Question

qt bisects ∠rtv. complete the proof that △qtu ≅ △qts.
1 qt bisects ∠rtv given
2 ∠s ≅ ∠u given
3 ∠utv ≅ ∠rts vertical angle theorem
4 ∠qtv ≅ ∠qtr definition of angle bisector
5 m∠qtu = m∠qtv + m∠utv additive property of angle measure
6 m∠qts = m∠qtr + m∠rts additive property of angle measure
7 m∠qtu = m∠qtr + m∠rts substitution
8 m∠qts = m∠qtu transitive property of equality
9 qt ≅ qt reflexive property of congruence
10 △qtu ≅ △qts

Explanation:

Step1: Identify congruent angles and sides

We have \(\angle S\cong\angle U\) (given), \(\angle QTU\cong\angle QTS\) (from steps 5 - 8: \(m\angle QTU=m\angle QTS\) so \(\angle QTU\cong\angle QTS\)), and \(\overline{QT}\cong\overline{QT}\) (Reflexive Property of Congruence).

Step2: Apply AAS (Angle - Angle - Side) congruence criterion

In \(\triangle QTU\) and \(\triangle QTS\), we have two angles (\(\angle S\cong\angle U\) and \(\angle QTU\cong\angle QTS\)) and a non - included side (\(\overline{QT}\)) that are congruent.

Answer:

By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle QTU\cong\triangle QTS\). So the reason for statement 10 is "AAS (Angle - Angle - Side) Congruence Theorem".