Question

$overline{ux} cong overline{vw}$, $overline{uw} perp overline{ux}$, and $overline{vw} perp overline{uw}$. complete the proof that $overline{wy} cong overline{uy}$.

	statement	reason
2	$overline{uw} perp overline{ux}$
3	$overline{vw} perp overline{uw}$
4	$angle u cong angle w$
5	$angle uyx cong angle vyw$
6	$\triangle uxy cong \triangle wvy$
7	$overline{wy} cong overline{uy}$

Explanation:

Step1: Reason for \( \overline{UX} \cong \overline{VW} \)

Given (as stated in the problem statement).

Step2: Reason for \( \overline{UW} \perp \overline{UX} \)

Given (from the problem's information).

Step3: Reason for \( \overline{VW} \perp \overline{UW} \)

Given (from the problem's information).

Step4: Reason for \( \angle U \cong \angle W \)

All right angles are congruent (since \( \overline{UW} \perp \overline{UX} \) and \( \overline{VW} \perp \overline{UW} \), so \( \angle U \) and \( \angle W \) are right angles).

Step5: Reason for \( \angle UYX \cong \angle VYW \)

Vertical angles are congruent (they are formed by the intersection of lines \( VX \) and \( UW \)).

Step6: Reason for \( \triangle UXY \cong \triangle WVY \)

AAS (Angle - Angle - Side) Congruence Criterion (we have \( \angle U \cong \angle W \), \( \angle UYX \cong \angle VYW \), and \( \overline{UX} \cong \overline{VW} \)).

Step7: Reason for \( \overline{WY} \cong \overline{UY} \)

Corresponding parts of congruent triangles are congruent (CPCTC) (since \( \triangle UXY \cong \triangle WVY \), the sides \( \overline{WY} \) and \( \overline{UY} \) are corresponding parts).

Answer:

1. Given
2. Given
3. Given
4. All right angles are congruent
5. Vertical angles are congruent
6. AAS (Angle - Angle - Side) Congruence
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)