Question

consider the figure shown. classify each of the following statements as always true, sometimes true, or never true. the statement m∠1 + m∠4 = 180 is
m∠1 + m∠4 = 180
m∠1 + m∠2 + m∠3 = 180
m∠2 + m∠4 = 180
∠2 ≅ ∠3
∠2 ≅ ∠4
m∠3 = m∠4

Explanation:

Step1: Recall angle - relationship in intersecting lines

When two lines intersect, vertical angles are congruent and linear - pair angles are supplementary.

Step2: Analyze $\angle1$ and $\angle4$

$\angle1$ and $\angle4$ form a linear pair. By the definition of a linear pair of angles (two adjacent angles whose non - common sides are opposite rays), the sum of the measures of two angles in a linear pair is always 180 degrees. So, $m\angle1 + m\angle4=180$ is always true.

Step3: Analyze $\angle2\cong\angle3$ and $\angle2\cong\angle4$

$\angle2$ and $\angle3$ are vertical angles, so $\angle2\cong\angle3$ is always true. $\angle2$ and $\angle4$ are not vertical angles and there is no general rule that makes them congruent, so $\angle2\cong\angle4$ is sometimes true.

Step4: Analyze $m\angle2 + m\angle4=180$

$\angle2$ and $\angle4$ are not a linear pair. There is no general rule that makes their sum 180 degrees, so $m\angle2 + m\angle4 = 180$ is sometimes true.

Step5: Analyze $m\angle1 + m\angle2+m\angle3=180$

The sum of the measures of angles around a point is 360 degrees. $\angle1+\angle2+\angle3+\angle4 = 360$. Since $\angle1 + \angle4=180$ and $\angle2+\angle3$ is not necessarily 180, $m\angle1 + m\angle2+m\angle3 = 180$ is sometimes true.

Step6: Analyze $m\angle3=m\angle4$

$\angle3$ and $\angle4$ are not vertical angles and there is no general rule that makes them congruent, so $m\angle3=m\angle4$ is sometimes true.

Answer:

• $m\angle1 + m\angle4 = 180$: Always true
• $\angle2\cong\angle3$: Always true
• $\angle2\cong\angle4$: Sometimes true
• $m\angle2 + m\angle4=180$: Sometimes true
• $m\angle1 + m\angle2+m\angle3 = 180$: Sometimes true
• $m\angle3=m\angle4$: Sometimes true