Question

de = 6 ef = 4 fd = 5
(c) find the following ratios.
write your answers in simplest form.
ab/de = 2 bc/ef = 2 ca/fd = 2
(d) three pairs of statements are shown below.
for each pair, select the true statement.
the angles in △abc have the same measures as the angles in △def.
the angles in △abc do not have the same measures as the angles in △def.
the ratios of corresponding side lengths are all equal to each other.
the ratios of corresponding side lengths are not all equal to each other.
the triangles are similar.
the triangles are not similar.

Explanation:

Step1: Check angle measures

For similar triangles, corresponding angles are equal. Since the ratios of corresponding side lengths are equal (from part c, \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD} = 2\)), by the Side - Side - Side (SSS) similarity criterion, \(\triangle ABC\sim\triangle DEF\). So, the angles in \(\triangle ABC\) have the same measures as the angles in \(\triangle DEF\).

Step2: Check side - length ratios

From part (c), \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD} = 2\). So, the ratios of corresponding side lengths are all equal to each other.

Step3: Check similarity

Since the ratios of all corresponding side lengths are equal (\(\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD} = 2\)), by the SSS similarity criterion, \(\triangle ABC\) and \(\triangle DEF\) are similar.

Answer:

1. The angles in \(\triangle ABC\) have the same measures as the angles in \(\triangle DEF\).
2. The ratios of corresponding side lengths are all equal to each other.
3. The triangles are similar.