Chapter 3.5, Problem 31PPSE

Solution

Tocalculate:Themeasure of each angle in the problem and what system of equation represent the situation. Also determine the easiest method of solving. It is given that in triangle PQR the measure of angle Q is three times the measure of angle P . The measure of angle R is $20°$ more than the measure of angle P .

The measure of each angle is $P=32°,Q=96°\text{ and }R=52°$ . The easiest method is the substitution method.

Given information:

In triangle PQR the measure of angle Q is three times the measure of angle P . The measure of angle R is $20°$ more than the measure of angle P .

Formula used:

The angle sum property of triangle is: The sum of all angles in a triangle is $180°$ .

Calculation:

Consider the given information.

Let the measure of angle P in triangle PQR is $x$ , the measure of angle Q is $y$ and the measure of angle R is $z$ .

By the angle sum property $x+y+z=180°$

It is given that the measure of angle Q is three times the measure of angle P .

  $y=3x$

Also, the measure of angle R is $20°$ more than the measure of angle P .

  $z=x+20°$

Here substitution method looks easier to solve because all the equations is already solved for x .

Substitute $y=3x$ and $z=x+20°$ into the equation $x+y+z=180°$ .

  $\begin{array}{c}x+3x+x+20°=180°\\ 5x=160°\\ x=32°\end{array}$

Solve for y .

  $\begin{array}{l}y=3\left(32°\right)\\ y=96°\end{array}$

Solve for z .

  $\begin{array}{c}z=32°+20°\\ z=52°\end{array}$

The measure of each angle is $P=32°,Q=96°\text{ and }R=52°$ .