Question

(note that figures are not necessarily drawn to scale.)
answer
the triangles are similar because
sss: three sides proportionate
sss: three sides congruent
sas: two sides proportionate, included angle congruent
sas: two sides + included angle congruent
aa: two angles congruent

Explanation:

Step1: Check triangle sides

First, identify the side lengths of both triangles. For triangle \( CED \): sides are \( 20 \), \( 27 \), \( 31 \)? Wait, no, wait—wait, the first triangle: \( CE = 20 \), \( CD = 27 \), \( ED = 31 \)? Wait, no, wait the second triangle: \( CI = 155 \)? Wait, no, the second triangle has sides \( 100 \), \( 135 \), \( 155 \)? Wait, let's check the ratios.

Wait, first triangle: sides \( 20 \), \( 27 \), \( 31 \)? Wait no, wait the first triangle: \( CE = 20 \), \( CD = 27 \), \( ED = 31 \)? Wait, no, maybe I misread. Wait the second triangle: sides \( 100 \), \( 135 \), \( 155 \). Let's check the ratios: \( 100 / 20 = 5 \), \( 135 / 27 = 5 \), \( 155 / 31 = 5 \)? Wait, no, \( 31 \times 5 = 155 \), \( 27 \times 5 = 135 \), \( 20 \times 5 = 100 \)? Wait, no, the second triangle's sides: wait the first triangle: \( CE = 20 \), \( CD = 27 \), \( ED = 31 \)? Wait, no, the first triangle: \( CE = 20 \), \( CD = 27 \), \( ED = 31 \)? Wait, the second triangle: \( IF = 100 \), \( CF = 135 \), \( CI = 155 \)? Wait, no, maybe the first triangle is \( CED \) with \( CE = 20 \), \( CD = 27 \), \( ED = 31 \), and the second triangle is \( CIF \) with \( CI = 155 \), \( CF = 135 \), \( IF = 100 \)? Wait, no, let's compute the ratios:

\( 100 / 20 = 5 \), \( 135 / 27 = 5 \), \( 155 / 31 = 5 \). Wait, \( 20 \times 5 = 100 \), \( 27 \times 5 = 135 \), \( 31 \times 5 = 155 \). So all three sides are in proportion (ratio 5). So the SSS similarity criterion (three sides proportionate) applies.

Step2: Determine the similarity criterion

Since all three corresponding sides are proportional (ratio \( 5 \) in this case), the SSS (three sides proportionate) criterion for similarity is met.

Answer:

The triangles are similar because SSS: three sides proportionate.