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Question 1191303: Complete the formal proof of the following theorem.
The bisectors of two adjacent supplementary angles form a right angle.
There is a line and three rays. There are six labeled points and four labeled angles.
Line A D is horizontal. B is on this line and between A and D.
Ray B C goes up and to the right.
Ray B F goes up and to the right. Point F is below and to the right of point C.
Ray B E goes up and to the left.
Angle A B E is labeled 1 and is marked with one arc.
Angle C B E is labeled 2 and is marked with one arc.
Angle C B F is labeled 3 and is marked with two arcs.
Angle D B F is labeled 4 and is marked with two arcs.
Given:
∠ABC is supplementary to ∠CBD.
BE bisects ∠ABC.
BF bisects ∠CBD.
Prove: ∠EBF is a right angle.
Statements Reasons
1.
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1. Given
2.
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2. The sum of the measures of supplementary ∠s is 180°.
3. m∠ABC = m∠1 + m∠2;
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3.
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4.
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4. Substitution
5.
BE bisects ∠ABC;
BF bisects ∠CBD.
5.
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6.
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6. If a ray bisects an ∠, then two ∠s of = measure are formed.
7. m∠2 + m∠2 + m∠3 + m∠3 = 180° 7.
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8.
2 · m∠2 + 2 · m∠3 = 180°
8. Combine like terms.
9. m∠2 + m∠3 = 90° 9.
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10.
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10. Angle-Addition Postulate
11.
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11. Substitution
12.
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12.
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Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the completed two-column proof:
**Given:**
∠ABC is supplementary to ∠CBD.
BE bisects ∠ABC.
BF bisects ∠CBD.
**Prove:**
∠EBF is a right angle.
| Statements | Reasons |
|---|---|
| 1. ∠ABC is supplementary to ∠CBD. | 1. Given |
| 2. m∠ABC + m∠CBD = 180° | 2. The sum of the measures of supplementary ∠s is 180°. |
| 3. m∠ABC = m∠1 + m∠2; m∠CBD = m∠3 + m∠4 | 3. Angle-Addition Postulate |
| 4. m∠1 + m∠2 + m∠3 + m∠4 = 180° | 4. Substitution |
| 5. BE bisects ∠ABC; BF bisects ∠CBD. | 5. Given |
| 6. m∠1 = m∠2; m∠3 = m∠4 | 6. If a ray bisects an ∠, then two ∠s of = measure are formed. |
| 7. m∠2 + m∠2 + m∠3 + m∠3 = 180° | 7. Substitution |
| 8. 2 · m∠2 + 2 · m∠3 = 180° | 8. Combine like terms. |
| 9. m∠2 + m∠3 = 90° | 9. Division Property of Equality |
| 10. m∠EBF = m∠2 + m∠3 | 10. Angle-Addition Postulate |
| 11. m∠EBF = 90° | 11. Substitution |
| 12. ∠EBF is a right angle. | 12. Definition of a right angle |
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