Chapter 7, Problem 18RP

To determine

To calculate : The possible values of the measure of the second largest angle

Answer to Problem 18RP

The values of the measure of the second largest angle are ${72}^0$ and ${75}^0$

Explanation of Solution

Given information : The following information has been given

Triangle ABC (let’s say) is a right triangle

Measure of one angle (A) is five times the measure of another angle (B)

Formula used : The sum of the angles of any triangle is ${180}^0$

Calculation : We know that in a right triangle, the largest angle will always be ${90}^0$ . Now, the second largest has to be one of the acute angles of the triangle.

Now, we have two possibilities for the given situation of − Measure of one angle is five times the measure of another angle

First case, $m\angle A={90}^0$ . Thus, we will have

  $m\angle B=\frac{1}{5}({90}^0)={18}^0$

Hence, the $m\angle C={180}^0-({90}^0+{18}^0)={72}^0$

Thus, second largest angle is $m\angle C={72}^0$

Second case,

  $\angle A+\angle B={90}^0$ , where

  $A=5B$

Thus, we get

  $m\angle A={75}^0$ and $m\angle B={15}^0$

Thus, second largest angle is $m\angle A={75}^0$