Question

which rule explains why these triangles are similar?
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Calculate the third angle of triangle \( \triangle JKI \)

The sum of angles in a triangle is \( 180^{\circ} \). For \( \triangle JKI \), if \( \angle K = 90^{\circ} \) and \( \angle I=31^{\circ} \), then \( \angle J=180^{\circ}-(90^{\circ} + 31^{\circ})=59^{\circ} \).

Step2: Calculate the third angle of triangle \( \triangle HFG \)

For \( \triangle HFG \), if \( \angle F = 90^{\circ} \) and \( \angle G = 39^{\circ} \), then \( \angle H=180^{\circ}-(90^{\circ}+39^{\circ}) = 51^{\circ} \).

Step3: Check similarity rules

• SSS (Side - Side - Side): We are not given any side - length information, so we cannot use the SSS similarity rule.
• SAS (Side - Angle - Side): We are not given any side - length information, so we cannot use the SAS similarity rule.
• AA (Angle - Angle): The AA similarity rule states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Here, \( \angle K=\angle F = 90^{\circ} \), but \( \angle I

eq\angle G\) and \( \angle J
eq\angle H\).

Answer:

none of the above; the triangles cannot be proven similar