Question

analyzing angle relationships of triangles
which statements are always true regarding the diagram? choose three
correct answers.
$m\angle 1+m\angle 2=90{}^{\circ }$
$m\angle 2+m\angle 4=m\angle 5$
$m\angle 2+m\angle 6=m\angle 5$
$m\angle 2+m\angle 4+m\angle 6=180{}^{\circ }$
$m\angle 3+m\angle 4=180{}^{\circ }$

Explanation:

Step1: Analyze the first statement

The sum of \(m\angle1 + m\angle2\) is not necessarily \(90^{\circ}\) as there is no indication of a right - angle in the problem.

Step2: Analyze the second statement

By the exterior angle theorem, in a triangle, an exterior angle is equal to the sum of the two non - adjacent interior angles. Here, \(\angle5\) is an exterior angle of the triangle with interior angles \(\angle2\) and \(\angle4\). So, \(m\angle2 + m\angle4=m\angle5\).

Step3: Analyze the third statement

\(m\angle2 + m\angle6
eq m\angle5\) because \(\angle5\) is related to \(\angle2\) and \(\angle4\) (from the exterior angle theorem of the triangle).

Step4: Analyze the fourth statement

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

Step5: Analyze the fifth statement

\(\angle3\) and \(\angle4\) form a linear pair. A linear pair of angles is supplementary. So, \(m\angle3 + m\angle4=180^{\circ}\).

Answer:

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