## SAT SUBJECT TEST MATH LEVEL 1

## PLANE GEOMETRY

## CHAPTER 9 Triangles

• Sides and Angles of a Triangle

• Right Triangles

• Pythagorean Theorem

• Special Right Triangles

• Perimeter and Area

• Similar Triangles

• Exercises

• Answers Explained

**S**ince 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.

### SIDES AND ANGLES OF A TRIANGLE

**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

**, and by KEY FACT H3, it is the longest side. The other two sides are called the**

*hypotenuse***.**

*legs*