Question

In a triangle $ABC$, angle $A$ $-$ angle $B$ $=15$ degrees and angle $B$ - angle $C$ $=30$ degrees. Find angles $A,B$ and $C$.

Answer 1

Step $1:$ Obtain the equations for each angel.

It is given that, angle $A$ $-$ angle $B$ $=15$ degrees.

$$\begin{gathered} \Rightarrow \angle A-\angle B=15° \\ \Rightarrow \angle A=15°+\angle B...1 \end{gathered}$$

Also, angle $B$ - angle $C$ $=30$ degrees.

$$\begin{gathered} \Rightarrow \angle B-\angle C=30° \\ \Rightarrow \angle C=\angle B-30°...2 \end{gathered}$$

According to the angle sum property of the triangle.

$\angle A+\angle B+\angle C=180°...3$

Step $2:$ Compute the values of all the angles.

From equation $1$, equation $2$, and equation $3$ we get,

$$\begin{gathered} 15°+\angle B+\angle B+\angle B-30°=180° \\ \Rightarrow 3\angle B-15°=180° \\ \Rightarrow 3\angle B=195° \\ \Rightarrow \angle B=65° \end{gathered}$$

Therefore, the value of $\angle B$ is $65°$.

Now, substitute the value of $\angle B$ in equation $1$.

$$\begin{gathered} \Rightarrow \angle A=65°+15° \\ \Rightarrow \angle A=80° \end{gathered}$$

Therefore, the value of $\angle A$ is $80°$.

Now, substitute the value of $\angle B$ in equation $2$.

$$\begin{gathered} \Rightarrow \angle C=65°-30° \\ \Rightarrow \angle C=35° \end{gathered}$$

Therefore, the value of $\angle C$ is $35°$.

Hence, the value of angle $A,B$ and $C$ are $80°,65°$ and $35°$ respectively.