Question

(a) in \\( \triangle a b c, m \angle a = 41 ^ { \circ } \\) and \\( m \angle b = 56 ^ { \circ } \\).
in \\( \triangle d e f, m \angle d = 41 ^ { \circ } \\) and \\( m \angle e = 56 ^ { \circ } \\).
use this information to find the following angle measures.
\\( m \angle c = 83 ^ { \circ } \\)
\\( m \angle f = 83 ^ { \circ } \\)
(b) use the ruler to find all side lengths for each triangle.
assume all lengths are whole numbers.
\\( a b = 97 \\)
\\( b c = 139 \\)
\\( c a = 124 \\)
\\( d e = 97 \\)
\\( e f = \square \\)
\\( f d = \square \\)
(c) find the following ratios.
write your answers in simplest form
\\( \frac { a b } { d e } = \square \\)
\\( \frac { b c } { e f } = \square \\)
\\( \frac { c a } { f d } = \square \\)
(d) three pairs of statements are shown below.
for each pair, select the true statement.
\\( \bigcirc \\) the angles in \\( \triangle a b c \\) have the same measures as the angles in \\( \triangle d e f \\).
\\( \bigcirc \\) the angles in \\( \triangle a b c \\) do not have the same measures as the angles in \\( \triangle d e f \\).

Explanation:

Step1: Determine the relationship between the triangles

Since \(m\angle A = m\angle D = 41^{\circ}\) and \(m\angle B=m\angle E = 56^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle DEF\). For similar triangles, the ratios of corresponding sides are equal.

Step2: Find the ratios

• For \(\frac{AB}{DE}\), since \(AB = 97\) and \(DE=97\), \(\frac{AB}{DE}=\frac{97}{97} = 1\).
• Because \(\triangle ABC\sim\triangle DEF\), \(\frac{BC}{EF}=\frac{AB}{DE}\) and \(\frac{CA}{FD}=\frac{AB}{DE}\).

Answer:

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