Basic Mathematics for College Students

818 Chapter 9 An Introduction to Geometry

The sum of the measures of the angles of any

triangle is 180°.

We can use algebra to find unknown angle

measures of a triangle.

Find the measure of each angle of.

The sum of the angle measures of any

triangle is 180°:

ABC

In an isosceles triangle, the angles opposite the sides

of equal length are called base angles.The third

angle is called the vertex angle.The third side is

called the base.

Isosceles Triangle Theorem: If two sides of a

triangle are congruent, then the angles opposite

those sides are congruent.

Converse of the Isosceles Triangle Theorem:If two

angles of a triangle are congruent, then the sides

opposite the angles have the same length, and the

triangle is isosceles.

The longest side of a right triangle is called the

hypotenuse,and the other two sides are called legs.

The hypotenuse of a right triangle is always opposite

the 90° (right) angle. The legs of a right triangle are

adjacent to (next to) the right angle.

Triangles can be classified by their angles.

C

B

A

x

3 x – 25°

x – 5°

Right triangle

Leg

Leg

Hypotenuse

(longest side)

Isosceles triangles

Base angle Base angle

Base

Vertex angle

Acute triangle

(has three acute angles)

Obtuse triangle

(has an obtuse angle)

Right triangle

(has one right angle)

We can use algebra to find unknown angle measures

of an isosceles triangle.

If the vertex angle of an isosceles triangle measures 26°, what is the

measure of each base angle?

If we let xrepresent the measure of

one base angle, the measure of the

other base angle is also x.(See the figure.) Since the sum of the

measures of the angles of any triangle is 180°, we have

On the left side, combine like terms.

To isolate 2x,subtract 26° from both sides.

To isolate x,divide both sides by 2.

The measure of each base angle is 77°.

x77°

2 x154°

2 x26°180°

xx26°180°

Combine like terms.

Add 30° to both sides.

Divide both sides by 5.

To find the measures of and , we evaluate the expressions

and for.

Thus,m(A)42°,m(B)101°, and .m(C)37°

101°

126°25° 37°

3 x25°3( 42 °)25° x 5° 42 °5°

3 x25° x5° x42°

B C

x42°

5 x210°

5 x30°180°

x 3 x 25 °x 5 °180°

x x

26°