Question

which two equations relate the measure of ∠4 to the measures of ∠1, ∠2, and/or ∠3.
a. m∠4 = m∠1 + m∠2
b. m∠4 = m∠2 + m∠3
c. m∠4 = 180° - (m∠1 + m∠2)
d. m∠4 = 180° - m∠3

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, \(m\angle1 + m\angle2+m\angle3=180^{\circ}\).

Step2: Recall linear - pair property

\(\angle3\) and \(\angle4\) form a linear - pair. Since the sum of angles in a linear - pair is 180°, \(m\angle3 + m\angle4=180^{\circ}\), which implies \(m\angle4 = 180^{\circ}-m\angle3\).

Step3: Substitute \(\angle3\) from angle - sum property

From \(m\angle1 + m\angle2+m\angle3=180^{\circ}\), we get \(m\angle3=180^{\circ}-(m\angle1 + m\angle2)\). Substituting this into \(m\angle4 = 180^{\circ}-m\angle3\), we have \(m\angle4=m\angle1 + m\angle2\).

Answer:

A. \(m\angle4 = m\angle1 + m\angle2\)
D. \(m\angle4 = 180^{\circ}-m\angle3\)