Question

which statements are always true regarding the diagram? check all that apply. □ $m\angle3 + m\angle4 = 180°$ □ $m\angle2 + m\angle4 + m\angle6 = 180°$ □ $m\angle2 + m\angle4 = m\angle5$ □ $m\angle1 + m\angle2 = 90°$ □ $m\angle4 + m\angle6 = m\angle2$ □ $m\angle2 + m\angle6 = m\angle5$

Explanation:

Step1: Analyze \( m\angle3 + m\angle4 \)

\(\angle3\) and \(\angle4\) form a linear pair (they are adjacent and supplementary), so \( m\angle3 + m\angle4 = 180^\circ \). This statement is true.

Step2: Analyze \( m\angle2 + m\angle4 + m\angle6 \)

The sum of the interior angles of a triangle is \( 180^\circ \). The triangle here has angles \(\angle2\), \(\angle4\), and the angle adjacent to \(\angle1\) (but \(\angle1\) and that angle form a linear pair, so focusing on the triangle's interior angles: \(\angle2\), \(\angle4\), and the angle that, with \(\angle6\), forms a linear pair? Wait, no—actually, the triangle's interior angles are \(\angle2\), \(\angle4\), and the angle opposite? Wait, looking at the diagram, the triangle has angles \(\angle2\), \(\angle4\), and the angle that is supplementary to \(\angle6\)? No, better: the sum of the interior angles of a triangle is \( 180^\circ \), so \( m\angle2 + m\angle4 + \) (the third interior angle) \( = 180^\circ \). But the third interior angle and \(\angle6\) form a linear pair? Wait, no—actually, the three angles \(\angle2\), \(\angle4\), and the angle at the vertex with \(\angle6\) (let's call it \(\angle x\)) sum to \( 180^\circ \), and \(\angle x + \angle6 = 180^\circ \), so \( m\angle2 + m\angle4 + (180^\circ - m\angle6) = 180^\circ \), which simplifies to \( m\angle2 + m\angle4 - m\angle6 = 0 \), so \( m\angle2 + m\angle4 = m\angle6 \)? No, that's not right. Wait, maybe I misread. Wait, the triangle's interior angles: looking at the diagram, the triangle has vertices at the three points, so the interior angles are \(\angle2\), \(\angle4\), and the angle that is between the sides with \(\angle1\) and \(\angle6\). Wait, actually, the correct approach: the sum of the interior angles of a triangle is \( 180^\circ \), so \( m\angle2 + m\angle4 + m\) (the third interior angle) \( = 180^\circ \). But the third interior angle and \(\angle6\) are supplementary? No, \(\angle6\) and that third interior angle are adjacent and form a linear pair? Wait, no—\(\angle6\) is an exterior angle? Wait, no, \(\angle5\) is a straight line, so \(\angle5\) is \( 180^\circ \), and \(\angle6\) and the angle adjacent to \(\angle2\) and \(\angle4\) form a triangle. Wait, maybe I made a mistake. Let's re-express: the triangle has angles \(\angle2\), \(\angle4\), and the angle that is equal to \( 180^\circ - \angle6 \) (since \(\angle6\) and that angle are supplementary). So \( m\angle2 + m\angle4 + (180^\circ - m\angle6) = 180^\circ \), which simplifies to \( m\angle2 + m\angle4 - m\angle6 = 0 \), so \( m\angle2 + m\angle4 = m\angle6 \). Wait, that would mean \( m\angle2 + m\angle4 + m\angle6 = m\angle6 + m\angle6 = 2m\angle6 \), which is not \( 180^\circ \) unless \( m\angle6 = 90^\circ \), so this statement is false.

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

\(\angle5\) is an exterior angle of the triangle, and by the exterior angle theorem, the exterior angle is equal to the sum of the two non-adjacent interior angles. The two non-adjacent interior angles to \(\angle5\) are \(\angle2\) and \(\angle4\), so \( m\angle2 + m\angle4 = m\angle5 \). This statement is true.

Step4: Analyze \( m\angle1 + m\angle2 = 90^\circ \)

There's no information given that \(\angle1\) and \(\angle2\) are complementary; they form a linear pair? Wait, \(\angle1\) and \(\angle2\) are adjacent and form a linear pair? Wait, no—\(\angle1\) and \(\angle2\) are on a straight line? Wait, the diagram shows \(\angle1\) and \(\angle2\) at a vertex, with a line going through, so they form a linear pair, so \( m\angle1 + m\angle2 = 180^\circ \), not \( 90^\circ \). So this statement is false.

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

There's no reason for this to be true; from the exterior angle theorem, we know \( m\angle2 + m\angle4 = m\angle5 \), and \(\angle5\) and \(\angle6\) form a linear pair? Wait, no—\(\angle5\) and \(\angle6\) are adjacent, so \( m\angle5 + m\angle6 = 180^\circ \), but that doesn't imply \( m\angle4 + m\angle6 = m\angle2 \). So this is false.

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

From the exterior angle theorem, it's \( m\angle2 + m\angle4 = m\angle5 \), not \( m\angle2 + m\angle6 \), so this is false.

Answer:

• \( m\angle3 + m\angle4 = 180^\circ \) (True)
• \( m\angle2 + m\angle4 + m\angle6 = 180^\circ \) (False)
• \( m\angle2 + m\angle4 = m\angle5 \) (True)
• \( m\angle1 + m\angle2 = 90^\circ \) (False)
• \( m\angle4 + m\angle6 = m\angle2 \) (False)
• \( m\angle2 + m\angle6 = m\angle5 \) (False)

So the true statements are \( m\angle3 + m\angle4 = 180^\circ \) and \( m\angle2 + m\angle4 = m\angle5 \).