Triangle ABC is shown below:
Triangle ABC. Line passes through points D, B, and E.
Given: ΔABC
Prove: All three angles of ΔABC add up to 180°.
The flowchart with missing reason proves the measures of the interior angles of ΔABC total 180°:
Top path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle EBC is congruent to angle BCA. By Substitution, the sum of the measures of angles BCA, CBA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle DBA is congruent to angle BAC. By Substitution, the sum of the measures of angles BCA, BCA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Definition of a Straight Angle, the measure of angle EBD equals 180 degrees. By space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees. Bottom path, by Construction, line segment DE is parallel to line segment AC. By Angle Addition Postulate, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees.
Which reason can be used to fill in the numbered blank space?
Definition of Supplementary Angles
Substitution
Definition of Complementary Angles
Angle Addition Postulate