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Given: [tex]$\angle ABC$[/tex] is bisected by ray [tex]$BD$[/tex], [tex]$m \angle ABD = 7x + 12$[/tex], and [tex]$m \angle DBC = 48 - 2x$[/tex]

Prove: [tex]$m \angle ABD = 40$[/tex]

Statement:

1. Ray [tex]$BD$[/tex] bisects [tex]$\angle ABC$[/tex]
2. [tex]$m \angle ABD = m \angle DBC$[/tex]
3. [tex]$m \angle ABD = 7x + 12$[/tex]
4. [tex]$m \angle DBC = 48 - 2x$[/tex]
5. [tex]$7x + 12 = 48 - 2x$[/tex]
6. [tex]$7x + 2x = 48 - 12$[/tex]
7. [tex]$9x = 36$[/tex]
8. [tex]$x = 4$[/tex]
9. [tex]$m \angle ABD = 7(4) + 12$[/tex]
10. [tex]$m \angle ABD = 40$[/tex]

Reason:

1. Given
2. Definition of Angle Bisector
3. Given
4. Given
5. Substitution
6. Addition Property of Equality
7. Simplification
8. Division Property of Equality
9. Substitution
10. Simplification



Answer :

GinnyAnswer
Let's provide a detailed, step-by-step proof for the given problem, clearly explaining each statement and its reason.

Given:
[tex]\(\angle ABC\)[/tex] is bisected by ray [tex]\(BD\)[/tex].
 [tex]\(m \angle ABD = 7x + 12\)[/tex].
* [tex]\(m \angle DBC = 48 - 2x\)[/tex].

To Prove:
[tex]\(m \angle ABD = 40\)[/tex].

Statements and Reasons:

1. Statement: Ray [tex]\(BD\)[/tex] bisects [tex]\(\angle ABC\)[/tex].
Reason: Given.

2. Statement: [tex]\(m \angle ABD = m \angle DBC\)[/tex].
Reason: Definition of angle bisector. Since ray [tex]\(BD\)[/tex] bisects [tex]\(\angle ABC\)[/tex], it means [tex]\(\angle ABD\)[/tex] and [tex]\(\angle DBC\)[/tex] are equal in measure.

3. Statement: [tex]\(m \angle ABD = 7x + 12\)[/tex].
Reason: Given.

4. Statement: [tex]\(m \angle DBC = 48 - 2x\)[/tex].
Reason: Given.

5. Statement: [tex]\(7x + 12 = 48 - 2x\)[/tex].
Reason: Substitution. Since [tex]\(m \angle ABD = m \angle DBC\)[/tex], we substitute the given expressions for [tex]\(m \angle ABD\)[/tex] and [tex]\(m \angle DBC\)[/tex] into the equation [tex]\(7x + 12 = 48 - 2x\)[/tex].

6. Statement: [tex]\(7x + 2x + 12 = 48\)[/tex].
Reason: Addition Property of Equality. We add [tex]\(2x\)[/tex] to both sides to start isolating [tex]\(x\)[/tex].

7. Statement: [tex]\(9x + 12 = 48\)[/tex].
Reason: Combine like terms.

8. Statement: [tex]\(9x = 36\)[/tex].
Reason: Subtraction Property of Equality. Subtract [tex]\(12\)[/tex] from both sides to further isolate [tex]\(x\)[/tex].

9. Statement: [tex]\(x = 4\)[/tex].
Reason: Division Property of Equality. Divide both sides by [tex]\(9\)[/tex] to solve for [tex]\(x\)[/tex].

10. Statement: [tex]\(m \angle ABD = 7(4) + 12\)[/tex].
Reason: Substitution. Substitute [tex]\(x = 4\)[/tex] back into the expression for [tex]\(m \angle ABD\)[/tex].

11. Statement: [tex]\(m \angle ABD = 28 + 12\)[/tex].
Reason: Perform the arithmetic operation [tex]\(7 \times 4\)[/tex].

12. Statement: [tex]\(m \angle ABD = 40\)[/tex].
Reason: Complete the arithmetic operation [tex]\(28 + 12\)[/tex].

Conclusion: We have proved that [tex]\(m \angle ABD\)[/tex] is indeed [tex]\(40\)[/tex].

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