Question

consider △abc and △def shown below.
answer the following questions.
(a) in △abc, m∠a = 40° and m∠b = 70°.
in △def, m∠d = 40° and m∠e = 70°.
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 \(m\angle C\) and \(m\angle F\)

The sum of angles in a triangle is \(180^{\circ}\).
For \(\triangle ABC\):
\(m\angle C=180^{\circ}-(m\angle A + m\angle B)\)
\(=180^{\circ}-(40^{\circ}+70^{\circ})\)
\(=180^{\circ}-110^{\circ}\)
\(=70^{\circ}\)
For \(\triangle DEF\):
\(m\angle F=180^{\circ}-(m\angle D + m\angle E)\)
\(=180^{\circ}-(40^{\circ}+70^{\circ})\)
\(=180^{\circ}-110^{\circ}\)
\(=70^{\circ}\)

Step2: Assume side - length measurement (hypothetical as no ruler data, but if we assume similar - triangle side - ratio concept)

Let's assume after measuring (in a real - case with a ruler):
Suppose \(AB = 6\), \(BC = 8\), \(CA = 10\) (a \(6 - 8 - 10\) triangle)
Since \(\triangle ABC\sim\triangle DEF\) (by AA similarity, \(\angle A=\angle D = 40^{\circ}\), \(\angle B=\angle E = 70^{\circ}\)), if \(DE = 3\), \(EF = 4\), \(FD = 5\)

Step3: Calculate ratios

\(\frac{AB}{DE}=\frac{6}{3}=2\)
\(\frac{BC}{EF}=\frac{8}{4}=2\)
\(\frac{CA}{FD}=\frac{10}{5}=2\)

Step4: Analyze statements

• For angles: Since \(m\angle A=m\angle D = 40^{\circ}\), \(m\angle B=m\angle E = 70^{\circ}\), \(m\angle C=m\angle F = 70^{\circ}\), the angles in \(\triangle ABC\) have the same measures as the angles in \(\triangle DEF\)
• For side - length ratios: If \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}\), the ratios of corresponding side lengths are all equal to each other
• Since angles are equal and side - length ratios are equal, the triangles are similar

Answer:

a. \(m\angle C = 70^{\circ}\), \(m\angle F = 70^{\circ}\)
b. (Hypothetical example) \(AB = 6\), \(BC = 8\), \(CA = 10\), \(DE = 3\), \(EF = 4\), \(FD = 5\)
c. \(\frac{AB}{DE}=2\), \(\frac{BC}{EF}=2\), \(\frac{CA}{FD}=2\)
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