Question

△vwy ≅ △xuy. complete the proof that \\(\overline{wx} \cong \overline{uv}\\).
image of a quadrilateral with vertices v, w, x, u and diagonals intersecting at y

	statement	reason
2	\\(\overline{uy} \cong \overline{wy}\\)
3	\\(\overline{vy} \cong \overline{xy}\\)
4	\\(\angle uvy \cong \angle wyx\\)
5	\\(\triangle wxy \cong \triangle uvy\\)	sas
6	\\(\overline{wx} \cong \overline{uv}\\)	cpctc

Explanation:

Step1: Analyze Statement 2

We know that \(\triangle VWY \cong \triangle XUY\) (given). By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, the corresponding sides \(UY\) and \(WY\) should be congruent. So the reason for \(\overline{UY} \cong \overline{WY}\) is CPCTC (from \(\triangle VWY \cong \triangle XUY\)).

Step2: Analyze Statement 3

Similarly, from \(\triangle VWY \cong \triangle XUY\), the corresponding sides \(VY\) and \(XY\) should be congruent. So the reason for \(\overline{VY} \cong \overline{XY}\) is also CPCTC (from \(\triangle VWY \cong \triangle XUY\)).

Step3: Analyze Statement 4

From \(\triangle VWY \cong \triangle XUY\), the corresponding angles \(\angle UVY\) and \(\angle WYX\) should be congruent. Wait, actually, we can also get \(\angle UVY\) and \(\angle WYX\) from the congruent triangles \(\triangle VWY\) and \(\triangle XUY\) (CPCTC for angles). But also, when we look at \(\triangle WXY\) and \(\triangle UVY\), we can use the congruent sides from steps 2 and 3. Alternatively, since \(\triangle VWY \cong \triangle XUY\), \(\angle VYW \cong \angle XYU\), but maybe more directly, from the congruent triangles, the angles \(\angle UVY\) and \(\angle WYX\) can be shown congruent via CPCTC or by the fact that the triangles are congruent. But actually, for statement 4, since we have \(\overline{UY} \cong \overline{WY}\) (step 2), \(\overline{VY} \cong \overline{XY}\) (step 3), and we can consider the included angles. Wait, no, statement 4 is \(\angle UVY \cong \angle WYX\). From \(\triangle VWY \cong \triangle XUY\), \(\angle VWY \cong \angle XUY\), but maybe a better way: since \(\triangle VWY \cong \triangle XUY\), then \(\angle VYW \cong \angle XYU\), and if we look at the angles around point \(Y\), but actually, the key is that for \(\triangle WXY \cong \triangle UVY\) (SAS in step 5), we need the included angle. Wait, step 5 is SAS, so we have two sides and the included angle. We have \(\overline{WY} \cong \overline{UY}\) (step 2), \(\overline{XY} \cong \overline{VY}\) (step 3), and the included angle between them: \(\angle WYX\) and \(\angle UVY\). So to show \(\angle UVY \cong \angle WYX\), we can use the fact that from \(\triangle VWY \cong \triangle XUY\), \(\angle VWY \cong \angle XUY\), but maybe more directly, the vertical angles or the corresponding angles from the congruent triangles. Wait, actually, the reason for \(\angle UVY \cong \angle WYX\) is also CPCTC (from \(\triangle VWY \cong \triangle XUY\))? No, maybe not. Wait, let's re - examine.

Wait, the main idea is:

For statement 2: Reason is CPCTC (because \(\triangle VWY\cong\triangle XUY\), so corresponding sides \(UY\) and \(WY\) are congruent).

For statement 3: Reason is CPCTC (because \(\triangle VWY\cong\triangle XUY\), so corresponding sides \(VY\) and \(XY\) are congruent).

For statement 4: Reason is CPCTC (because \(\triangle VWY\cong\triangle XUY\), so corresponding angles \(\angle UVY\) and \(\angle WYX\) are congruent) or maybe the fact that the angles are equal because of the congruent triangles. But let's focus on statement 2 first.

So for statement 2:

Since \(\triangle VWY\cong\triangle XUY\), by CPCTC, \(UY = WY\), so \(\overline{UY}\cong\overline{WY}\). So the reason is "CPCTC (from \(\triangle VWY\cong\triangle XUY\))".

For statement 3:

Since \(\triangle VWY\cong\triangle XUY\), by CPCTC, \(VY = XY\), so \(\overline{VY}\cong\overline{XY}\). So the reason is "CPCTC (from \(\triangle VWY\cong\triangle XUY\))".

For statement 4:

Since \(\triangle VWY\cong\triangle XUY\), by CPCTC, \(\angle VWY\cong\angle XUY\), but we need \(\angle UVY\cong\angle WYX\). Wait, maybe another approach. The triangles \(\triangle WXY\) and \(\triangle UVY\): we have \(WY = UY\) (step 2), \(XY = VY\) (step 3), and the included angle between \(WY\) and \(XY\) is \(\angle WYX\), and the included angle between \(UY\) and \(VY\) is \(\angle UVY\). For SAS congruence (step 5), we need these included angles to be congruent. So the reason for \(\angle UVY\cong\angle WYX\) is "CPCTC (from \(\triangle VWY\cong\triangle XUY\))" or "Corresponding angles of congruent triangles are congruent (from \(\triangle VWY\cong\triangle XUY\))". But let's get back to the first empty box (statement 2).

Answer:

For statement 2, the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent) (from \(\triangle VWY \cong \triangle XUY\))"

For statement 3, the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent) (from \(\triangle VWY \cong \triangle XUY\))"

For statement 4, the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent) (from \(\triangle VWY \cong \triangle XUY\))" (or "Corresponding angles of congruent triangles are congruent")