Question

triangle sum theorem proof: given: \\( \triangle abc \\) with \\( \overleftrightarrow{ad}\parallel\overline{bc} \\) prove: \\( m\angle1 + m\angle2 + m\angle3 = 180^{\circ} \\)

Explanation:

Step1: Analyze the given information

We are given a triangle \(ABC\) with \(AD\parallel BC\). We need to prove \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\).

Step2: Use the properties of parallel lines

Since \(AD\parallel BC\), by the Alternate - Interior Angles Theorem, \(\angle4\cong\angle1\) and \(\angle5\cong\angle2\). By the definition of congruent angles, \(m\angle4 = m\angle1\) and \(m\angle5=m\angle2\).

Step3: Use the Linear - Pair Postulate

\(\angle4\) and \(\angle CAD\) form a linear pair. So, \(m\angle4 + m\angle CAD=180^{\circ}\) (by the Linear - Pair Postulate: if two angles form a linear pair, then they are supplementary).

Step4: Use the Angle - Addition Postulate

By the Angle - Addition Postulate, \(m\angle CAD=m\angle3 + m\angle5\) (since \(\angle CAD\) is composed of \(\angle3\) and \(\angle5\)).

Step5: Substitute

Substitute \(m\angle CAD=m\angle3 + m\angle5\) into \(m\angle4 + m\angle CAD=180^{\circ}\). We get \(m\angle4+m\angle3 + m\angle5=180^{\circ}\).

Step6: Substitute again

Since \(m\angle4 = m\angle1\) and \(m\angle5=m\angle2\), substitute these into \(m\angle4+m\angle3 + m\angle5=180^{\circ}\). We have \(m\angle1+m\angle2 + m\angle3=180^{\circ}\).

Answer:

The proof of the Triangle Sum Theorem \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\) is completed as shown above.