Question

in the figure below, lines l and k are parallel. suppose that m∠4 = 46° and m∠5 = 80°. complete the statements below. by the angle - addition property, m∠4 + m∠2 + m∠5 = °. we are given that m∠4 = 46° and m∠5 = 80°, so m∠2 = °. we see that ∠4 and ∠1 are select and since lines l and k are parallel, ∠4 and ∠1 are select so, m∠1 = °. we see that ∠3 and ∠5 are select and since lines l and k are parallel, ∠3 and ∠5 are select so, m∠3 = °. therefore, m∠1 + m∠2 + m∠3 = °. the relationship between ∠1, ∠2, and ∠3 is an example of the following rule. the sum of the interior angle measures of a triangle is °.

Explanation:

Step1: Recall angle - addition property for a straight - line

The sum of angles on a straight - line is 180°. So, \(m\angle4 + m\angle2 + m\angle5=180^{\circ}\).

Step2: Solve for \(m\angle2\)

Given \(m\angle4 = 46^{\circ}\) and \(m\angle5 = 80^{\circ}\), we substitute into the equation \(m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\). Then \(m\angle2=180-(m\angle4 + m\angle5)=180-(46 + 80)=54^{\circ}\).

Step3: Identify the relationship between \(\angle4\) and \(\angle1\)

\(\angle4\) and \(\angle1\) are vertical angles. Vertical angles are equal. Also, since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle1\) are corresponding angles (by considering the transversal that forms these angles with \(l\) and \(k\)). So, \(m\angle1 = m\angle4=46^{\circ}\).

Step4: Identify the relationship between \(\angle3\) and \(\angle5\)

\(\angle3\) and \(\angle5\) are vertical angles. Vertical angles are equal. Also, since lines \(l\) and \(k\) are parallel, \(\angle3\) and \(\angle5\) are alternate - interior angles. So, \(m\angle3 = m\angle5 = 80^{\circ}\).

Step5: Calculate the sum of \(\angle1\), \(\angle2\), and \(\angle3\)

\(m\angle1 + m\angle2 + m\angle3=46 + 54+80 = 180^{\circ}\). The sum of the interior angle measures of a triangle is \(180^{\circ}\).

Answer:

\(m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\); \(m\angle2 = 54^{\circ}\); \(\angle4\) and \(\angle1\) are vertical angles and corresponding angles, \(m\angle1 = 46^{\circ}\); \(\angle3\) and \(\angle5\) are vertical angles and alternate - interior angles, \(m\angle3 = 80^{\circ}\); \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\); The sum of the interior angle measures of a triangle is \(180^{\circ}\)