Question

prove the triangle angle-sum theorem. given: △abc prove: ( mangle 1 + mangle 2 + mangle 3 = 180 ) plan: draw a line through c, because a straight angle measures 180. this line should be parallel to the line containing ( overline{ab} ) so that an alternate interior angle relationship is formed. proof: use theorem 2-10, which states that through a point not on a line, there is one and only one line parallel to the given line, to justify drawing a line through c parallel to ( overline{ab} ). step 2 (selected), step 3, conclusion.

Explanation:

Step1: Draw Parallel Line

Draw a line \( l \) through point \( C \) parallel to \( \overline{AB} \). By Theorem 2 - 10, this line exists. Let the angles formed at \( C \) with this line be \( \angle 4 \) (between \( l \) and \( \overline{AC} \)) and \( \angle 5 \) (between \( l \) and \( \overline{BC} \)).

Step2: Alternate Interior Angles

Since \( l \parallel AB \) and \( \overline{AC} \) is a transversal, \( \angle 1\cong\angle 4 \) (Alternate Interior Angles Theorem). Similarly, since \( l \parallel AB \) and \( \overline{BC} \) is a transversal, \( \angle 2\cong\angle 5 \) (Alternate Interior Angles Theorem).

Step3: Straight Angle Sum

\( \angle 4+\angle 3+\angle 5 = 180^{\circ} \) (Definition of a straight angle, as \( l \) is a straight line).

Step4: Substitute Angles

Substitute \( \angle 4 \) with \( \angle 1 \) and \( \angle 5 \) with \( \angle 2 \) (from Step 2) into the equation from Step 3. We get \( \angle 1+\angle 3+\angle 2=180^{\circ} \), which is \( m\angle 1 + m\angle 2+m\angle 3 = 180^{\circ} \).

Answer:

The Triangle Angle - Sum Theorem is proven as shown above, and \( m\angle 1 + m\angle 2 + m\angle 3=180^{\circ} \) holds.