Question

\overrightarrow{rt} bisects \angle sru and \angle stu. complete the proof that \triangle rst \cong \triangle rut.

1 \overrightarrow{rt} bisects \angle sru given
2 \overrightarrow{rt} bisects \angle stu given
3 \angle srt \cong \angle tru definition of angle bisector
4 \angle rts \cong \angle rtu definition of angle bisector
5 \overline{rt} \cong \overline{rt} reflexive property of congruence
6 \triangle rst \cong \triangle rut

Explanation:

Step1: Recall the ASA (Angle - Side - Angle) congruence criterion

Two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle.

Step2: Identify the congruent parts

• From step 3: \(\angle SRT\cong\angle TRU\)
• From step 4: \(\angle RTS\cong\angle RTU\)
• From step 5: \(\overline{RT}\cong\overline{RT}\) (common side, which is the included side for the two pairs of angles)

Answer:

By the ASA (Angle - Side - Angle) congruence criterion.