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°
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°
Answers
GPT-4o mini
To prove that line segment AD is parallel to line segment BC using the provided information, we can utilize the properties of angles and the relationships established by parallel lines. Here’s a step-by-step layout for the proof in a 2-column format similar to the one you provided:
### Proof:
| **Statements** | **Reasons** |
|---------------------------------------------------------|------------------------------------------------------|
| 1. Line segment AB is parallel to line segment DC. | Given |
| 2. Measure of angle 2 = measure of angle 4. | Given |
| 3. Angle 1 and angle 4 are supplementary. | Definition of supplementary angles |
| 4. Measure of angle 1 + measure of angle 4 = 180°. | Definition of supplementary angles |
| 5. Measure of angle 1 + measure of angle 2 = 180°. | Substitution (from statement 2 into statement 4) |
| 6. Angle 1 and angle 2 are supplementary. | Definition of supplementary angles |
| 7. Line segment AD is parallel to line segment BC. | Converse of the Same-Side Interior Angles Theorem |
### Explanation of Steps:
1. **Given Information**: We start with the given that line segment AB is parallel to line segment DC.
2. **Angle Equality**: By assumption, it is given that the measure of angle 2 equals the measure of angle 4.
3. **Supplementary Angles**: Since angle 1 and angle 4 are on the same side of the transversal and form a linear pair, they are supplementary.
4. **Supplementary Sum**: This means that the sum of their measures is 180 degrees.
5. **Substitution**: Since we know the measures are equal (step 2), we can substitute the measure of angle 4 with the measure of angle 2 in the equation from step 4.
6. **Supplementary Angles**: From step 5, we have established that angle 1 and angle 2 sum to 180 degrees, which means that they are also supplementary.
7. **Parallel Lines**: As a result of step 6 indicating that angle 1 and angle 2 are supplementary while lines AD and BC are cut by a transversal (line AB), we can use the Converse of the Same-Side Interior Angles Theorem. This theorem states that if two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel.
Thus, it is proven that line segment AD is indeed parallel to line segment BC.
### Proof:
| **Statements** | **Reasons** |
|---------------------------------------------------------|------------------------------------------------------|
| 1. Line segment AB is parallel to line segment DC. | Given |
| 2. Measure of angle 2 = measure of angle 4. | Given |
| 3. Angle 1 and angle 4 are supplementary. | Definition of supplementary angles |
| 4. Measure of angle 1 + measure of angle 4 = 180°. | Definition of supplementary angles |
| 5. Measure of angle 1 + measure of angle 2 = 180°. | Substitution (from statement 2 into statement 4) |
| 6. Angle 1 and angle 2 are supplementary. | Definition of supplementary angles |
| 7. Line segment AD is parallel to line segment BC. | Converse of the Same-Side Interior Angles Theorem |
### Explanation of Steps:
1. **Given Information**: We start with the given that line segment AB is parallel to line segment DC.
2. **Angle Equality**: By assumption, it is given that the measure of angle 2 equals the measure of angle 4.
3. **Supplementary Angles**: Since angle 1 and angle 4 are on the same side of the transversal and form a linear pair, they are supplementary.
4. **Supplementary Sum**: This means that the sum of their measures is 180 degrees.
5. **Substitution**: Since we know the measures are equal (step 2), we can substitute the measure of angle 4 with the measure of angle 2 in the equation from step 4.
6. **Supplementary Angles**: From step 5, we have established that angle 1 and angle 2 sum to 180 degrees, which means that they are also supplementary.
7. **Parallel Lines**: As a result of step 6 indicating that angle 1 and angle 2 are supplementary while lines AD and BC are cut by a transversal (line AB), we can use the Converse of the Same-Side Interior Angles Theorem. This theorem states that if two lines are cut by a transversal and the same-side interior angles are supplementary, then the lines are parallel.
Thus, it is proven that line segment AD is indeed parallel to line segment BC.