Question

proof of the triangle sum theorem
given: △abc and the diagram
prove: ( mangle 4 + mangle 2 + mangle 5 = 180^circ )

Explanation:

Step1: Identify Parallel Lines and Transversal

From the diagram, \( AB \parallel EC \) (marked with parallel arrows) and \( AC \) is a transversal. So, \( \angle 4 \) and \( \angle 1 \) are alternate interior angles, so \( m\angle 4 = m\angle 1 \) (Alternate Interior Angles Theorem). Also, \( BC \) is a transversal, so \( \angle 5 \) and \( \angle 3 \) are alternate interior angles, so \( m\angle 5 = m\angle 3 \) (Alternate Interior Angles Theorem).

Step2: Straight Angle Property

\( \angle 1 + \angle 2 + \angle 3 \) form a straight angle, so \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \) (Definition of Straight Angle).

Step3: Substitute Equal Angles

Substitute \( m\angle 4 \) for \( m\angle 1 \) and \( m\angle 5 \) for \( m\angle 3 \) in the equation from Step 2: \( m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ \).

Answer:

By using the Alternate Interior Angles Theorem (for \( \angle 4 \cong \angle 1 \) and \( \angle 5 \cong \angle 3 \)) and the Straight Angle Property (\( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \)), we substitute to get \( m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ \), proving the statement.