Question

triangle sum theorem:
the sum of the interior angle measures of a
triangle is 180°
given: \\( \triangle a b c \\) with \\( \overleftrightarrow { d e } \parallel \overline { a c } \\)
prove: \\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\)
proof:

1. \\( \overleftrightarrow { d e } \parallel \overline { a c } \\) sele
2. \\( \angle 1 \cong \angle d b a \\)

\\( \angle 3 \cong \angle e b c \\) sele

1. \\( m \angle 1 = m \angle d b a \\)

\\( m \angle 3 = m \angle e b c \\) sele

1. \\( m \angle d b a + m \angle 2 = m \angle d b c \\) sele
2. \\( m \angle d b c + m \angle e b c = 180 ^ { \circ } \\) sele

6.
\\( m \angle d b a + m \angle 2 + m \angle e b c = 180 ^ { \circ } \\)
\\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\) sele

Explanation:

Step1: Given

This is the given condition in the problem.

Step2: Alternate interior angles

When two parallel lines are cut by a transversal, alternate interior angles are congruent. Here, \(\overleftrightarrow{DE}\parallel\overline{AC}\), so \(\angle1\cong\angle DBA\) and \(\angle3\cong\angle EBC\)

Step3: Congruent angles have equal measures

If two angles are congruent, then their measures are equal. So \(m\angle1 = m\angle DBA\) and \(m\angle3=m\angle EBC\)

Step4: Angle - addition postulate

The measure of an angle formed by two adjacent angles is the sum of the measures of the two adjacent angles. So \(m\angle DBA + m\angle2=m\angle DBC\)

Step5: Linear - pair postulate

If two angles form a linear pair (are adjacent and their non - common sides are opposite rays), then the sum of their measures is \(180^{\circ}\). So \(m\angle DBC + m\angle EBC = 180^{\circ}\)

Step6: Substitution

Substitute \(m\angle DBA\) with \(m\angle1\) and \(m\angle EBC\) with \(m\angle3\) in the equation \(m\angle DBA + m\angle2+m\angle EBC=180^{\circ}\) (from step 5 and step 4 substitution)

Answer:

1. Given
2. Alternate interior angles
3. Congruent angles have equal measures
4. Angle - addition postulate
5. Linear - pair postulate
6. Substitution