Question

$overline{vw} perp overline{tw}$, $angle vtw cong angle tvu$, and $overline{uv} perp overline{tu}$. complete the proof that $\triangle tvw cong \triangle vtu$.

(image of a quadrilateral with vertices v, w, t, u and a diagonal from v to t)

statement	reason
2 $angle vtw cong angle tvu$	given
3 $overline{uv} perp overline{tu}$	given
4 $angle u cong angle w$
5 $overline{tv} cong overline{tv}$
6 $\triangle tvw cong \triangle vtu$	aas

Explanation:

Step1: Identify right angles

Perpendicular lines form right angles, so $\angle W = 90^\circ$ and $\angle U = 90^\circ$, thus $\angle U \cong \angle W$.

Step2: Recognize shared side

$\overline{TV}$ is a side of both triangles, so it is congruent to itself by reflexive property.

Step3: Apply AAS congruence

We have two pairs of congruent angles ($\angle VTW \cong \angle TVU$, $\angle U \cong \angle W$) and one pair of congruent non-included sides ($\overline{TV} \cong \overline{TV}$), so $\triangle TVW \cong \triangle VTU$ by AAS.

Answer:

1. $\overline{VW} \perp \overline{TW}$ (Given)
2. $\angle VTW \cong \angle TVU$ (Given)
3. $\overline{UV} \perp \overline{TU}$ (Given)
4. $\angle U \cong \angle W$ (Definition of perpendicular lines)
5. $\overline{TV} \cong \overline{TV}$ (Reflexive Property of Congruence)
6. $\triangle TVW \cong \triangle VTU$ (AAS)