Question

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

Explanation:

Step1: Recall angle - pair relationships

When two lines intersect, vertical angles are congruent and adjacent angles are supplementary.

Step2: Analyze \(m\angle1 + m\angle4=180\)

\(\angle1\) and \(\angle4\) are adjacent angles formed by two intersecting lines. Adjacent angles on a straight - line sum to 180 degrees. So \(m\angle1 + m\angle4 = 180\) is always true.

Step3: Analyze \(m\angle1 + m\angle2 + m\angle3=180\)

The sum of angles around a point is 360 degrees. \(\angle1+\angle2+\angle3+\angle4 = 360\) and \(\angle1 + \angle4=180\), \(\angle2+\angle3 = 180\). So \(m\angle1 + m\angle2 + m\angle3=180\) is never true.

Step4: Analyze \(m\angle2 + m\angle4=180\)

\(\angle2\) and \(\angle4\) are vertical angles. Vertical angles are congruent. They are equal in measure. Only when \(\angle2=\angle4 = 90^{\circ}\) (i.e., the two intersecting lines are perpendicular), \(m\angle2 + m\angle4=180\). So \(m\angle2 + m\angle4 = 180\) is sometimes true.

Step5: Analyze \(\angle2\cong\angle3\)

\(\angle2\) and \(\angle3\) are adjacent angles. They are supplementary (\(m\angle2 + m\angle3=180\)). They are congruent only when \(m\angle2=m\angle3 = 90^{\circ}\). So \(\angle2\cong\angle3\) is sometimes true.

Step6: Analyze \(\angle2\cong\angle4\)

\(\angle2\) and \(\angle4\) are vertical angles. Vertical angles are always congruent. So \(\angle2\cong\angle4\) is always true.

Step7: Analyze \(m\angle3=m\angle4\)

\(\angle3\) and \(\angle4\) are adjacent angles. They are supplementary (\(m\angle3 + m\angle4=180\)). They are equal only when \(m\angle3=m\angle4 = 90^{\circ}\). So \(m\angle3=m\angle4\) is sometimes true.

Answer:

The statement \(m\angle2 + m\angle4 = 180\) is sometimes true.
The statement \(\angle2\cong\angle3\) is sometimes true.
The statement \(\angle2\cong\angle4\) is always true.
The statement \(m\angle3=m\angle4\) is sometimes true.