Question

which statements are always true regarding the diagram? check all that apply

m∠3 + m∠4 = 180°

m∠2 + m∠4 + m∠6 = 180°

m∠2 + m∠4 = m∠5

m∠1 + m∠2 = 90°

m∠4 + m∠6 = m∠2

m∠2 + m∠6 = m∠5

Explanation:

Step1: Analyze \(m\angle3 + m\angle4 = 180^{\circ}\)

\(\angle3\) and \(\angle4\) are adjacent angles forming a linear - pair. By the linear - pair postulate, if two angles form a linear pair, then their measures add up to \(180^{\circ}\). So \(m\angle3 + m\angle4=180^{\circ}\) is always true.

Step2: Analyze \(m\angle2 + m\angle4 + m\angle6 = 180^{\circ}\)

In a triangle, the sum of the interior angles of a triangle is \(180^{\circ}\). The angles \(\angle2\), \(\angle4\), and \(\angle6\) are the interior angles of a triangle. So \(m\angle2 + m\angle4 + m\angle6 = 180^{\circ}\) is always true.

Step3: Analyze \(m\angle2 + m\angle6=m\angle5\)

By the exterior - angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. \(\angle5\) is an exterior angle of the triangle with non - adjacent interior angles \(\angle2\) and \(\angle6\). So \(m\angle2 + m\angle6=m\angle5\) is always true.

Step4: Analyze \(m\angle2 + m\angle4=m\angle5\)

Since \(m\angle5=m\angle2 + m\angle6\) (from the exterior - angle theorem) and \(m\angle4
eq m\angle6\) (in general, there is no information to suggest they are equal), \(m\angle2 + m\angle4=m\angle5\) is not always true.

Step5: Analyze \(m\angle1 + m\angle2 = 90^{\circ}\)

There is no information in the diagram to suggest that \(\angle1\) and \(\angle2\) are complementary. They are just two angles of a polygon (not necessarily a right - angled polygon), so \(m\angle1 + m\angle2 = 90^{\circ}\) is not always true.

Step6: Analyze \(m\angle4 + m\angle6=m\angle2\)

From the sum of angles in a triangle \(m\angle2 + m\angle4 + m\angle6 = 180^{\circ}\), if \(m\angle4 + m\angle6=m\angle2\), then \(2m\angle2=180^{\circ}\) (substituting \(m\angle4 + m\angle6\) with \(m\angle2\) in \(m\angle2 + m\angle4 + m\angle6 = 180^{\circ}\)), which would mean \(m\angle2 = 90^{\circ}\). But there is no information to suggest that \(\angle2\) is a right - angle. So \(m\angle4 + m\angle6=m\angle2\) is not always true.

Answer:

• \(m\angle3 + m\angle4=180^{\circ}\)
• \(m\angle2 + m\angle4 + m\angle6 = 180^{\circ}\)
• \(m\angle2 + m\angle6=m\angle5\)