Which of the following best completes the proof showing that ΔWXZ ~ ΔXYZ?

Question

triangles WXZ and XYZ that share side XZ with right angle XZW, WZ equals 10, XZ equals 5, and YZ equals 2 and one half

Since segment XZ is perpendicular to segment WY, angles WZX and XZY are both right angles and congruent. The proportion ________ shows the corresponding sides are proportional, so the triangles are similar by the SAS Similarity Postulate.

1. 5 over 10 equals 5 over 2 and one half

2. 5 over 10 equals 5 over 5

3.10 over 5 equals 5 over 2 and one half

4.10 over 5 equals 2 and one half over 5

Answer 1

To show that triangles WXZ and XYZ are similar by the SAS Similarity Postulate, we need to establish that two pairs of corresponding sides are in proportion, along with the angle that is included between those sides being congruent.

We know the following:

• Side WZ = 10
• Side XZ = 5
• Side YZ = 2.5 (which is the same as 2 and one-half)

Since segment XZ is common to both triangles and angles WZX and XZY are right angles, we can compare the sides adjacent to these angles.

To specifically answer which proportional relationship confirms the similarity:

We would write the ratio of the corresponding sides:

• From triangle WXZ, WZ = 10 and XZ = 5: \( \frac{XZ}{WZ} = \frac{5}{10} \)
• We can then compare this side ratio to a corresponding side ratio in triangle XYZ: \( \frac{XZ}{YZ} = \frac{5}{2.5} = \frac{5}{5} \)

The correct option that expresses this proportional relationship would be Option 2: \( \frac{5}{10} = \frac{5}{5} \)

Therefore, the correct answer is:

2. 5 over 10 equals 5 over 5