Question

given: δwxy is isosceles with legs \\(\overline{wx}\\) and \\(\overline{wy}\\); δwvz is isosceles with legs \\(\overline{wv}\\) and \\(\overline{wz}\\).
prove: δwxy ~ δwvz

complete the steps of the proof.

(image of a triangle diagram with points w, z, y, v, x and a two - column proof table. the table has statements and reasons columns. statements: 1. \\(\triangle wxy\\) is isosceles with legs \\(wx\\) and \\(wy\\); 2. \\(\triangle wvz\\) is isosceles with legs \\(wv\\) and \\(wz\\); 3. \\(\overline{wx}\cong\overline{wy}\\); \\(\overline{wv}\cong\overline{wz}\\); 4. \\(\blacktriangle\\); 5. \\((wz)(wx)=(wy)(wz)\\); 6. \\((wz)(wx)=(wy)(wv)\\); 7. \\(\frac{wy}{wz}=\frac{wx}{wv}\\); 8. \\(\angle w\cong\angle w\\); 9. \\(\triangle wxy\sim\triangle wvz\\). reasons: 1. given; 2. given; 3. def. of isosceles triangle; 4. def. of \\(\cong\\); 5. multiplication property; 6. \\(\blacktriangle\\); 7. property of proportion; 8. reflexive property; 9. \\(\blacktriangle\\). also, there are dropdown options for the blanks: \\(wx = wy; wv = wz\\); \\(wx = wz; wv = wy\\); \\(wy = wy; wz = wz\\))

Explanation:

Step 1: Analyze Step 4

The reason for step 4 is "def. of $\cong$", which means we use the definition of congruent segments (if two segments are congruent, their lengths are equal). From step 3, we have $\overline{WX} \cong \overline{WY}$ and $\overline{WV} \cong \overline{WZ}$, so by definition of congruence, $WX = WY$ and $WV = WZ$. So the statement for step 4 is $WX = WY; WV = WZ$.

Step 2: Analyze Step 6

Step 5 is $(WZ)(WX) = (WY)(WZ)$, and step 6 is $(WZ)(WX) = (WY)(WV)$. We know from step 4 that $WV = WZ$ (since $\overline{WV} \cong \overline{WZ}$ implies $WV = WZ$), so we substitute $WZ$ with $WV$ in the right - hand side of step 5. This is the substitution property.

Step 3: Analyze Step 9

To prove that $\triangle WXY \sim \triangle WVZ$, we have two sides in proportion ($\frac{WY}{WZ}=\frac{WX}{WV}$ from step 7) and the included angle $\angle W$ congruent (from step 8, $\angle W\cong\angle W$). The theorem that states if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar is the SAS (Side - Angle - Side) similarity theorem. So the reason for step 9 is SAS similarity theorem.

Answer:

• Step 4 Statement: $WX = WY; WV = WZ$
• Step 6 Reason: Substitution Property
• Step 9 Reason: SAS (Side - Angle - Side) Similarity Theorem