Question

w 21 cm x diagram with right angles at x and z, z 21 cm y is δwxz ≅ δyzx? why or why not? ○ yes, they are congruent by sas. ○ yes, they are both right triangles. ○ no, the triangles share side xz. ○ no, there is only one set of congruent sides.

Explanation:

To determine if \(\triangle WXZ \cong \triangle YZX\), we analyze the given information:

1. \(WX = YZ = 21\) cm (given side lengths).
2. \(\angle WXZ\) and \(\angle YZX\) are right angles (marked with right angle symbols), so \(\angle WXZ=\angle YZX = 90^\circ\).
3. The triangles share the side \(XZ\) (common side), so \(XZ = ZX\) (reflexive property of congruence).

Using the SAS (Side - Angle - Side) congruence criterion, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Here, we have:

• Side \(WX\cong YZ\) (both 21 cm).
• Included angle \(\angle WXZ\cong\angle YZX\) (both right angles).
• Side \(XZ\cong ZX\) (common side).

Now let's analyze the other options:

• Option "Yes, they are both right triangles": Just being right triangles does not guarantee congruence. There are many right triangles with different side lengths that are not congruent.
• Option "No, the triangles share side \(\overline{XZ}\)": Sharing a side (which is a congruent side) is actually a reason for congruence, not against it.
• Option "No, there is only one set of congruent sides": We have two sets of congruent sides (\(WX\cong YZ\) and \(XZ\cong ZX\)) and a congruent included angle, so this is incorrect.

Answer:

A. Yes, they are congruent by SAS.