Question

lines ab and cd are parallel. find the measures of the three angles in triangle afb. enter your answers in the boxes below. m∠a: ° m∠f: ° m∠b: °

Explanation:

Step1: Find the angle adjacent to 157°

The adjacent - angle to 157° forms a linear - pair. Let this angle be \(x\). Since the sum of angles in a linear - pair is 180°, we have \(x = 180^{\circ}-157^{\circ}=23^{\circ}\).

Step2: Use the property of parallel lines

Since \(AB\parallel CD\), \(\angle A\) and the 42° angle are corresponding angles. So \(m\angle A = 42^{\circ}\).

Step3: Use the angle - sum property of a triangle

In \(\triangle AFB\), we know that the sum of the interior angles of a triangle is 180°. We already know one angle is \(m\angle A = 42^{\circ}\), and the angle corresponding to the 23° angle (alternate interior angles for parallel lines \(AB\) and \(CD\)) is also 23°. Let \(\angle F\) be the third angle of the triangle.
We use the formula \(m\angle A+m\angle B + m\angle F=180^{\circ}\).
We know \(m\angle A = 42^{\circ}\), and \(m\angle B = 23^{\circ}\). Then \(m\angle F=180^{\circ}-(42^{\circ}+23^{\circ}) = 115^{\circ}\).

Answer:

\(m\angle A:42^{\circ}\)
\(m\angle B:23^{\circ}\)
\(m\angle F:115^{\circ}\)