Question

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 = 94° m∠f = 94° (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.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is $180^{\circ}$. For $\triangle ABC$, we have $m\angle A + m\angle B+m\angle C=180^{\circ}$. Given $m\angle A = 30^{\circ}$ and $m\angle B = 56^{\circ}$, then $m\angle C=180^{\circ}-(30^{\circ}+ 56^{\circ})=94^{\circ}$. Similarly, for $\triangle DEF$ with $m\angle D = 30^{\circ}$ and $m\angle E = 56^{\circ}$, $m\angle F=180^{\circ}-(30^{\circ}+56^{\circ}) = 94^{\circ}$.

Step2: Measure side - lengths (not possible in this text - based response)

Since we are not given actual ruler - measurement capabilities here, we assume we would measure the lengths of the sides $AB$, $BC$, $CA$, $DE$, $EF$, and $FD$ with a ruler in a real - world scenario. Let's say in a real measurement, $AB = a$, $BC = b$, $CA = c$, $DE = d$, $EF = e$, $FD = f$.

Step3: Calculate ratios

The ratios are $\frac{AB}{DE}=\frac{a}{d}$, $\frac{BC}{EF}=\frac{b}{e}$, $\frac{CA}{FD}=\frac{c}{f}$. In the case of similar triangles (since $\angle A=\angle D$, $\angle B=\angle E$, $\angle C=\angle F$), these ratios are equal.

Answer:

(a) $m\angle C = 94^{\circ}$, $m\angle F = 94^{\circ}$
(b) Without actual ruler - measurement, we cannot provide values.
(c) Without actual side - length values, we cannot provide values.
(d) No statements are shown for part (d) so no answer can be given.