1.1 Angle Measure and Special Triangles

Objective 1

•Use the vocabulary associated with a study of angles and triangles.

Angle

• An angle is the joining of two rays at a common endpoint called the vertex.

Angle Measurement

Complement and Supplement Angles

Exercises

• What is the complement of a 57° angle?

Since complementary angles add up to 90°, the complement of 57° will be 90° − 57° = 33°.

• What is the supplement of a 132° angle?

Since complementary angles add up to 180°, the supplement of 132° will be 180° − 132° = 48°.

Objective 2

•Find and recognize coterminal angles.

Initial and Terminal Sides

Positive and Negative Angles

• Angles formed by a counterclockwise rotation from the initial side are considered positive angles, and angles formed by a clockwise rotation are negative angles.

Examples

• Positive Angle

• Negative Angle

Coterminal Angles

• Any angle (positive or negative) share the same initial and terminal sides are called coterminal angles. Coterminal angles will always differ by multiples of 360°, meaning that for any interger k, angles ϴ and ϴ + 360°k will be coterminal.

Exercises

• Find two positive angles and two negative angles that are coterminal with 60°.

For 𝑘 = −2, 60° + 360° −2 = −660°.

For 𝑘 = −1, 60° + 360° −1 = −300°.

For 𝑘 = 1, 60° + 360° 1 = 420°.

For 𝑘 = 2, 60° + 360° 2 = 780°.

Note that many other answers are possible.

Objective 3

•Find fixed ratios of the sides of special Triangles.

45-45-90 Triangle

• 45-45-90 triangle is an isosceles right triangle with the longest side (opposite the 90° angle) called the hypotenuse and two equal legs and two 45° angles.

• The length of the legs are equal and the hypotenuse is 2 times the length of either leg.

30-60-90 Triangle

• 30-60-90 triangle is a right triangle with hypotenuse opposite from the 90° angle, longer leg opposite from the 60° angle, and shorter leg opposite from the 30° angle.

• The length of the hypotenuse is double of the shorter leg.

• The length of the longer leg is 3 times the shorter leg.

Exercises

• Find the unknown side lengths of the following 30-60-90 triangles if the hypotenuse measures 12 miles.

Since the hypotenuse is double the length of the shorter side and the hypotenuse is 10m, the shorter leg is 12 ÷ 2 = 6 miles.

If the longer leg is 3 times the shorter leg, then the longer leg

is 3 × 6 = 6 3 miles.

Exercises

• A masonry cut 18 2 in. by 18 2 in. tiles diagonally so the longest side of the tile will line up with the wall. How many of those tiles does he need to cut if the wall is 432 inches long?

Since the tile is square in shape, if you cut the tile diagonally,

you will have created two 45-45-90 triangle with sides 18 2 in. long. The measurement of the diagonal is found by multiplying

2 to the side length which is 2 x 18 2 = 36 in.

The wall is 432 in. long and diagonal

side of the tile is to face the wall, then

you will need 432 ÷ 36 = 12 pieces.