Chapter 6.1, Problem 8PSB

To determine

To show that $\angle VSOand\angle VTO$ are congruent.

Explanation of Solution

Given information:

A circle with center O lies in plane p .

  $\begin{array}{l}\overleftrightarrow{VO}\perp \overleftrightarrow{OS}\\ \overleftrightarrow{VO}\perp \overleftrightarrow{OT} \end{array}$

Formula used:

The below properties are used:

Radii of a circle are congruent.

Perpendicular lines form right angles.

Right angles are congruent.

Two triangles are congruent by SAS congruence rule.

Proof:

It is given that,

A circle with center O lies in plane p .

  

Radii of a circle are congruent.

  $\overline{OS} \cong \overline{OT}$

It is given that,

  $\begin{array}{l}\overleftrightarrow{VO}\perp \overleftrightarrow{OS}\\ \overleftrightarrow{VO}\perp \overleftrightarrow{OT} \end{array}$

Perpendicular lines form right angles.

  $\angle VOSand \angle VOTarerightangles.$

Right angles are congruent.

  $\angle VOS\cong \angle VOT$

By reflexive property, we get

  $\overline{VO} \cong \overline{VO}$

By SAS congruence rule, we get

  $\Delta VOS \cong \Delta VOT$

As corresponding parts of congruent triangles are congruent, we get

  $\angle VSO\cong \angle VTO$