# Geom Hspecialtri

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```					 5-8 Applying Special Right Triangles
5-8 Applying Special Right Triangles

Warm Up
Lesson Presentation
Lesson Quiz

Holt Geometry
Holt Geometry
5-8 Applying Special Right Triangles

Warm Up
For Exercises 1 and 2, find the value of x.

1.                   2.

Simplify each expression.

3.                   4.

Holt Geometry
5-8 Applying Special Right Triangles

Objectives
Justify and apply properties of
45°-45°-90° triangles.
Justify and apply properties of
30°- 60°- 90° triangles.

Holt Geometry
5-8 Applying Special Right Triangles
A diagonal of a square divides it into two congruent
isosceles right triangles. Since the base angles of an
isosceles triangle are congruent, the measure of
each acute angle is 45°. So another name for an
isosceles right triangle is a 45°-45°-90° triangle.
45

45

A 45°-45°-90° triangle is one type of special right
triangle. You can use the Pythagorean Theorem to
find a relationship among the side lengths of a 45°-
45°-90° triangle.

Holt Geometry
5-8 Applying Special Right Triangles
The Isosceles Right
50

Why?            a2 + b2 = c2
l2 + l2 = h 2
2l2 = h2
2l = h

Holt Geometry
5-8 Applying Special Right Triangles

Corollary to Theorem 50:
Each diagonal of a square is 2 times the length of
one side.

45

45

Holt Geometry
5-8 Applying Special Right Triangles
Example 1A: Finding Side Lengths in a 45°- 45º- 90º
Triangle

Find the value of x. Give your

By the Triangle Sum Theorem, the
measure of the third angle in the
triangle is 45°. So it is a 45°-45°-
90° triangle with a leg length of 8.

Holt Geometry
5-8 Applying Special Right Triangles
Example 1B: Finding Side Lengths in a 45º- 45º- 90º
Triangle

Find the value of x. Give your
The triangle is an isosceles right
triangle, which is a 45°-45°-90°
triangle. The length of the hypotenuse
is 5.

Rationalize the denominator.

Holt Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 1a

By the Triangle Sum Theorem, the
measure of each of the two
congruent angles in the triangle is
45°. So it is a 45°-45°-90° triangle
with a leg length of

x = 20          Simplify.

Holt Geometry
5-8 Applying Special Right Triangles

A 30°-60°-90° triangle is another special right
triangle. You can use an equilateral triangle to find
a relationship between its side lengths.

30

60             60

Holt Geometry
5-8 Applying Special Right Triangles
Right

51     ^

This can also be proven algebraically, but
let’s move on to some problems.

Theorem 51. This corollary involves
equilateral triangles and a familiar formula.

Holt Geometry
5-8 Applying Special Right Triangles
Example 3A: Finding Side Lengths in a 30º-60º-90º
Triangle

Find the values of x and y. Give

22 = 2x     Hypotenuse = 2(shorter leg)
11 = x      Divide both sides by 2.

Substitute 11 for x.

Holt Geometry
5-8 Applying Special Right Triangles
Example 3B: Finding Side Lengths in a 30º-60º-90º
Triangle

Find the values of x and y. Give your

Rationalize the denominator.

y = 2x     Hypotenuse = 2(shorter leg).

Simplify.
Holt Geometry
5-8 Applying Special Right Triangles
Example 3C: Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y.

24 = 2x     Hypotenuse = 2(shorter leg)
12 = x      Divide both sides by 2.

Substitute 12 for x.

Holt Geometry

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