Thread: More angles

In the right-angled triangle ABC opposite, CAB = 90 degrees
The bisector of ABC meets AC at D.
Let ABD = z, ACB = y and ADB = x
Give reasons why 2z = 90 degrees - y and x = z + y
Hence show that x = 45 degrees + 1/2 y

Looking at angle B, you see that it is equal to 2z. Looking at the triangle ABC, we have the 3 angles of the triangle: y, 2z, and 90 (this is angle CAB). We know that the sum of a triangle's angles add up to 180, so:
Solve for 2z.

To prove x = 2z + y, note from the first question:
.

Looking at triangle ABD, we see that . Using the fact that , substitute it in and we get:


Solve for x.

For the last one, from the first question we have that . Substitute that into the expression for x in the 2nd question and you should be done proving it.