Chapter 2.4, Problem 22WE

(a)

To determine

To calculate: The measure of angle 2 and angle 4

Answer to Problem 22WE

  $\begin{array}{l}m\angle 2=153°\\ m\angle 4=153°\end{array}$

Explanation of Solution

Given information: $\angle 1and\angle 2$are supplements

  $\angle 3and\angle 4$are supplements

  $m\angle 1=\angle 3=27°$

Formula used: Sum of two supplementary angle is equal to 180

Calculation:

Add the angles 1 and 2 and set it equal to 180. Substitute 27 for angle 1 and solve for angle 2. Substitute 27 for angle 3 as well.

  $\begin{array}{l}m\angle 1+m\angle 2=180°\\ m\angle 1=27°\end{array}$

Substitute it in the first equation

  $\begin{array}{l}27°+m\angle 2=180°\\ m\angle 2=180°-27°\\ m\angle 2=153°\end{array}$

Substitute 27 degree for angle 3 and solve for angle 4

  $\begin{array}{l}m\angle 3+m\angle 4=180°\\ Substitute,m\angle 3=27°\\ 27°+m\angle 4=180°\\ m\angle 4=180°-27°\\ m\angle 4=153°\end{array}$

(b)

To determine

To calculate: The measure of angle 2 and angle 4 in terms of x

Answer to Problem 22WE

  $\begin{array}{l}m\angle 2=180°-x\\ m\angle 4=180°-x\end{array}$

Explanation of Solution

Given information: $\angle 1and\angle 2$are supplements

  $\angle 3and\angle 4$are supplements

  $m\angle 1=\angle 3=x$

Formula used: Sum of two supplementary angle is equal to 180

Calculation:

Add the angles 1 and 2 and set it equal to 180 degree. Substitute x for angle 1 and solve for angle 2. Substitute x for angle 3 as well.

  $\begin{array}{l}m\angle 1+m\angle 2=180°\\ Substitue,m\angle 1=x\\ x+m\angle 2=180°\\ m\angle 2=180°-x\end{array}$

Substitute x for angle 3 and do the same

  $\begin{array}{l}m\angle 3+m\angle 4=180°\\ m\angle 3=x\\ x+m\angle 4=180°\\ m\angle 4=180°-x\end{array}$

(c )

To determine

To prove: if two angles are congruent then their supplements are congruent

Answer to Problem 22WE

Yes , if two angles are congruent then their supplements are congruent

Explanation of Solution

Given information: : $\angle 1and\angle 2$are supplements

  $\angle 3and\angle 4$are supplements

Proof: Suppose angles 1 and 3 are congruent

We know that the sum of supplementary angles are 180 degree.

  $\begin{array}{l}m\angle 1+m\angle 2=180°\\ m\angle 3+m\angle 4=180°\\ \angle 1\cong \angle 3\\ \end{array}$

When two angles are congruent then they are equal

  $\begin{array}{l}m\angle 1=m\angle 3\\ m\angle 3+m\angle 2=180°\\ m\angle 3=180°-m\angle 2\end{array}$

We know that ,

  $\begin{array}{l}m\angle 3+m\angle 4=180°\\ 180°-m\angle 2+m\angle 4=180°\\ 180°-180°+m\angle 4=m\angle 2\\ m\angle 4=m\angle 2\end{array}$

So the supplements angle 4 and angle 2 are congruent

Therefore, if two angles are congruent then their supplements are congruent