Relative angles
Can you work out the angles of a triangle from the relationships between them?
Problem
In triangle ABC, angle B is $\frac{3}{4}$ of the size of angle C and $1 \frac{1}{2}$ times the size of angle A.
What is the size in degrees of angle B?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: $60^\circ$
Finding angle B in terms of angle A and angle C
$B=\frac{3}{4}C$ and $B =\frac{3}{2}A$, so $$\begin{align} \tfrac{3}{4}C&=\tfrac{3}{2}A \\ \Rightarrow \tfrac{1}{4}C&=\tfrac{1}{2}A \\ \Rightarrow C&=2A\end{align}$$.
Since $A$, $B$ and $C$ are the angles of a triangle, $A+B+C=180$.
So, substituting in $B$ and $C$ in terms of $A$, $$\begin{align}A+\tfrac32A+2A&=180\\
\Rightarrow \tfrac{9}{2}A&=180\\
\Rightarrow \tfrac{1}{2}A&=20\\
\Rightarrow A&=40\end{align}$$ So $B=\frac{3}{2}\times40=60$.
Finding angles A and C in terms of angle B
$ B=\frac{3}{4}C$ so $C=\frac{4}{3}B$.
And $B=\frac{3}{2}A$ so $A=\frac{2}{3}B$.
Since A, B and C are the angles of a triangle, $A+B+C=180$, so $$\begin{align}\tfrac{2}{3}B+B+\tfrac{4}{3}B&=180\\
\Rightarrow 3B&=180\\
\Rightarrow B&=60\end{align}$$