in triangle ABC, BD bisects <ABC, <ABD = 4x + 6 and m<DBC = 5x -5. find the m<abc
then 4x + 6 = 5x - 5
-x = -11
x = 11
angle ABC = 9x + 1 = 100
To find the measure of angle ABC (m<ABC), we need to use the fact that BD bisects angle ABC.
Let's go step by step:
1. According to the angle bisector theorem, when a line bisects an angle in a triangle, it divides the opposite side in a ratio that is equal to the ratio of the adjacent sides. In other words, we have the following proportion: AB / AC = BD / DC.
2. We need to find the measure of angle ABC, which is opposite to side AC. So, let's call the measure of angle ABC as y.
3. We are given the measure of angle ABD, which is 4x + 6.
4. We are also given the measure of angle DBC, which is 5x - 5.
5. Since BD bisects angle ABC, the ratio of AB to AC should be equal to the ratio of BD to DC. We can set up the proportion as follows: AB / AC = BD / DC.
6. Substituting the values we have, we get: (4x + 6) / AC = BD / DC.
7. Now, let's simplify the equation: (4x + 6) / AC = 1 / 1.
8. Cross-multiplying, we get: AC = 4x + 6.
9. Now, we have the value of AC (opposite to angle ABC) in terms of x. We can use this information to find the measure of angle ABC (m<ABC).
10. We also know that the sum of the angles in a triangle is 180 degrees. So, y + (4x + 6) + (5x - 5) = 180.
11. Simplifying the equation: 9x + y + 1 = 180.
12. To find the measure of angle ABC, we need to isolate "y" in the equation. Subtracting 1 from both sides: 9x + y = 179.
13. Since we have the value of AC (4x + 6) in terms of x, we can substitute it into the equation: 9x + (4x + 6) = 179.
14. Simplifying the equation: 13x + 6 = 179.
15. Solving for x: 13x = 173.
16. Dividing both sides by 13: x = 13.
17. Now that we have the value of x, we can substitute it into the equation to find the measure of angle ABC.
18. Plugging in x = 13 into the equation: y = 179 - 9(13) = 38.
Therefore, the measure of angle ABC (m<ABC) is 38 degrees.