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.

statements | reasons
--- | ---

1. \\(\triangle wxy\\) is isosceles with legs \\(\overline{wx}\\) and \\(\overline{wy}\\) | 1. given
2. \\(\triangle wvz\\) is isosceles with legs \\(\overline{wv}\\) and \\(\overline{wz}\\) | 2. given
3. \\(\overline{wx} \cong \overline{wy}\\); \\(\overline{wv} \cong \overline{wz}\\) | 3. def. of isosceles traingle
4. \\(\boldsymbol{\cdot}\\) | 4. def. of \\(\cong\\)
5. \\((wz)(wx) = (wy)(wz)\\) | 5. multiplication property
6. \\((wz)(wx) = (wy)(wv)\\) | 6. \\(\boldsymbol{\cdot}\\)
7. \\(\frac{wy}{wz} = \frac{wx}{wv}\\) | 7. property of proportion
8. \\(\angle w \cong \angle w\\) | 8. reflexive property
9. \\(\triangle wxy \sim \triangle wvz\\) | \\(\boldsymbol{\cdot}\\)

Explanation:

Step1: Fill step 4 (def of ≅)

Since congruent segments have equal lengths, step 4 is $WX = WY$; $WV = WZ$.

Step2: Fill step 6 (substitution)

Replace $WZ$ with $WV$ (from step 3: $WV ≅ WZ$) in step 5: $(WZ)(WX) = (WY)(WV)$.

Step3: Fill step 9 (similarity rule)

SAS similarity: $\frac{WY}{WZ} = \frac{WX}{WV}$ (step7) and $\angle W ≅ \angle W$ (step8), so $\triangle WXY \sim \triangle WVZ$.

Answer:

Step 4: $WX = WY$; $WV = WZ$
Step 6: substitution property
Step 9: SAS similarity criterion