In triangle ABC,angle B=2times angle A. The bisectors of these angles meet at D,while BD extended meet AC at E. If angle CEB=70 degrees, find angle BAC.
To find angle BAC, let's break down the problem step by step:
1. Start by assigning variables to the angles in triangle ABC. Let angle A be x degrees, then angle B is 2x degrees.
2. We know that the bisectors of angles A and B meet at point D. This means that angle ADB is a right angle (90 degrees).
3. Angle ADB can also be expressed as the sum of angles ADE and EDB. Since angle ADE is a bisector of angle A, we can assume that angle ADE is equal to angle EDB, so both angles are y degrees.
4. Therefore, we can express angle ADB as 2y degrees, since the bisector of angle B also goes through point D.
5. Angle ADB can also be expressed as (180 - x) degrees, since the sum of angles in a triangle is 180 degrees.
6. Equating the two expressions for angle ADB, we get the equation: 2y = (180 - x).
7. From the given information, we know that angle CEB is 70 degrees. Since angle CEB is an exterior angle of triangle BDE, it can be expressed as the sum of angles BDE and BED. Since angle BDE is equal to angle EDB (both y degrees), we can set them equal to each other and express angle CEB as 2y degrees.
8. Thus, we have 2y = 70.
9. Solving this equation, we find that y = 35 degrees.
10. Now we can substitute the value of y back into our equation 2y = (180 - x) to solve for x: 2(35) = (180 - x). Simplifying, we get 70 = 180 - x.
11. Subtracting 70 from both sides, we have x = 110 degrees.
12. Therefore, angle BAC is equal to x, which is 110 degrees.