Question

analyzing angle relationships of triangles
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 \(m\angle3 + m\angle4=180^{\circ}\)

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

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

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 the triangle. So \(m\angle2 + m\angle4 + m\angle6=180^{\circ}\) is True.

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

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 False.

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

There is no information given that \(\angle1\) and \(\angle2\) are complementary. So \(m\angle1 + m\angle2 = 90^{\circ}\) is False.

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

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. \(\angle2\) is not an exterior angle in a way that \(m\angle4 + m\angle6=m\angle2\). In fact, \(m\angle2+m\angle4 + m\angle6=180^{\circ}\) (sum of triangle angles). So \(m\angle4 + m\angle6=m\angle2\) is False.

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

By the exterior - angle theorem, \(m\angle2 + m\angle4=m\angle5\) (where \(\angle4\) and \(\angle6\) are related as \(m\angle4 + m\angle6+(m\angle2)=180^{\circ}\)). Since \(m\angle2 + m\angle4=m\angle5\) and \(m\angle4 + m\angle6+(m\angle2)=180^{\circ}\), \(m\angle2 + m\angle6=m\angle5\) is False.

Answer:

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