Question

given: δwxy is isosceles with legs \\(\overline{wx}\\) and \\(\overline{wy}\\), δwvz is isosceles with legs \\(\overline{wv}\\) and \\(\overline{wz}\\).
prove: \\(\delta wxy \sim \delta wvz\\)

complete the steps of the proof.

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

options (for one blank): sas similarity theorem, ssa similarity theorem, sss similarity theorem

Explanation:

Step 1: Analyze Step 4

From step 3, we know \(\overline{WX}\cong\overline{WY}\) and \(\overline{WV}\cong\overline{WZ}\). By the definition of congruent segments, their lengths are equal. So if \(WX = WY\) (from \(\overline{WX}\cong\overline{WY}\)) and \(WV = WZ\) (from \(\overline{WV}\cong\overline{WZ}\)), then step 4 should state \(WX = WY\) and \(WV = WZ\) (or \(WZ = WV\)).

Step 2: Analyze Step 6

In step 5, we have \((WZ)(WX)=(WY)(WZ)\). In step 6, the right - hand side becomes \((WY)(WV)\). Since from step 3, \(\overline{WV}\cong\overline{WZ}\), so \(WV = WZ\) (by definition of congruence). So we can substitute \(WZ\) with \(WV\) in the right - hand side of the equation from step 5. The reason for step 6 is substitution (because we substitute \(WZ\) with \(WV\) since \(WV = WZ\)).

Step 3: 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 in proportion to two sides of another triangle and the included angle is congruent, then the two triangles are similar. Here, in \(\triangle WXY\) and \(\triangle WVZ\), the sides around the common angle \(\angle W\) are in proportion (\(\frac{WY}{WZ}=\frac{WX}{WV}\)) and \(\angle W\) is congruent to itself. So the reason for step 9 is the SAS similarity theorem.

Step 4 (Filling in the blanks):

• Step 4: \(WX = WY\), \(WV = WZ\) (or \(WZ = WV\))
• Step 6 Reason: substitution (because \(WV = WZ\) from step 3, so we substitute \(WZ\) with \(WV\) in \((WY)(WZ)\) to get \((WY)(WV)\))
• Step 9 Reason: SAS similarity theorem

Final Answers for the Drop - downs:

• For the first blank (step 4 content): \(WX = WY\), \(WV = WZ\) (or equivalent based on congruence)
• For the second blank (step 6 reason): substitution
• For the third blank (step 9 reason): SAS similarity theorem

Answer:

Step 1: Analyze Step 4

From step 3, we know \(\overline{WX}\cong\overline{WY}\) and \(\overline{WV}\cong\overline{WZ}\). By the definition of congruent segments, their lengths are equal. So if \(WX = WY\) (from \(\overline{WX}\cong\overline{WY}\)) and \(WV = WZ\) (from \(\overline{WV}\cong\overline{WZ}\)), then step 4 should state \(WX = WY\) and \(WV = WZ\) (or \(WZ = WV\)).

Step 2: Analyze Step 6

In step 5, we have \((WZ)(WX)=(WY)(WZ)\). In step 6, the right - hand side becomes \((WY)(WV)\). Since from step 3, \(\overline{WV}\cong\overline{WZ}\), so \(WV = WZ\) (by definition of congruence). So we can substitute \(WZ\) with \(WV\) in the right - hand side of the equation from step 5. The reason for step 6 is substitution (because we substitute \(WZ\) with \(WV\) since \(WV = WZ\)).

Step 3: 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 in proportion to two sides of another triangle and the included angle is congruent, then the two triangles are similar. Here, in \(\triangle WXY\) and \(\triangle WVZ\), the sides around the common angle \(\angle W\) are in proportion (\(\frac{WY}{WZ}=\frac{WX}{WV}\)) and \(\angle W\) is congruent to itself. So the reason for step 9 is the SAS similarity theorem.

Step 4 (Filling in the blanks):

• Step 4: \(WX = WY\), \(WV = WZ\) (or \(WZ = WV\))
• Step 6 Reason: substitution (because \(WV = WZ\) from step 3, so we substitute \(WZ\) with \(WV\) in \((WY)(WZ)\) to get \((WY)(WV)\))
• Step 9 Reason: SAS similarity theorem

Final Answers for the Drop - downs:

• For the first blank (step 4 content): \(WX = WY\), \(WV = WZ\) (or equivalent based on congruence)
• For the second blank (step 6 reason): substitution
• For the third blank (step 9 reason): SAS similarity theorem