Question

1. which similarity criterion is used to prove that △xyw ~ △xyz ~ △yzw when yw is the altitude to the hypotenuse?

a. sas
b. aa
c. hl
d. sss

Explanation:

To determine the similarity criterion for right triangles when an altitude is drawn to the hypotenuse:

• Recall that in a right triangle, the altitude to the hypotenuse creates two smaller right triangles, all similar to the original and each other.
• The AA (Angle - Angle) similarity criterion applies because:

1. All triangles are right - angled (one right angle, so one pair of equal angles).
2. They share a common acute angle (e.g., $\angle X$ is common to $\triangle XYW$ and $\triangle XYZ$; $\angle Z$ is common to $\triangle YZW$ and $\triangle XYZ$), giving a second pair of equal angles.

• SAS similarity would require two sides in proportion and the included angle, but here AA is more direct. HL (Hypotenuse - Leg) is for congruence, not similarity. SSS similarity would require all three sides in proportion, but AA is sufficient here.

Answer:

b. AA