Not so easy as it seems!
Geometry
Level
2
In triangle ABC, angle B is obtuse. D is the mid-point of BC. angle ADB = 45 degree and angle ACD = 30 degree. Find angle BAD. (Give the answer in degrees.)
The answer is 30.
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1 solution
Draw BL perpendicular to AC and join L to D.
Since, angle BCL = 30 degree, we get angle CBL = 60 degree. Since BLC is a right-angled triangle with angle BCL = 30 degree, we have BL = BC/2 = BD. Thus in triangle BLD, we see that BL = BD and angle DBL = 60 degree. This implies that BLD is an equilateral triangle and hence LB = LD. Using angle LBD = 60 degree and angle ADB = 45 degree, we get angle ADL = 15 degree. But angle DAL = 15 degree. So, LD = LA. We hence have LD = LA = LB. So, L is the circumcentre of triangle BDA. Thus, angle BAD = half of angle BLD = 30 degree.