Question

please find the measure of the missing angles. draw a picture to help organize your thoughts.

1. the m∠a is complementary to the m∠b. the m∠c is complementary to the m∠b. if m∠a = 62°, what is the m∠b and the m∠c?
2. the m∠d is supplementary to the m∠e. the m∠f is supplementary to the m∠e. if m∠f = 113°, what is the m∠d and the m∠e?

Explanation:

Problem 16

Step1: Recall complementary angles sum

Complementary angles sum to \( 90^\circ \). So, if \( m\angle A \) and \( m\angle B \) are complementary, \( m\angle A + m\angle B = 90^\circ \).
Given \( m\angle A = 62^\circ \), then \( m\angle B = 90^\circ - 62^\circ \)

Step2: Calculate \( m\angle B \)

\( m\angle B = 90 - 62 = 28^\circ \)

Step3: Recall \( m\angle C \) is complementary to \( m\angle B \)

So, \( m\angle B + m\angle C = 90^\circ \), then \( m\angle C = 90^\circ - m\angle B \)

Step4: Calculate \( m\angle C \)

Substitute \( m\angle B = 28^\circ \), \( m\angle C = 90 - 28 = 62^\circ \)

Step1: Recall supplementary angles sum

Supplementary angles sum to \( 180^\circ \). \( \angle D \) and \( \angle E \) are supplementary, so \( m\angle D + m\angle E = 180^\circ \). \( \angle F \) and \( \angle E \) are supplementary, so \( m\angle F + m\angle E = 180^\circ \).

Step2: Analyze equal angles

From \( m\angle D + m\angle E = 180^\circ \) and \( m\angle F + m\angle E = 180^\circ \), we can see \( m\angle D = m\angle F \) (since both equal \( 180^\circ - m\angle E \)). But we know \( m\angle F = 113^\circ \), so \( m\angle D = 113^\circ \).

Step3: Calculate \( m\angle E \)

Using \( m\angle F + m\angle E = 180^\circ \), \( m\angle E = 180^\circ - m\angle F \)

Step4: Substitute \( m\angle F \)

\( m\angle E = 180 - 113 = 67^\circ \)

Answer:

\( m\angle B = 28^\circ \), \( m\angle C = 62^\circ \)

Problem 17