Question

Lines bc and ed are parallel. They are intersected by transversal ae, in which point b lies between points a and e. They are also intersected by transversal ec. Angle abc measures 70 degrees. Angle ced measures 30 degrees. Given: Line bc is parallel to line ed, m∠abc = 70°, m∠ced = 30°. Prove: m∠bec = 40°. Statement Justification: Line bc is parallel to line ed (given), m∠abc = 70° (given), m∠ced = 30° (given), m∠bec = m∠ced (corresponding angles theorem), m∠bec = 40° (subtraction property of equality). Which of the following accurately completes the missing statement and justification of the two-column proof?

1) m∠abc = m∠ced; corresponding angles theorem
2) m∠abc = m∠ced; alternate interior angles theorem
3) m∠abc = m∠bed; corresponding angles theorem
4) m∠abc = m∠bed; alternate interior angles theorem

Answer 1

The correct answer to complete the two-column proof is option 4: m∠abc = m∠bed, which is justified by the Alternate Interior Angles Theorem. This allows us to calculate the measure of angle BEC as 40 degrees using the Subtraction Property of Equality.

Step-by-step explanation:

To prove that the measure of angle BEC is 40 degrees, given that lines BC and ED are parallel and intersected by transversals AE and EC, we can utilize geometric theorems involving parallel lines and angles. Since angle ABC is given as 70 degrees and angle CED as 30 degrees, we can use the properties of parallel lines to find the relationship between the angles.

Out of the options provided, the correct statement and justification are: angle ABC is equal to angle BED, according to the Alternate Interior Angles Theorem. This is because when two parallel lines are intersected by a transversal, the alternate interior angles are congruent. Thus, the missing statement and justification in the two-column proof are option 4: m∠abc = m∠bed; alternate interior angles theorem.

By establishing the equality of these angles, we can use the Subtraction Property of Equality to determine that the measure of angle BEC, which is the difference between angle BED and angle CED, is indeed 40 degrees (70 degrees - 30 degrees).