Question

Given: Line segment A B is parallel to line segment D C and Measure of angle 2 equals measure of angle 4
Prove: Line segment A D is parallel to line segment B C

A parallelogram has points A B C D. Angle D A B is angle 1, Angle A B C is angle 2, Angle B C D is angle 3, and angle C D A is angle 4.

A 2-column table with 7 rows. Column 1 is labeled statements and has entries line segment A B is parallel to line segment D C, measure of angle 2 = measure of angle 4, angle 1 and angle 4 are supplements, question mark, measure of angle 1 + measure of angle 2 = 180 degrees, angle 1 and angle 2 are supplements, line segment A D is parallel to line segment B C. Column 2 is labeled reasons with entries given, given, same side interior angles theorem, definition of supplementary angles, substitution, definition of supplementary angles, converse same side interior angles theorem.



m∠1 = m∠4
m∠2 = m∠3
m∠1 + m∠4 = 180°
m∠2 + m∠3 = 180°

Answer 1

The missing statement (row 4) is
m∠1 + m∠4 = 180°.

Complete 2-column proof:

Statements — Reasons
1. AB ∥ DC — Given
2. m∠2 = m∠4 — Given
3. ∠1 and ∠4 are supplementary — Same-side interior angles theorem (AB ∥ DC with transversal AD)
4. m∠1 + m∠4 = 180° — Definition of supplementary angles
5. m∠1 + m∠2 = 180° — Substitution (replace m∠4 by m∠2 from step 2)
6. ∠1 and ∠2 are supplementary — Definition of supplementary angles
7. AD ∥ BC — Converse of the same-side interior angles theorem (if same-side interior angles are supplementary, the lines are parallel)

Therefore AD ∥ BC.