Question

which statements are always true regarding the diagram? check all that apply.
$m\angle 3 + m\angle 4 = 180^{circ}$
$m\angle 2 + m\angle 4 + m\angle 6 = 180^{circ}$
$m\angle 2 + m\angle 4 = m\angle 5$
$m\angle 1 + m\angle 2 = 90^{circ}$
$m\angle 4 + m\angle 6 = m\angle 2$
$m\angle 2 + m\angle 6 = m\angle 5$

Explanation:

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

\(\angle3\) and \(\angle4\) are adjacent and form a linear pair. By the linear - pair postulate, if two angles form a linear pair, then they are supplementary. So \(m\angle3 + m\angle4 = 180^{\circ}\) is True.

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

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

Step3: Analyze \(m\angle2 + m\angle4=m\angle5\)

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 \(\angle4\). So \(m\angle2 + m\angle4=m\angle5\) is True.

Step4: Analyze \(m\angle1 + m\angle2 = 90^{\circ}\)

There is no information given that \(\angle1\) and \(\angle2\) are complementary. They are adjacent, but we don't know if their sum is \(90^{\circ}\). So \(m\angle1 + m\angle2 = 90^{\circ}\) is False.

Step5: Analyze \(m\angle4 + m\angle6=m\angle2\)

From the triangle interior - angle sum (\(m\angle2 + m\angle4 + m\angle6=180^{\circ}\)), if \(m\angle4 + m\angle6=m\angle2\), then \(2m\angle2=180^{\circ}\), \(m\angle2 = 90^{\circ}\), which is not necessarily True. So \(m\angle4 + m\angle6=m\angle2\) is False.

Step6: Analyze \(m\angle2 + m\angle6=m\angle5\)

By the exterior - angle theorem, \(m\angle5=m\angle2 + m\angle4
eq m\angle2 + m\angle6\) (since \(m\angle4
eq m\angle6\) in general). So \(m\angle2 + m\angle6=m\angle5\) is False.

Answer:

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