Question

uv ⊥ tw and ∠v ≅ ∠u. complete the proof that tu ≅ tv.

(image of triangle with points u, t, v, w)

statement | reason
--- | ---

1. uv ⊥ tw | given
2. ∠v ≅ ∠u | given
3. ∠twu ≅ ∠twv |
4. tw ≅ tw |
5. △tuw ≅ △tvw |
6. tu ≅ tv | cpctc

Explanation:

Step1: Justify right angles congruence

Perpendicular lines form right angles, and all right angles are congruent. So $\angle TWU \cong \angle TWV$ because $\overline{UV} \perp \overline{TW}$ creates two right angles, which are always congruent.

Step2: Justify reflexive property

A segment is congruent to itself. $\overline{TW} \cong \overline{TW}$ by the Reflexive Property of Congruence.

Step3: Justify triangle congruence

We have $\angle U \cong \angle V$, $\angle TWU \cong \angle TWV$, and $\overline{TW} \cong \overline{TW}$. This matches the AAS (Angle-Angle-Side) Congruence Theorem, so $\triangle TUW \cong \triangle TVW$.

Answer:

Statement	Reason
2 $\angle V \cong \angle U$	Given
3 $\angle TWU \cong \angle TWV$	All right angles are congruent
4 $\overline{TW} \cong \overline{TW}$	Reflexive Property of Congruence
5 $\triangle TUW \cong \triangle TVW$	AAS Congruence Theorem
6 $\overline{TU} \cong \overline{TV}$	CPCTC