Chapter 8.5, Problem 24PSB

To determine

To show two ratios are equal.

Explanation of Solution

Given information:

  $\overleftrightarrow{GD}$ is parallel to $\overleftrightarrow{FE}$ .

  $\overleftrightarrow{BD}$ is parallel to $\overleftrightarrow{CE}$ .

Formula used:

The below properties are used:

If lines are parallel then corresponding angles are congruent.

Corresponding sides of similar triangles are proportional.

Calculation:

  

It is given that,

  $\overleftrightarrow{GD}$ is parallel to $\overleftrightarrow{FE}$ .

  $\overleftrightarrow{BD}$ is parallel to $\overleftrightarrow{CE}$ .

If lines are parallel then corresponding angles are congruent.

  $\angle BDA\cong \angle CEA$

By Reflexive property, we get

  $\angle BAD\cong \angle BAD$

Two triangles are similar by AA similarity rule.

  $\Delta BAD~\Delta CAE$

Corresponding sides of similar triangles are proportional.

  $\frac{AB}{AC}=\frac{AD}{AE}$

If lines are parallel then corresponding angles are congruent.

  $\angle AGD\cong \angle AFE$

By Reflexive property, we get

  $\angle GAD\cong \angle GAD$

Two triangles are similar by AA similarity rule.

  $\Delta AGD~\Delta AFE$

Corresponding sides of similar triangles are proportional.

  $\frac{AG}{AF}=\frac{AD}{AE}$

By Substitution property, we get

  $\frac{AB}{AC}=\frac{AG}{AF}$