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: Check linear - pair relationship

Angles \(∠3\) and \(∠4\) form a linear pair. By the linear - pair postulate, \(m∠3 + m∠4=180^{\circ}\).

Step2: Use triangle - angle sum property

In a triangle, the sum of interior angles is \(180^{\circ}\). But \(∠2\), \(∠4\), and \(∠6\) are not all interior angles of the same triangle.

Step3: Use exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
For the triangle with angles \(∠2\), \(∠4\), and the third angle (let's call it \(x\)), \(∠5\) is an exterior angle. \(m∠5=m∠2 + m∠4\) (because \(m∠5=m∠x + m∠4\) and \(m∠x=m∠2\) by vertical angles or parallel - line properties if we assume the lines are parallel as per the arrow - head markings).

Step4: Analyze \(m∠1 + m∠2\)

There is no information to suggest that \(∠1\) and \(∠2\) are complementary (\(m∠1 + m∠2 = 90^{\circ}\)). They could be supplementary if the lines are parallel (but not necessarily \(90^{\circ}\)).

Step5: Analyze \(m∠4 + m∠6=m∠2\)

This does not follow from the triangle angle sum or exterior - angle theorem.

Step6: Analyze \(m∠2 + m∠6=m∠5\)

By the exterior - angle theorem, \(m∠5=m∠2 + m∠4\) (not \(m∠2 + m∠6\))

Answer:

\(m∠3 + m∠4 = 180^{\circ}\), \(m∠2 + m∠4=m∠5\)