Answer :
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].
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].