Chapter 8.5, Problem 23PSB

To determine

To show two ratios are equal.

Explanation of Solution

Given information:

  $\overleftrightarrow{AC}$ is perpendicular to $\overline{BE}$

  $\angle 1\cong \angle 4$

Formula used:

The below properties and theorems are used:

Perpendicular lines form right angles.

Complementary angles form right angles.

Complementary of congruent angles are congruent.

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.

Calculation:

  

It is given that,

  $\overleftrightarrow{AC}$ is perpendicular to $\overline{BE}$

Perpendicular lines form right angles.

  $\angle ABE$ and $\angle CBE$ are right angles.

It is given that,

  $\angle 1\cong \angle 4$

Complementary angles form right angles.

  $\angle 2$ is complementary to $\angle 1$

  $\angle 3$ is complementary to $\angle 4$

Complementary of congruent angles are congruent.

  $\angle 2\cong \angle 3$

Bisector divides angles into two congruent angles.

  $\stackrel{\rightharpoonup }{BE}$ bisects $\angle DEF$

Using Angle Bisector Theorem, we get

  $\frac{BD}{BF}=\frac{DE}{EF}$