Question

completing a proof of similar triangles

statements	reasons
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 triangle
4. $\boldsymbol{\bullet}$	4. def. of $\cong$
5. $(wz)(wx) = (wy)(wz)$	5. multiplication property
6. $(wz)(wx) = (wy)(wv)$	6. $\boldsymbol{\bullet}$
7. $\frac{wy}{wz} = \frac{wx}{wv}$	7. property of proportion
8. $\angle w \cong \angle w$	8. reflexive property
9. $\triangle wxy \sim \triangle wvz$	9. $\boldsymbol{\bullet}$

given: $\triangle wxy$ is isosceles with legs $\overline{wx}$ and $\overline{wy}$; $\triangle wvz$ is isosceles with legs $\overline{wv}$ and $\overline{wx}$.

prove: $\triangle wxy \sim \triangle wvz$

image of triangle diagram with points w, v, x, z, y

complete the steps of the proof.

dropdown options for step 4, step 6, step 9 (step 9 dropdown shows: sas similarity theorem, ssa similarity theorem, sss similarity theorem)

Explanation:

Step1: Analyze Step 4

From step 3, we have $\overline{WX} \cong \overline{WY}$ and $\overline{WV} \cong \overline{WZ}$. By the definition of congruent segments, congruent segments have equal lengths. So, $WX = WY$ and $WV = WZ$. So step 4 should be $WX = WY$; $WV = WZ$.

Step2: Analyze Step 6

In step 5, we have $(WZ)(WX)=(WY)(WZ)$, and from step 3, $WV = WZ$ (since $\overline{WV} \cong \overline{WZ}$). So we can substitute $WZ$ with $WV$ in the right - hand side of the equation in step 5. This is the substitution property. So the reason for step 6 is substitution property.

Step3: Analyze Step 9

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

Answer:

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