Chapter 3.4, Problem 31WE

To determine

To find: The congruent angles of the given figure.

Answer to Problem 31WE

The congruent angles of the given figure are $\angle \text{1}\cong \angle 2,\angle 4\cong \angle 6,\angle 3\cong \angle 4$

Explanation of Solution

Given information:

  $\text{Given:}\begin{array}{c}\overline{AB}\perp \overline{BF};\overline{HD}\perp \overline{BF}\\ \overline{GF}\perp \overline{BF};\angle A\cong \angle G\end{array}$

  

Consider the given figure and conditions.

Given,

  $\begin{array}{l}AB\perp BF,HD\perp BF,GF\perp BF,\angle A\cong \angle G\hfill \\ AB\perp BF\text{ And HD}\perp BF\hfill \\ B\Vert HD\hfill \end{array}$

  $AB\left|\right|HD$ and $AH$ is a transversal line.

  $\angle 1\cong \angle 2$ (Vertically opposite angles are equal ) and

  $\angle A\cong \angle 3$ (Alternative angles are equal) ........(1)

Given,

  $HD\perp BF\text{And}FG\perp BF$

So,

  $HD\left|\right|FG$

Since, if two lines are perpendicular to same line then they are parallel to each other.

  $HD\left|\right|FG$ and $HG$ is a transversal line.

  $\angle 4\cong \angle 6$ (Alternative angles)

  $\angle 4\cong \angle G(\angle G\cong \angle 6)$ ........(2)

Given,

  $\begin{array}{l}\angle A\cong \angle G\hfill \\ \text{So,}\hfill \\ \angle 3=\angle 4\hfill \\ \angle \text{1}\cong \angle 2,\angle 4\cong \angle 6,\angle 3\cong \angle 4\hfill \end{array}$