Question

consider $\triangle abc$ and $\triangle def$ shown below.

answer the following questions.
(a) in $\triangle abc$, $m\angle a = 22^\circ$ and $m\angle b = 50^\circ$.
in $\triangle def$, $m\angle d = 22^\circ$ and $m\angle e = 50^\circ$.
use this information to find the following angle measures.
$m\angle c = \square^\circ$
$m\angle f = \square^\circ$
(b) use the ruler to find all side lengths for each triangle.
assume all lengths are whole numbers.
$ab = \square$
$bc = \square$
$ca = \square$
$de = \square$
$ef = \square$
$fd = \square$
(c) find the following ratios.
write your answers in simplest form.
$\frac{ab}{de} = \square$
$\frac{bc}{ef} = \square$
$\frac{ca}{fd} = \square$
(d) three pairs of statements are shown below.
for each pair, select the true statement.
\bigcirc the angles in $\triangle abc$ have the same measures as the angles in $\triangle def$.
\bigcirc the angles in $\triangle abc$ do not have the same measures as the angles in $\triangle def$.
\bigcirc the ratios of corresponding side lengths are all equal to each other.
\bigcirc the ratios of corresponding side lengths are not all equal to each other.
\bigcirc the triangles are similar.
\bigcirc the triangles are not similar.

Explanation:

Step1: Calculate $\angle C$ (triangle sum)

The sum of angles in a triangle is $180^\circ$.
$m\angle C = 180^\circ - 22^\circ - 50^\circ = 108^\circ$

Step2: Calculate $\angle F$ (triangle sum)

The sum of angles in a triangle is $180^\circ$.
$m\angle F = 180^\circ - 22^\circ - 50^\circ = 108^\circ$

Step3: Measure sides (whole number estimates)

Using the ruler and whole number assumption:
$AB=2$, $BC=1$, $CA=2$
$DE=4$, $EF=2$, $FD=4$

Step4: Compute side ratios (simplify)

$\frac{AB}{DE} = \frac{2}{4} = \frac{1}{2}$
$\frac{BC}{EF} = \frac{1}{2}$
$\frac{CA}{FD} = \frac{2}{4} = \frac{1}{2}$

Step5: Evaluate similarity statements

• Angle measures match, so first true statement: The angles in $\triangle ABC$ have the same measures as the angles in $\triangle DEF$.
• Ratios are all equal to $\frac{1}{2}$, so second true statement: The ratios of corresponding side lengths are all equal to each other.
• Equal angles + proportional sides mean triangles are similar, so third true statement: The triangles are similar.

Answer:

(a) $m\angle C = 108^\circ$
$m\angle F = 108^\circ$

(b) $AB=2$, $BC=1$, $CA=2$
$DE=4$, $EF=2$, $FD=4$

(c) $\frac{AB}{DE} = \frac{1}{2}$, $\frac{BC}{EF} = \frac{1}{2}$, $\frac{CA}{FD} = \frac{1}{2}$

(d)

• The angles in $\triangle ABC$ have the same measures as the angles in $\triangle DEF$.
• The ratios of corresponding side lengths are all equal to each other.
• The triangles are similar.