Question

part 6: classifying angles & relationships

1. classify each angle as acute, right, obtuse, or straight:

32° is
90° is
147° is
180° is

1. classify each pair of angles as vertical, complementary, supplementary, and/or a linear pair.
2. state whether each statement below is always, sometimes, or never true.

a. if x and y are supplementary angles, then x is obtuse.
b. if x and y are right angles, then x and y are supplementary angles.
c. if x and y are complementary angles, then y is a right angle.

Explanation:

11. Classify each angle

Step1: Classify \(32^{\circ}\)

An acute angle is an angle whose measure is between \(0^{\circ}\) and \(90^{\circ}\). Since \(0^{\circ}<32^{\circ}<90^{\circ}\), \(32^{\circ}\) is acute.

Step2: Classify \(90^{\circ}\)

A right - angle is an angle whose measure is exactly \(90^{\circ}\). So \(90^{\circ}\) is right.

Step3: Classify \(147^{\circ}\)

An obtuse angle is an angle whose measure is between \(90^{\circ}\) and \(180^{\circ}\). Since \(90^{\circ}<147^{\circ}<180^{\circ}\), \(147^{\circ}\) is obtuse.

Step4: Classify \(180^{\circ}\)

A straight angle is an angle whose measure is exactly \(180^{\circ}\). So \(180^{\circ}\) is straight.

13. State whether each statement is always, sometimes, or never true

a. If \(x\) and \(y\) are supplementary angles (\(x + y=180^{\circ}\))

Supplementary angles: If \(x = 30^{\circ}\), then \(y = 150^{\circ}\) ( \(x\) is acute and \(y\) is obtuse); if \(x=90^{\circ}\), then \(y = 90^{\circ}\) (both are right - angles). So the statement “If \(x\) and \(y\) are supplementary angles, then \(x\) is obtuse” is sometimes true.

b. If \(x\) and \(y\) are right angles (\(x = y=90^{\circ}\))

Since \(x + y=90^{\circ}+90^{\circ}=180^{\circ}\), and supplementary angles are two angles whose sum is \(180^{\circ}\). So the statement “If \(x\) and \(y\) are right angles, then \(x\) and \(y\) are supplementary angles” is always true.

c. If \(x\) and \(y\) are complementary angles (\(x + y=90^{\circ}\))

Complementary angles: If \(x = 30^{\circ}\), then \(y=60^{\circ}\) (both are acute); if \(x = 0^{\circ}\), then \(y = 90^{\circ}\). But in the general case of complementary angles \(x+y = 90^{\circ}\), \(y\) is not necessarily \(90^{\circ}\). So the statement “If \(x\) and \(y\) are complementary angles, then \(y\) is a right angle” is never true.

Answer:

1. \(32^{\circ}\) is acute; \(90^{\circ}\) is right; \(147^{\circ}\) is obtuse; \(180^{\circ}\) is straight.
2. a. sometimes; b. always; c. never.