Question

consider △abc and △def shown below. answer the following questions. (a) in △abc, m∠a = 30° and m∠b = 56°. in △def, m∠d = 30° and m∠e = 56°. use this information to find the following angle measures. m∠c = ° m∠f = ° (b) use the ruler to find all side - lengths for each triangle. assume all lengths are whole numbers. ab = bc = ca = de = ef = fd = (c) find the following ratios. write your answers in simplest form. $\frac{ab}{de}=$ $\frac{bc}{ef}=$ $\frac{ca}{fd}=$ (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: Find angle C in △ABC

The sum of angles in a triangle is 180°. So $m\angle C=180^{\circ}-m\angle A - m\angle B$. Substituting $m\angle A = 30^{\circ}$ and $m\angle B=56^{\circ}$, we get $m\angle C=180^{\circ}-30^{\circ}-56^{\circ}=94^{\circ}$.

Step2: Find angle F in △DEF

Similarly, for △DEF with $m\angle D = 30^{\circ}$ and $m\angle E = 56^{\circ}$, $m\angle F=180^{\circ}-m\angle D - m\angle E=180^{\circ}-30^{\circ}-56^{\circ}=94^{\circ}$.

Step3: Side - length measurement (not possible to calculate without actual ruler measurement, assume we have measured them and got values)

Let's assume we measure the sides with a ruler and get: $AB = a$, $BC = b$, $CA = c$, $DE = d$, $EF = e$, $FD = f$.

Step4: Calculate side - length ratios

$\frac{AB}{DE}=\frac{a}{d}$, $\frac{BC}{EF}=\frac{b}{e}$, $\frac{CA}{FD}=\frac{c}{f}$

Step5: Analyze triangle similarity

Since $m\angle A=m\angle D = 30^{\circ}$, $m\angle B=m\angle E = 56^{\circ}$ and $m\angle C=m\angle F = 94^{\circ}$, the angles in △ABC have the same measures as the angles in △DEF. Also, if the ratios of corresponding side - lengths $\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$, the triangles are similar.

Answer:

(a) $m\angle C = 94^{\circ}$, $m\angle F = 94^{\circ}$
(b) (Values depend on actual ruler - measurement, assume $AB = a$, $BC = b$, $CA = c$, $DE = d$, $EF = e$, $FD = f$)
(c) $\frac{AB}{DE}=\frac{a}{d}$, $\frac{BC}{EF}=\frac{b}{e}$, $\frac{CA}{FD}=\frac{c}{f}$
(d) The angles in △ABC have the same measures as the angles in △DEF. If the side - length ratios are equal, the triangles are similar.