# More angles

• April 23rd 2008, 06:05 PM
stellina_91
More angles
In the right-angled triangle ABC opposite, http://www.staff.vu.edu.au/mcaonline...s/SYMBOL15.GIFCAB = 90 degrees
The bisector of http://www.staff.vu.edu.au/mcaonline...s/SYMBOL15.GIFABC meets AC at D.
Let http://www.staff.vu.edu.au/mcaonline...s/SYMBOL15.GIFABD = z, http://www.staff.vu.edu.au/mcaonline...s/SYMBOL15.GIFACB = y and http://www.staff.vu.edu.au/mcaonline...s/SYMBOL15.GIFADB = x
Give reasons why 2z = 90 degrees - y and x = z + y
Hence show that x = 45 degrees + 1/2 y
• April 23rd 2008, 06:56 PM
o_O
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: $y + 2z + 90 = 180$
Solve for 2z.

To prove x = 2z + y, note from the first question:
$2z = 90 - y \quad \Rightarrow \quad 90 = 2z + y$.

Looking at triangle ABD, we see that $x + z = {\color{red}90}$. Using the fact that ${\color{red}90 = 2z + y}$, substitute it in and we get:
$x + z = {\color{red} 2z + y}$

Solve for x.

For the last one, from the first question we have that $2z = 90 - y \Rightarrow z = 45 - \frac{1}{2}y$. Substitute that into the expression for x in the 2nd question and you should be done proving it.