Question

which statements are always true regarding the diagram? select three options.
( m angle 5 + m angle 3 = m angle 4 )
( m angle 3 + m angle 4 + m angle 5 = 180 ^ { circ } )
( m angle 5 + m angle 6 = 180 ^ { circ } )
( m angle 2 + m angle 3 = m angle 6 )
( m angle 2 + m angle 3 + m angle 5 = 180 ^ { circ } )

Explanation:

Step1: Check the angle - sum property of a triangle and linear - pair and exterior - angle theorems

• For \(m\angle5 + m\angle3=m\angle4\):

By the exterior - angle theorem, in a triangle, an exterior angle is equal to the sum of the two non - adjacent interior angles. But here, this is not a valid application of the exterior - angle theorem.

• For \(m\angle3 + m\angle4 + m\angle5 = 180^{\circ}\):

This is incorrect. The sum of angles in a triangle is \(180^{\circ}\), but \(\angle3,\angle4,\angle5\) do not form a triangle.

• For \(m\angle5 + m\angle6=180^{\circ}\):

Since \(\angle5\) and \(\angle6\) form a linear pair (they are adjacent angles and their non - common sides form a straight line). By the linear - pair postulate, if two angles form a linear pair, then \(m\angle A + m\angle B=180^{\circ}\). So \(m\angle5 + m\angle6 = 180^{\circ}\) is True.

• For \(m\angle2 + m\angle3=m\angle6\):

By the exterior - angle theorem (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). Let's consider the triangle with angles \(\angle2,\angle3\) and the angle opposite to \(\angle6\) (the non - adjacent interior angles to \(\angle6\)). So \(m\angle2 + m\angle3=m\angle6\) is True.

• For \(m\angle2 + m\angle3 + m\angle5=180^{\circ}\):

Since \(m\angle6=180^{\circ}-m\angle5\) (from \(m\angle5 + m\angle6 = 180^{\circ}\)) and \(m\angle2 + m\angle3=m\angle6\) (from the exterior - angle theorem), then substituting \(m\angle6\) in \(m\angle2 + m\angle3 + m\angle5\), we get \((180^{\circ}-m\angle5)+m\angle5=180^{\circ}\). So \(m\angle2 + m\angle3 + m\angle5 = 180^{\circ}\) is True.

Answer:

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