Question

state if the triangles in each pair are similar. if so, state how you know they are similar and complete the similarity statement.
2)
△cba ~ ______

Explanation:

Step1: Check Proportional Sides

First, find the lengths of corresponding sides. For \( \triangle CBA \) and \( \triangle CKL \) (assuming \( K \) and \( L \) form \( \triangle CKL \)):
- \( CB = 168 \), \( CK = 60 \)? Wait, no, \( CB \) is total length? Wait, \( CK = 60 \), \( KB = 168 - 60 = 108 \)? Wait, no, the vertical side: \( CK = 60 \), \( CB = 60 + 168 \)? Wait, no, the diagram: \( C \) to \( K \) is 60, \( K \) to top (B?) Wait, maybe \( CB = 60 + 168 \)? No, the left side: from \( C \) to the top (let's say \( B \)) is 168? Wait, no, the label is 168 from \( C \) to the top (maybe \( B \) is at the top, \( C \) at bottom left, \( A \) at bottom right. Then \( CK = 60 \), \( KL = 70 \), \( CB = 60 + 168 \)? No, maybe \( CB = 60 + 168 \) is wrong. Wait, the side \( BA = 196 \), \( KL = 70 \), \( CA = 112 \), \( LA = 72 \), so \( CL = 112 - 72 = 40 \)? Wait, no, the horizontal side: \( C \) to \( L \) to \( A \), with \( C \) to \( A \) is 112, \( L \) to \( A \) is 72, so \( CL = 112 - 72 = 40 \). Wait, maybe I misread. Let's re-express:

Wait, the triangles: \( \triangle CBA \) (vertices \( C \), \( B \), \( A \)) and \( \triangle CKL \) (vertices \( C \), \( K \), \( L \)). Let's check the ratios of sides:

- \( CK / CB \): \( CK = 60 \), \( CB = 60 + 168 \)? No, maybe \( CB = 60 + 168 \) is incorrect. Wait, the left side: from \( C \) to \( K \) is 60, from \( K \) to \( B \) is 168? So \( CB = 60 + 168 = 228 \)? No, that doesn't match. Wait, the side \( BA = 196 \), \( KL = 70 \). Let's check \( KL / BA = 70 / 196 = 5/14 \). \( CL / CA \): \( CA = 112 \), \( CL = 112 - 72 = 40 \)? No, \( CL \) is from \( C \) to \( L \), \( LA = 72 \), so \( CL = 112 - 72 = 40 \). Then \( CL / CA = 40 / 112 = 5/14 \). \( CK / CB \): \( CB \) is the vertical side. If \( CK = 60 \), \( CB = 60 + 168 = 228 \)? No, that can't be. Wait, maybe the vertical side is \( CB = 60 + 168 \) is wrong. Wait, the label 168 is from \( C \) to the top (maybe \( B \) is at the top, \( K \) is a point on \( CB \), so \( CK = 60 \), \( KB = 168 \)? So \( CB = 60 + 168 = 228 \)? Then \( CK / CB = 60 / 228 = 5/19 \), which doesn't match. Wait, maybe I got the sides wrong. Let's look at the other sides: \( KL = 70 \), \( BA = 196 \). \( 70/196 = 5/14 \). \( CL = 112 - 72 = 40 \), \( CA = 112 \). \( 40/112 = 5/14 \). \( CK = 60 \), \( CB = 60 + 168 = 228 \)? No, that's not 5/14. Wait, maybe \( CB = 60 + 168 \) is wrong. Wait, the vertical side: \( CK = 60 \), \( CB = 60 + 168 \)? No, maybe the 168 is from \( K \) to \( B \), so \( CB = 60 + 168 = 228 \), but \( 60/228 = 5/19 \), not 5/14. Wait, maybe the vertical side is \( CB = 60 + 168 \) is incorrect. Wait, perhaps the diagram has \( CB = 60 + 168 \) as \( CK = 60 \), \( KB = 168 \), so \( CB = 228 \), \( BA = 196 \), \( KL = 70 \), \( CA = 112 \), \( CL = 112 - 72 = 40 \). Then \( KL/BA = 70/196 = 5/14 \), \( CL/CA = 40/112 = 5/14 \), and \( CK/CB = 60/228 = 5/19 \). That's not equal. Wait, maybe I misread the vertical side. Wait, maybe \( CB = 60 + 168 \) is wrong, and the vertical side is \( CB = 60 + 168 \) is actually \( CK = 60 \), \( CB = 60 + 168 = 228 \), but \( BA = 196 \), \( KL = 70 \), \( 70/196 = 5/14 \), \( CL = 112 - 72 = 40 \), \( 40/112 = 5/14 \), and \( CK = 60 \), \( CB = 60 + 168 = 228 \), \( 60/228 = 5/19 \). Not equal. Wait, maybe the vertical side is \( CB = 60 + 168 \) is wrong, and the 168 is \( CB \), and \( CK = 60 \), so \( KB = 168 - 60 = 108 \). Then \( CK/CB = 60/168 = 5/14 \), \( KL/BA = 70/196 = 5/14 \), \( CL/CA = (112 - 72)/112 = 40/112 = 5/14 \). Ah! That makes sense. So \( CB = 168 \), \( CK = 60 \)? No, \( CB = 168 \), \( CK = 60 \) would mean \( KB = 168 - 60 = 108 \), but \( BA = 196 \), \( KL = 70 \), \( 70/196 = 5/14 \), \( CK/CB = 60/168 = 5/14 \), \( CL/CA = (112 - 72)/112 = 40/112 = 5/14 \). Yes! So \( CL = 112 - 72 = 40 \), \( CA = 112 \), so \( CL/CA = 40/112 = 5/14 \), \( CK/CB = 60/168 = 5/14 \), \( KL/BA = 70/196 = 5/14 \). So by SSS similarity (all sides proportional), \( \triangle CBA \sim \triangle CKL \).

Step2: Identify Similar Triangle

Since the ratios of corresponding sides \( CK/CB = 60/168 = 5/14 \), \( CL/CA = 40/112 = 5/14 \), \( KL/BA = 70/196 = 5/14 \), the triangles \( \triangle CBA \) and \( \triangle CKL \) (where \( K \) is on \( CB \) and \( L \) is on \( CA \)) are similar by SSS similarity. So the similarity statement is \( \triangle CBA \sim \triangle CKL \).

Answer:

\( \triangle CKL \) (assuming \( K \) and \( L \) are the vertices of the smaller triangle, so \( \triangle CBA \sim \triangle CKL \))