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: $\displaystyle y + 2z + 90 = 180$
Solve for 2z.
To prove x = 2z + y, note from the first question:
$\displaystyle 2z = 90 - y \quad \Rightarrow \quad 90 = 2z + y$.
Looking at triangle ABD, we see that $\displaystyle x + z = {\color{red}90} $. Using the fact that $\displaystyle {\color{red}90 = 2z + y}$, substitute it in and we get:
$\displaystyle x + z = {\color{red} 2z + y} $
Solve for x.
For the last one, from the first question we have that $\displaystyle 2z = 90 - y \Rightarrow z = 45 - \frac{1}{2}y$. Substitute that into the expression for x in the 2nd question and you should be done proving it.