Question

su bisects ∠tuv and ∠tsv. complete the proof that △suv ≅ △sut. 1 su bisects ∠tuv given 2 su bisects ∠tsv given 3 ∠sut ≅ ∠suv 4 ∠tsu ≅ ∠usv definition of angle bisector 5 su ≅ su 6 △suv ≅ △sut

Explanation:

Step1: Analyze the given information

We know that \(SU\) bisects \(\angle TUV\) (given), so \(\angle TUS=\angle VUS\) by the definition of an angle - bisector. Also, \(SU\) bisects \(\angle TSV\) (given), so \(\angle TSU = \angle VSU\) by the definition of an angle - bisector. And \(SU = SU\) (common side).

Step2: Use the ASA (Angle - Side - Angle) congruence criterion

In \(\triangle SUV\) and \(\triangle SUT\), we have \(\angle TUS=\angle VUS\), \(SU = SU\), and \(\angle TSU=\angle VSU\). According to the ASA congruence criterion (\(ASA\): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent), \(\triangle SUV\cong\triangle SUT\).

Answer:

The reason for statement \(3\) (\(\angle SUT=\angle SUV\)) is "Definition of angle bisector" (since \(SU\) bisects \(\angle TSV\)). The reason for statement \(6\) (\(\triangle SUV\cong\triangle SUT\)) is "ASA (Angle - Side - Angle) congruence criterion".