Chapter 8.5, Problem 16PSB

To determine

To show that two ratios are equal.

Explanation of Solution

Given information:

  $\angle 3 \cong \angle 5$

Formula used:

The below properties and theorem are used:

Vertical angles are congruent.

If a ray divides an angle into two congruent angles, it is a bisector.

Angle bisector theorem states that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle.

Proof:

  

It is given that,

  $\angle 3 \cong \angle 5$

Vertical angles are congruent.

  $\angle 3 \cong \angle 4$

By transitive property, we get

  $\angle 4 \cong \angle 5$

If a ray divides an angle into two congruent angles, it is a bisector.

  $\overleftrightarrow{VS}bisects\angle RVT$

By Angle Bisector Theorem, we get

  $\frac{RV}{VT}=\frac{RS}{ST}$