Question

which rule explains why these scalene triangles are similar?
g
28
h
k
54
64
j
36
i
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Identify Vertical Angles

Angles at \( K \) ( \( \angle GKH \) and \( \angle IKJ \)) are vertical angles, so they are equal.

Step2: Check Side Ratios

First, find the ratios of the sides around the equal angle. For \( \triangle GKH \) and \( \triangle IKJ \):

• \( \frac{GK}{IK}=\frac{28}{36}=\frac{7}{9} \)
• \( \frac{HK}{JK}=\frac{54}{64}=\frac{27}{32} \)

Wait, no, maybe I mixed up. Wait, actually, let's re - check. Wait, the sides: \( GK = 28 \), \( IK = 36 \), \( HK = 54 \), \( JK = 64 \). Wait, no, maybe the correct pairs: \( \frac{GK}{JK}=\frac{28}{64}=\frac{7}{16} \), \( \frac{HK}{IK}=\frac{54}{36}=\frac{3}{2} \). No, that's not right. Wait, vertical angles are equal, so \( \angle GKH=\angle IKJ \). Now, check the ratios of the sides adjacent to the vertical angle. So for \( \triangle GKH \) and \( \triangle IKJ \), the sides around \( \angle K \):

• \( \frac{GK}{IK}=\frac{28}{36}=\frac{7}{9} \)
• \( \frac{HK}{JK}=\frac{54}{64}=\frac{27}{32} \)

Wait, these ratios are not equal. Wait, maybe I made a mistake in the side labels. Wait, no, maybe the triangles are \( \triangle GKH \) and \( \triangle IKJ \). Wait, \( GK = 28 \), \( IK = 36 \), \( HK = 54 \), \( JK = 64 \). Wait, \( \frac{GK}{JK}=\frac{28}{64}=\frac{7}{16} \), \( \frac{HK}{IK}=\frac{54}{36}=\frac{3}{2} \). These are not equal. Wait, but maybe the other way. Wait, no, the SAS similarity rule states that if two sides of one triangle are in proportion to two sides of another triangle and the included angle is equal, then the triangles are similar. But here, the included angle is the vertical angle (equal), but the ratios of the sides around the angle are not equal. Wait, but wait, maybe I miscalculated. Wait, \( \frac{28}{36}=\frac{7}{9} \) and \( \frac{54}{64}=\frac{27}{32} \). These are not equal. Also, for SSS, we would need all three sides in proportion, which they aren't. For AA, we need two angles equal, but we only have one angle (vertical angle) equal, and no other information about angles. So the triangles cannot be proven similar.

Answer:

none of the above; the triangles cannot be proven similar