Question

7
select the correct answer from each drop - down menu
given: \\( \overline { st } \\) bisects \\( \angle utv \\)
prove: \\( su = sv \\)
construct segment \\( su \\) so that it is perpendicular to segment \\( ut \\) and segment \\( sv \\) so that it is perpendicular to segment \\( vt \\)
complete the proof

Explanation:

Step1: Angle Bisector Definition

Since \(\overline{ST}\) bisects \(\angle UTV\), by the definition of an angle bisector, \(\angle UTS\cong\angle VTS\).

Step2: Perpendicular Segments Definition

Segments \(SU\perp UT\) and \(SV\perp VT\). By the definition of perpendicular segments, \(\angle TUS\) and \(\angle TVS\) are right angles (\(90^{\circ}\)).

Step3: AAS Congruence Criterion

We have \(\angle UTS\cong\angle VTS\) (from angle - bisector), \(\angle TUS\cong\angle TVS\) (all right angles are congruent), and \(\overline{ST}\cong\overline{ST}\) (reflexive property). So, \(\triangle UTS\cong\triangle VTS\) by the AAS (Angle - Angle - Side) congruence criterion.

Step4: Congruent Segments Definition

Since \(\overline{SU}\cong\overline{SV}\) (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent), and by the definition of congruent segments, if two segments are congruent, then their lengths are equal. So \(SU = SV\).

Answer:

For \(\angle UTS\cong\angle VTS\): definition of an angle bisector.
For \(\angle TUS\) and \(\angle TVS\) are right angles: definition of perpendicular segments.
For \(\triangle UTS\cong\triangle VTS\): AAS (Angle - Angle - Side) congruence criterion.
For \(SU = SV\): definition of congruent segments.