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Chapter 13: Problem 1

Which of the following could be the degree measures of two angles in a right triangle? Indicate all such angles. a. \(20^{\circ}\) and \(70^{\circ}\) b. \(30^{\circ}\) and \(60^{\circ}\) c. \(45^{\circ}\) and \(45^{\circ}\) d. \(55^{\circ}\) and \(55^{\circ}\) e. \(75^{\circ}\) and \(75^{\circ}\)

Short Answer

Expert verified
The pairs of angle measures that could belong to a right triangle are: a. \(20^{\circ}\) and \(70^{\circ}\), b. \(30^{\circ}\) and \(60^{\circ}\), and c. \(45^{\circ}\) and \(45^{\circ}\). This is based on the rule for a right triangle that one angle measures \(90^{\circ}\) and the other two angles are acute with their sum equaling \(90^{\circ}\). The pairs d. \(55^{\circ}\) and \(55^{\circ}\) and e. \(75^{\circ}\) and \(75^{\circ}\) could not be in a right triangle as their sums are more than \(90^{\circ}\).

Step by step solution

01

Analyze the first pair of angles

The sum of \(20^{\circ}\) and \(70^{\circ}\) is \(90^{\circ}\). Since the sum of these angles is \(90^{\circ}\), and a right triangle also has a \(90^{\circ}\) angle, these angles could be in a right triangle. So, answer 'a' is possible.
02

Analyze the second pair of angles

The sum of \(30^{\circ}\) and \(60^{\circ}\) is \(90^{\circ}\). Similar to step one, as the sum equals \(90^{\circ}\), these angles could be in a right triangle. So, answer 'b' is also possible.
03

Analyze the third pair of angles

Similar to steps one and two, the sum of \(45^{\circ}\) and \(45^{\circ}\) is \(90^{\circ}\). Therefore, this pair of angles could be in a right triangle as well. Therefore, answer 'c' is another possible solution.
04

Analyze the fourth pair of angles

The sum of \(55^{\circ}\) and \(55^{\circ}\) is \(110^{\circ}\). As this sum is more than \(90^{\circ}\), these angles cannot be part of a right triangle. Thus, answer 'd' is not possible.
05

Analyze the fifth pair of angles

The sum of \(75^{\circ}\) and \(75^{\circ}\) is \(150^{\circ}\). This sum is more than \(90^{\circ}\), which means these angles cannot be part of a right triangle. Therefore, answer 'e' is not possible.

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