Question
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 Substitution, 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 space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By Substitution, 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?
Associative Property of Addition
Triangle Exterior Angle Theorem
Angle Addition Postulate
Commutative Property of Addition
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 Substitution, 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 space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By Substitution, 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?
Associative Property of Addition
Triangle Exterior Angle Theorem
Angle Addition Postulate
Commutative Property of Addition
Answers
GPT-4o mini
The correct reason to fill in the numbered blank space is the **Angle Addition Postulate**.
Here's the reasoning in context: Since \( DE \) is parallel to \( AC \), the angle \( EBD \) is a straight angle and measures \( 180^\circ \). The Angle Addition Postulate states that if a point lies on the interior of an angle, the measures of the angles formed are equal to the measure of the larger angle. Therefore, the sum of the measures of angles \( EBC \), \( CBA \), and \( DBA \) equals the measure of the straight angle \( EBD \).
Thus, the completed statement would be:
"By the Angle Addition Postulate, the sum of the measures of angles \( EBC \), \( CBA \), and \( DBA \) equals the measure of angle \( EBD \).
Here's the reasoning in context: Since \( DE \) is parallel to \( AC \), the angle \( EBD \) is a straight angle and measures \( 180^\circ \). The Angle Addition Postulate states that if a point lies on the interior of an angle, the measures of the angles formed are equal to the measure of the larger angle. Therefore, the sum of the measures of angles \( EBC \), \( CBA \), and \( DBA \) equals the measure of the straight angle \( EBD \).
Thus, the completed statement would be:
"By the Angle Addition Postulate, the sum of the measures of angles \( EBC \), \( CBA \), and \( DBA \) equals the measure of angle \( EBD \).