Question

which statements are always true regarding the diagram? select three options. □ ( mangle5 + mangle3 = mangle4 ) □ ( mangle3 + mangle4 + mangle5 = 180^circ ) □ ( mangle5 + mangle6 = 180^circ ) □ ( mangle2 + mangle3 = mangle6 ) □ ( mangle2 + mangle3 + mangle5 = 180^circ )

Explanation:

1. For \( m\angle5 + m\angle6 = 180^\circ \): Angles \( \angle5 \) and \( \angle6 \) form a linear pair (they are adjacent and supplementary), so their measures sum to \( 180^\circ \). This is always true.
2. For \( m\angle2 + m\angle3 = m\angle6 \): By the exterior angle theorem of a triangle, the exterior angle (\( \angle6 \)) is equal to the sum of the two non - adjacent interior angles (\( \angle2 \) and \( \angle3 \)) of the triangle. So this is always true.
3. For \( m\angle2 + m\angle3 + m\angle5 = 180^\circ \): The sum of the interior angles of a triangle is \( 180^\circ \). In the triangle, the interior angles are \( \angle2 \), \( \angle3 \), and \( \angle5 \), so their measures sum to \( 180^\circ \). This is always true.

• \( m\angle5 + m\angle3 = m\angle4 \): There is no geometric relationship that guarantees this.
• \( m\angle3 + m\angle4 + m\angle5 = 180^\circ \): These angles do not form the interior angles of a triangle or a supplementary set in a way that their sum is \( 180^\circ \) always.

Answer:

• \( m\angle5 + m\angle6 = 180^\circ \)
• \( m\angle2 + m\angle3 = m\angle6 \)
• \( m\angle2 + m\angle3 + m\angle5 = 180^\circ \)