Chapter 2.8, Problem 9PSB

To determine

One given angle is supplementary to another given angle

Answer to Problem 9PSB

  $\angle 2\text{ is supplement to }\angle 3$

Explanation of Solution

Given information :

  $\angle 1\text{ is congruent to }\angle 3$

Formula used:

Congruency means same size and same shape as per the definition of congruency

One angle is said to be supplementary to other, if sum of both the angles is equal to $180°$

Sum of the angles formed on one side of line is $180°$

Proof:

Since,

  $\begin{array}{l}\angle 1\cong \angle \mathrm{3......}\left(\text{given}\right)\\ \therefore \angle 1\text{ = }\angle \mathrm{3........}\left(i\right)\left(\text{same shape and same size, as per definition of congruency}\right)\end{array}$

And as per figure,

  $\angle 1\text{ + }\angle 2=180°$

Therefore, they are the only angles on same side of the straight line.

  $\begin{array}{l}\Rightarrow \angle 1\text{ \& }\angle 2\text{ are supplementary angles }\left(\text{because their sum is equal to 180}°\right)\\ \Rightarrow \angle 3\text{ \& }\angle 2\text{ are supplementary angles }\left(\text{because from}\left(i\right),\angle 1\text{ = }\angle 3\right)\\ \text{Thus, }\\ \angle 2\text{ is supplementary to }\angle 3\\ \end{array}$

Hence, Proved.