Question

complete the proof that $\triangle uwy \cong \triangle uxv$.
image of a triangle with points v, u, y, w, x and a table of statements and reasons for a proof
statement | reason
1 $\angle uxv \cong \angle uwy$ | given
2 $\angle xuy \cong \angle vuw$ | given
3 $\overline{ux} \cong \overline{uw}$ | given
4 $m\angle wuy = m\angle wux + m\angle xuy$ | additive property of angle measure
5 $m\angle vux = m\angle vuw + m\angle wux$ | additive property of angle measure
6 $m\angle wuy = m\angle wux + m\angle vuw$ | substitution
7 $m\angle vux = m\angle wuy$ | transitive property of equality
8 blank for statement | blank for reason

Explanation:

Step1: Identify congruent parts

We know $\angle UXV \cong \angle UWY$ (Statement 1), $\overline{UX} \cong \overline{UW}$ (Statement 3), and we just proved $\angle VUX \cong \angle WUY$ (from Step 7, since their measures are equal, so the angles are congruent).

Step2: Apply ASA congruence

To prove $\triangle UWY \cong \triangle UXV$, we can use the Angle - Side - Angle (ASA) congruence criterion. The ASA criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

For $\triangle UWY$ and $\triangle UXV$:

• $\angle UWY \cong \angle UXV$ (given, Statement 1)
• $\overline{UW} \cong \overline{UX}$ (given, Statement 3)
• $\angle WUY \cong \angle VUX$ (proven in Step 7)

So, by the ASA congruence postulate, $\triangle UWY \cong \triangle UXV$.

Looking at the table, Statement 8 should be the conclusion of the congruence of the triangles. So the statement for step 8 is $\triangle UWY \cong \triangle UXV$. The reason would be "ASA (Angle - Side - Angle) Congruence Postulate" (since we have two angles and the included side congruent).

Answer:

$\triangle UWY \cong \triangle UXV$ (by ASA Congruence Postulate, using the congruent angles $\angle UWY \cong \angle UXV$, $\angle WUY \cong \angle VUX$ and included side $\overline{UW} \cong \overline{UX}$)