Question

in the figure below, lines l and k are parallel. suppose that m∠5 = 78° and m∠4 = 45°. complete the statements below. by the angle - addition property, m∠4 + m∠2 + m∠5 = °. we are given that m∠4 = 45° and m∠5 = 78°, 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

The sum of angles around a point is $360^{\circ}$. But for a straight - line, the sum of adjacent angles is $180^{\circ}$. Since $\angle4$, $\angle2$, and $\angle5$ are adjacent angles on a straight - line, $m\angle4 + m\angle2 + m\angle5=180^{\circ}$.

Step2: Calculate $m\angle2$

We know that $m\angle4 = 45^{\circ}$ and $m\angle5 = 78^{\circ}$. Using the equation $m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$, we can solve for $m\angle2$ as follows:

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Step3: Analyze the relationship between $\angle4$ and $\angle1$

$\angle4$ and $\angle1$ are non - adjacent angles formed by a transversal intersecting two parallel lines $l$ and $k$. $\angle4$ and $\angle1$ are alternate interior angles. Since lines $l$ and $k$ are parallel, alternate interior angles are congruent. So $m\angle1 = m\angle4=45^{\circ}$.

Step4: Analyze the relationship between $\angle3$ and $\angle5$

$\angle3$ and $\angle5$ are non - adjacent angles formed by a transversal intersecting two parallel lines $l$ and $k$. $\angle3$ and $\angle5$ are corresponding angles. Since lines $l$ and $k$ are parallel, corresponding angles are congruent. So $m\angle3 = m\angle5 = 78^{\circ}$.

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

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Answer:

By the angle addition property, $m\angle4 + m\angle2 + m\angle5 = 180^{\circ}$.
We are given that $m\angle4 = 45^{\circ}$ and $m\angle5 = 78^{\circ}$, so $m\angle2 = 57^{\circ}$.
We see that $\angle4$ and $\angle1$ are alternate interior angles.
And since lines $l$ and $k$ are parallel, $\angle4$ and $\angle1$ are congruent. So, $m\angle1 = 45^{\circ}$.
We see that $\angle3$ and $\angle5$ are corresponding angles.
And since lines $l$ and $k$ are parallel, $\angle3$ and $\angle5$ are congruent. So, $m\angle3 = 78^{\circ}$.
Therefore, $m\angle1 + m\angle2 + m\angle3 = 180^{\circ}$.
The relationship between $\angle1$, $\angle2$, and $\angle3$ is an example of the following rule. The sum of the interior angle measures of a triangle is $180^{\circ}$.