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# Geometry Study Guide for McGraw-Hill's ASVAB (page 2)

By McGraw-Hill Professional
Updated on Jun 26, 2011

### Triangles

A polygon is a closed figure that can be drawn without lifting the pencil. It is made up of line segments (sides) that do not cross. A triangle is a polygon with three sides. Every triangle has three angles that total 180°.

Identifying the Longest Side of a Triangle The longest side of a triangle is always opposite the largest angle. So, if a triangle has angles of 45°, 55°, and 80°, the side opposite the 80° angle would be the longest.

Types of Triangles There are four main types of triangles. They are equilateral, isosceles, scalene, and right. Each has special characteristics that you should know.

Equilateral Triangle This kind of triangle has three congruent (equal) sides and three congruent (equal) angles. In an equilateral triangle, each angle measures 60°.

Isosceles Triangle This type of triangle has at least two congruent sides, and the angles opposite the congruent sides are also congruent. In the isosceles triangle shown below, sides AB and BC are congruent. BAC and BCA are also congruent. In an isosceles triangle, if you know the measure of any one angle, you can calculate the measures of the other two.

Examples

In this isosceles triangle, if 1 measures 30°, what is the measure of 3?

Since 1 and 2 are congruent, 2 must also measure 30°. Together, 1 and 2 add up to 60°. Since the sum of all three angles in any triangle is 180°, 3 must be 180 – 60 = 120°.

If 3 measures 100°, what are the measures of 1 and 2?

Since the sum of all three angles in any triangle is 180°, the sum of the measures of 1 and 2 must be 180 – 100 = 80°. Since angles 1 and 2 are congruent, each one must measure 80° ÷ 2 = 40°.

Scalene Triangle This kind of triangle has no equal sides or angles.

Right Triangle This kind of triangle has one angle that measures 90°. This angle is the right angle. It is identified in the figure by the little "box." Since the sum of all three angles in any triangle is 180°, the sum of the two remaining angles in a right triangle is 180 – 90 = 90°.

In a right triangle, there is a special relationship among the lengths of the three sides. This relationship is described by the Pythagorean Theorem.

In the right triangle below, C is the right angle. The side opposite the right angle is called the hypotenuse (c). It is always the longest side. The other two sides (a and b) are called legs.

According to the Pythagorean Theorem, in any right triangle, the sum of the squares of the legs equals the square of the hypotenuse. In symbols:

a2 + b2 = c2

So if you know the lengths of any two sides of a right triangle, you can calculate the length of the third side.

Examples

Base and Height of a Triangle Any side of a triangle can be called the base. The height is the length of a line segment that connects a base to the vertex opposite that base and is perpendicular to it.

Look at the triangle below. Dashed line CD is the height. Line CD is perpendicular to the base AB.

Where line CD meets base AB, it creates two right angles, CDA and CDB.

Median of a Triangle A median of a triangle is a line drawn from any vertex to the middle of the opposite side. This line splits the opposite side into two equal lengths.

Examples

Dashed line AD is a median of triangle ABC. It splits side BC into two equal lengths, .

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