is…
42°
Now use the triangle
angle sum theorem:
x + 42 + 42 = 180
x + 84 = 180
x = 96°
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 3
Your Turn.
Find x and y.
x = 65°
50°
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
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 4
Solve for x.
(2x)°
3x – 25 = 2x
x = 25
(3x – 25)°
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Theorem 5.7
Converse of the Base Angles Theorem.
If two angles of a triangle are congruent,
then the sides opposite them are congruent.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 5
4x + 52
4(8) + 52 = 84
2x + 68
Solve for x, then
find the length of
the legs.
Since base angles are
equal,
opposite sides are
equal.
4x + 52 = 2x + 68
2x + 52 = 68
2x = 16
x=8
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 6 Your Turn
Find the length of each side.
40
4x – 2
5x
30
5x = 3x + 16
2x = 16
x=8
3x + 16
40
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Equilateral Triangles
A special case of an isosceles triangle.
 If a triangle is equilateral, then it is also
equiangular.
 If a triangle is equiangular, then it is also
equilateral.

January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
All sides are congruent.
Example 7
3x – 10 = x + 10
Solve for x.
2x = 20
3x – 10
x + 10
x = 10
2x = x + 10
x = 10
2x
3x – 10 = 2x
x – 10 = 0
x = 10
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Prove triangles are congruent:
SSS
 SAS
 ASA
 AAS
 SSA does not work.

January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Special Case of SSA
Even though this is SSA, these triangles are congruent.
HL
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Theorem 5.9 HL (Rt.
(or Hyp-Leg)
)
If the hypotenuse and a leg of a right triangle
are congruent to corresponding parts of
another right triangle, then the triangles are
congruent.

January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Corollary LL (or Leg-Leg)
If the two legs of a right triangle are
congruent to corresponding parts of another
right triangle, then the triangles are
congruent. (This is just SAS.)

January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Example 8
Is ABD  CBD?
B
50°
10 cm
90°
?
A
10 cm
40°
D
C
1. ABD and CBD are both rt. s.
2. AB = CB (they’re both 10 cm)
3. The triangles share BD.
4. The hypotenuse and leg are
congruent  ABD  CBD by HL.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
One last problem. Solve for x and y.
y°
x°
50°
Solution…
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
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?
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Summary

The base angles of an isosceles triangle
are congruent.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Summary

Equilateral triangles are Equiangular.
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5.4 Isosceles, Equilateral, and Right Triangles
Summary

Two right triangles can be proved
congruent using HL or LL.
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles
Assignment
January 7, 2016
5.4 Isosceles, Equilateral, and Right Triangles