Question

which is a true statement about the diagram?
∘ ( mangle5 + mangle6 = mangle1 )
∘ ( mangle3 + mangle4 + mangle5 = 180^circ )
∘ ( mangle1 + mangle2 = 180^circ )
∘ ( mangle2 + mangle3 = mangle5 )

Explanation:

Step1: Analyze each option

• Option 1: \( m\angle5 + m\angle6 = m\angle1 \)

\( \angle5 \) and \( \angle6 \) are supplementary (form a linear pair), so \( m\angle5 + m\angle6 = 180^\circ \). \( \angle1 \) is an exterior angle, but this equality doesn’t hold. Eliminate.

• Option 2: \( m\angle3 + m\angle4 + m\angle5 = 180^\circ \)

The sum of angles in a triangle is \( 180^\circ \), but \( \angle3, \angle4, \angle5 \) are not the angles of a triangle. \( \angle3 + \angle2 + \angle5 = 180^\circ \) (triangle angle sum), not including \( \angle4 \). Eliminate.

• Option 3: \( m\angle1 + m\angle2 = 180^\circ \)

\( \angle1 \) and \( \angle2 \) form a linear pair (adjacent, supplementary), so their measures sum to \( 180^\circ \). This holds.

• Option 4: \( m\angle2 + m\angle3 = m\angle5 \)

By the exterior angle theorem, \( m\angle2 + m\angle3 = m\angle4 \) (exterior angle at vertex 4), not \( \angle5 \). Eliminate.

Answer:

\( m\angle1 + m\angle2 = 180^\circ \) (the third option)