Question

which statements are always true regarding the diagram? select three options.
$m\angle5 + m\angle3 = m\angle4$
$m\angle3 + m\angle4 + m\angle5 = 180^{\circ}$
$m\angle5 + m\angle6 = 180^{\circ}$
$m\angle2 + m\angle3 = m\angle6$
$m\angle2 + m\angle3 + m\angle5 = 180^{\circ}$

Explanation:

Step1: Analyze the first option

By the exterior - angle theorem, \(m\angle5 + m\angle3=m\angle4\) is incorrect. The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Here, \(\angle4\) is not an exterior angle formed by \(\angle3\) and \(\angle5\).

Step2: Analyze the second option

\(m\angle3 + m\angle4 + m\angle5 = 180^{\circ}\) is incorrect. In a triangle, the sum of interior angles is \(180^{\circ}\). But \(\angle3\), \(\angle4\), and \(\angle5\) are not the interior angles of a single triangle.

Step3: Analyze the third option

Since \(\angle5\) and \(\angle6\) are adjacent angles forming a linear pair. By the linear - pair postulate, \(m\angle5 + m\angle6=180^{\circ}\).

Step4: Analyze the fourth option

By the exterior - angle theorem, for the triangle with interior angles \(\angle2\) and \(\angle3\), the exterior angle \(\angle6\) is equal to the sum of the two non - adjacent interior angles. So \(m\angle2 + m\angle3=m\angle6\).

Step5: Analyze the fifth option

In the triangle with interior angles \(\angle2\), \(\angle3\), and the angle adjacent to \(\angle5\) (which is \(180 - m\angle5\)). By the triangle - angle sum theorem, \(m\angle2 + m\angle3+(180 - m\angle5)=180\), which simplifies to \(m\angle2 + m\angle3=m\angle5\). Also, since \(m\angle2 + m\angle3=m\angle6\) and \(m\angle5 + m\angle6 = 180^{\circ}\), substituting \(m\angle6=m\angle2 + m\angle3\) into \(m\angle5 + m\angle6 = 180^{\circ}\) gives \(m\angle2 + m\angle3 + m\angle5=180^{\circ}\).

Answer:

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