Let's solve the problem step by step and fill in the missing statements and reasons in the two-column proof.
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Given:
- ∠1 is supplementary to ∠2.
- $\overrightarrow{BD}$ bisects ∠ABC.
Prove:
- ∠1 is supplementary to ∠3.
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We are given a diagram with angles at point B:
- ∠1 is on the left, forming a straight line with ∠ABC (which is composed of ∠2 and ∠3).
- ∠2 and ∠3 are adjacent angles that make up ∠ABC.
- Ray BD bisects ∠ABC, so it splits it into two equal parts: ∠2 and ∠3.
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Now, let’s complete the proof table:
| Statement | Reason |
|---------|--------|
| 1. ∠1 is supplementary to ∠2. | Given |
| 2. $m∠1 + m∠2 = 180^\circ$ | Definition of supplementary |
| 3. $\overrightarrow{BD}$ bisects ∠ABC. | Given |
| 4. $m∠2 = m∠3$ | Definition of angle bisector |
| 5. $m∠1 + m∠3 = 180^\circ$ | Substitution Property of Equality |
| 6. ∠1 is supplementary to ∠3. | Definition of supplementary |
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Explanation of Each Step:
1.
Given: We start with what is provided.
2.
Definition of supplementary: Two angles are supplementary if their measures add up to 180°. So, since ∠1 and ∠2 are supplementary, their measures sum to 180°.
3.
Given: This is the second piece of information provided.
4.
Definition of angle bisector: If a ray bisects an angle, it divides it into two congruent (equal) angles. So, ∠2 ≅ ∠3 → $m∠2 = m∠3$.
5.
Substitution Property of Equality: From step 2: $m∠1 + m∠2 = 180^\circ$.
Since $m∠2 = m∠3$ (step 4), we can substitute $m∠3$ for $m∠2$:
→ $m∠1 + m∠3 = 180^\circ$
6.
Definition of supplementary: Since the sum of the measures of ∠1 and ∠3 is 180°, they are supplementary.
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✔ Final Answer:
Completed Table:
| Statement | Reason |
|---------|--------|
| 1. ∠1 is supplementary to ∠2. | Given |
| 2. $m∠1 + m∠2 = 180^\circ$ | Definition of supplementary |
| 3. $\overrightarrow{BD}$ bisects ∠ABC. | Given |
| 4. $m∠2 = m∠3$ | Definition of angle bisector |
| 5. $m∠1 + m∠3 = 180^\circ$ | Substitution Property of Equality |
| 6. ∠1 is supplementary to ∠3. | Definition of supplementary |
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This proves that ∠1 is supplementary to ∠3 using the given information and logical reasoning.
Parent Tip: Review the logic above to help your child master the concept of angle bisector conjecture.
