CHAPTER 9 Triangles

• Sides and Angles of a Triangle

• Right Triangles

• Pythagorean Theorem

• Special Right Triangles

• Perimeter and Area

• Similar Triangles

• Exercises

• Answers Explained

Since more geometry questions on the Math 1 test concern triangles than any other topic, it is imperative that you learn all the KEY FACTS in this section.


Key Fact H1

In any triangle, the sum of the measures of the three angles is 180º.

x + y + z = 180

EXAMPLE 1: To find the value of x in the figure below, use KEY FACT H1 twice. 130 + 30 + z = 180  z = 20  y = 20, then 90 + x + 20 = 180  x = 70.

In the figure below, , which is formed by extending side  of , is called an exterior angle.

Key Fact H2

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

In the diagram above, w = x + y.

Key Fact H3

In any triangle:

• The longest side is opposite the largest angle.

• The shortest side is opposite the smallest angle.

• Sides with equal lengths are opposite angles with equal measures (the angles opposite congruent sides are congruent).

EXAMPLE 2: By using KEY FACT H3, we can draw the following conclusions concerning the angles in the figure below. In , since angles D and E are opposite congruent sides ( and , respectively), m∠D = m∠E, angle C is the largest angle, so  is the longest side. Specifically, AB > 6. Also, since BC > AC, m∠A > m∠B.

• A triangle is called scalene if the three sides all have different lengths. Then by KEY FACT H3, the three angles all have different measures.

• A triangle is called isosceles if two sides are congruent. By KEY FACT H3, the two angles opposite the congruent sides are congruent.

• A triangle is called equilateral if all three sides are congruent. By KEY FACT H3, all three angles are congruent. Since the sum of the measures of the three angles is 180°, each angle measures 60°.

• Acute triangles are triangles in which all three angles are acute. An acute triangle could be scalene, isosceles, or equilateral.

• Obtuse triangles are triangles in which one angle is obtuse and two are acute. An obtuse triangle could be scalene or isosceles.

• Right triangles are triangles that have one right angle and two acute angles. A right triangle could be scalene or isosceles. The side opposite the 90° angle is called the hypotenuse, and by KEY FACT H3, it is the longest side. The other two sides are called the legs.