Question

triangle abc and triangle def are similar. in triangle abc, m∠a = 82°. in triangle def, m∠f = 67°. what are m∠e and m∠g? m∠e = □° m∠g = □°

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°.

Step2: Since the triangles are similar, corresponding angles are equal

Let's assume that \(\angle A\) corresponds to \(\angle D\), \(\angle B\) corresponds to \(\angle E\), and \(\angle C\) corresponds to \(\angle F\).
We know that in \(\triangle DEF\), we want to find \(\angle E\). Using the angle - sum property of a triangle: \(m\angle D + m\angle E+m\angle F=180^{\circ}\).
We know \(m\angle F = 67^{\circ}\). And since the triangles are similar, if we assume the correspondence as above, we first note that we are not given information about \(\angle D\) directly, but we know the sum of angles in a triangle.
We know that in \(\triangle ABC\), \(m\angle A = 82^{\circ}\). If \(\angle A\) corresponds to \(\angle D\), then \(m\angle D=82^{\circ}\).
So, \(m\angle E=180^{\circ}-m\angle D - m\angle F\).
Substitute \(m\angle D = 82^{\circ}\) and \(m\angle F = 67^{\circ}\) into the formula:
\[m\angle E=180-(82 + 67)=180 - 149=31^{\circ}\]

Step3: Find \(m\angle G\)

\(\angle G\) and \(\angle E\) are supplementary (a linear - pair). Since the sum of angles in a linear - pair is 180°, if \(m\angle E = 31^{\circ}\), then \(m\angle G=180 - m\angle E\).
\[m\angle G=180 - 31=149^{\circ}\]

Answer:

\(m\angle E = 31\)
\(m\angle G = 149\)