Question

answer the following questions.
(a) in △abc, m∠a = 41° and m∠b = 56°.
in △def, m∠d = 41° and m∠e = 56°.
use this information to find the following angle measures.
m∠c = 83°
m∠f = 83°

Explanation:

Step1: Recall the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle ABC\), we know that \(m\angle A + m\angle B+m\angle C=180^{\circ}\). Given \(m\angle A = 41^{\circ}\) and \(m\angle B = 56^{\circ}\).
Substitute the values into the formula: \(m\angle C=180^{\circ}-(m\angle A + m\angle B)\).

$$m\angle C=180^{\circ}-(41^{\circ}+ 56^{\circ})$$

$$m\angle C=180^{\circ}-97^{\circ}$$

$$m\angle C = 83^{\circ}$$

Step2: Apply the triangle angle - sum theorem to \(\triangle DEF\)

For \(\triangle DEF\), \(m\angle D + m\angle E+m\angle F=180^{\circ}\). Given \(m\angle D = 41^{\circ}\) and \(m\angle E = 56^{\circ}\).
Substitute the values into the formula: \(m\angle F=180^{\circ}-(m\angle D + m\angle E)\).

$$m\angle F=180^{\circ}-(41^{\circ}+ 56^{\circ})$$

$$m\angle F=180^{\circ}-97^{\circ}$$

$$m\angle F = 83^{\circ}$$

Answer:

\(m\angle C = 83^{\circ}\), \(m\angle F = 83^{\circ}\)