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: Calculate $m\angle2$

Given $m\angle4 = 46^{\circ}$ and $m\angle5 = 80^{\circ}$. We use the formula $m\angle2=180-(m\angle4 + m\angle5)$. Substitute the values: $m\angle2=180-(46 + 80)=180 - 126 = 54^{\circ}$.

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. Corresponding angles formed by parallel lines and a transversal are equal. 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 corresponding angles. Corresponding angles formed by parallel lines and a transversal are equal. So, $m\angle3 = m\angle5=80^{\circ}$.

Step5: Calculate $m\angle1 + m\angle2 + m\angle3$

$m\angle1 + m\angle2 + m\angle3=46+54 + 80=180^{\circ}$.

Step6: Recall the sum of interior angles of a triangle

The sum of the interior - angle measures of a triangle is 180°.

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 corresponding 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°.