Certainly! Let's walk through the problem step-by-step using the given information and filled in some of the missing parts of the solution.
### Given Information:
- [tex]$\angle ABC$[/tex] is bisected by ray BD, which means [tex]$\angle ABD = \angle DBC$[/tex]
- [tex]$m \angle ABD = 7x + 12$[/tex]
- [tex]$m \angle DBC = 48 - 2x$[/tex]
### To Prove:
- [tex]$m \angle ABD = 40$[/tex]
### Proof:
| Statement | Reason |
|--------------------------------------------|-----------------------------------------------------|
| 1. Ray BD bisects [tex]$\angle ABC$[/tex] | 1. Given |
| 2. [tex]$m \angle ABD = m \angle DBC$[/tex] | 2. Definition of Angle Bisector |
| 3. [tex]$m \angle ABD = 7x + 12$[/tex] | 3. Given |
| 4. [tex]$m \angle DBC = 48 - 2x$[/tex] | 4. Given |
| 5. [tex]$7x + 12 = 48 - 2x$[/tex] | 5. Substitution |
| 6. [tex]$7x + 12 = 48 - 2x$[/tex] | 6. Subtraction Property of Equality |
| 7. [tex]$7x + 2x = 48 - 12$[/tex] | 7. Addition Property of Equality |
| 8. [tex]$9x = 36$[/tex] | 8. Simplification |
| 9. [tex]$x = 4$[/tex] | 9. Division Property of Equality |
| 10. [tex]$m \angle ABD = 7(4) + 12$[/tex] | 10. Substitution |
| 11. [tex]$m \angle ABD = 28 + 12$[/tex] | 11. Arithmetic |
| 12. [tex]$m \angle ABD = 40$[/tex] | 12. Simplification |
### Explanation:
1. Given: Ray BD bisects [tex]$\angle ABC$[/tex]. This means that ray BD splits [tex]$\angle ABC$[/tex] into two equal parts: [tex]$\angle ABD$[/tex] and [tex]$\angle DBC$[/tex].
2. Definition of Angle Bisector: Since ray BD bisects [tex]$\angle ABC$[/tex], we have [tex]$m \angle ABD = m \angle DBC$[/tex].
3. Given: [tex]$m \angle ABD = 7x + 12$[/tex].
4. Given: [tex]$m \angle DBC = 48 - 2x$[/tex].
5. Substitution: Since [tex]$m \angle ABD = m \angle DBC$[/tex] (from step 2), we substitute the given expressions: [tex]$7x + 12 = 48 - 2x$[/tex].
6. Subtraction Property of Equality: To isolate [tex]$x$[/tex], first add [tex]$2x$[/tex] to both sides of the equation: [tex]$7x + 12 + 2x = 48 - 2x + 2x$[/tex] resulting in [tex]$7x + 2x + 12 = 48$[/tex].
8. Simplification: Combine like terms. This results in [tex]$9x + 12 = 48$[/tex].
9. Subtraction: Subtract 12 from both sides of the equation: [tex]$9x = 36$[/tex].
10. Division Property of Equality: Divide both sides by 9 to solve for [tex]$x$[/tex]: [tex]$x = 4$[/tex].
11. Substitution: Substitute [tex]$x = 4$[/tex] back into the expression for [tex]$m \angle ABD$[/tex]: [tex]$m \angle ABD = 7(4) + 12$[/tex].
12. Arithmetic: Multiply and then add to find [tex]$m \angle ABD$[/tex]: [tex]$m \angle ABD = 28 + 12 = 40$[/tex].
Thus, the measure of [tex]$\angle ABD$[/tex] is indeed 40 degrees.
