Question

1. classify each pair of angles as vertical, complementary, supplementary, and/or a linear

pair.

1. 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:

a. Analyze the statement about supplementary angles and obtuse angles

• Supplementary angles sum to \(180^{\circ}\). An obtuse angle is greater than \(90^{\circ}\) but less than \(180^{\circ}\).
• If \(x = 30^{\circ}\) and \(y=150^{\circ}\), \(x + y=180^{\circ}\) (supplementary), but \(x\) is acute. If \(x = 90^{\circ}\) and \(y = 90^{\circ}\), \(x + y=180^{\circ}\) (supplementary), but \(x\) is right - angled. So \(x\) is not always obtuse when \(x\) and \(y\) are supplementary.

b. Analyze the statement about right angles and supplementary angles

• A right angle is \(90^{\circ}\). If \(x = 90^{\circ}\) and \(y = 90^{\circ}\), then \(x + y=90^{\circ}+90^{\circ}=180^{\circ}\). By the definition of supplementary angles (two angles whose sum is \(180^{\circ}\)), if \(x\) and \(y\) are right angles, they are supplementary.

c. Analyze the statement about complementary angles and right angles

• Complementary angles sum to \(90^{\circ}\). A right angle is \(90^{\circ}\). If \(x + y=90^{\circ}\), and \(y = 90^{\circ}\), then \(x=0^{\circ}\), but angles in geometry are greater than \(0^{\circ}\). For example, if \(x = 30^{\circ}\) and \(y = 60^{\circ}\), \(x + y=90^{\circ}\) (complementary) and \(y\) is acute.

Answer:

a. Sometimes
b. Always
c. Never