Questions: S U bisects angle T V X and angle U S W. Complete the proof that S U is congruent to S W.

Transcript text: $\overleftrightarrow{S U}$ bisects $\angle T V X$ and $\angle U S W$. Complete the proof that $\overline{S U} \cong \overline{S W}$.

Solution

Solution Steps

Step 1: Vertical Angles

Angle WVX and angle TVU are vertical angles, so they are congruent.

Step 2: Definition of Angle Bisector

SV bisects angles TVX and USW. Therefore, angle SVX is congruent to angle SVT, and angle VSW is congruent to angle USV.

Step 3: Angle Addition Postulate

The measure of angle SVW is equal to the sum of the measures of angles SVX and WVX. The measure of angle SVU is equal to the sum of the measures of angles SVT and TVU.

Step 4: Substitution

Since angle WVX is congruent to angle TVU, and angle SVX is congruent to angle SVT, the measure of angle SVW can be written as the sum of the measures of angles SVT and TVU.

Step 5: Transitive Property of Equality

It has been established that the measure of angle SVU is equal to the sum of the measures of angles SVT and TVU, and the measure of angle SVW is also equal to the sum of the measures of angles SVT and TVU. Therefore, the measures of angles SVU and SVW are equal, so the angles are congruent.

Step 6: ASA Postulate

Triangles SVW and SVU share side SV, angle VSW is congruent to angle USV, and angle SVW is congruent to angle SVU. Therefore, the triangles are congruent by the Angle-Side-Angle (ASA) postulate.

Step 7: CPCTC

Because the two triangles are congruent, their corresponding parts are congruent (CPCTC). SU and SW are corresponding sides, so they are congruent.