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 \(HGF\)

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle HGF\), if \(\angle G = 36^{\circ}\) and \(\angle F=98^{\circ}\), then \(\angle H=180^{\circ}-(36^{\circ} + 98^{\circ})=46^{\circ}\)

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

For \(\triangle JKI\), if \(\angle K = 42^{\circ}\) and \(\angle I = 98^{\circ}\), then \(\angle J=180^{\circ}-(42^{\circ}+98^{\circ}) = 40^{\circ}\)

Step3: Check similarity rules

• SSS (Side - Side - Side) similarity: We have no information about the side lengths of the two triangles.
• SAS (Side - Angle - Side) similarity: We have no information about the sides including the angles.
• AA (Angle - Angle) similarity: The angles of \(\triangle HGF\) are \(36^{\circ},98^{\circ},46^{\circ}\) and the angles of \(\triangle JKI\) are \(42^{\circ},98^{\circ},40^{\circ}\). There are not two pairs of equal angles.

Answer:

none of the above; the triangles cannot be proven similar