Question

in the figure below, lines m and n are parallel, m∠2 = 62° and m∠3 = 73°. (a) part a what is m∠5? justify your reasoning using mathematical vocabulary. (b) part b based on your answer to part a, what is m∠4? justify your reasoning using mathematical vocabulary.

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. In the triangle formed by the intersection of the transversal and the parallel lines, if we consider the angles related to ∠2, ∠3 and ∠5. Let's assume the triangle has angles ∠2, ∠3 and ∠5. Then \(m\angle2 + m\angle3+m\angle5=180^{\circ}\).

Step2: Substitute the given values

We know that \(m\angle2 = 62^{\circ}\) and \(m\angle3 = 73^{\circ}\). Substituting these values into the equation \(62^{\circ}+73^{\circ}+m\angle5 = 180^{\circ}\).

Step3: Solve for \(m\angle5\)

First, add 62 and 73: \(62 + 73=135\). Then, \(m\angle5=180^{\circ}-(62^{\circ}+73^{\circ})=180 - 135=45^{\circ}\).

Step4: For part B, use corresponding - angles property

Since lines \(m\) and \(n\) are parallel, ∠4 and ∠5 are corresponding angles. Corresponding angles formed by a transversal intersecting two parallel lines are congruent. So \(m\angle4 = m\angle5\).

Answer:

(a) \(m\angle5 = 45^{\circ}\), because the sum of the interior angles of a triangle is 180°, and by substituting the given values of ∠2 and ∠3 into the angle - sum formula \(m\angle2 + m\angle3+m\angle5 = 180^{\circ}\), we get \(m\angle5=45^{\circ}\).
(b) \(m\angle4 = 45^{\circ}\), because ∠4 and ∠5 are corresponding angles and corresponding angles formed by a transversal intersecting two parallel lines are congruent.