Question

In the given figure $$\mathrm{\angle }b={70}^{\circ }$$. Find $$\mathrm{\angle }a,\mathrm{\angle }c,\mathrm{\angle }d$$.

Answer 1

Hint: In this problem, we are given a diagram, where one of the angles is given and we have to find the other angels. We can see that we have $$\mathrm{\angle }b={70}^{\circ }$$ which is opposite to $$\mathrm{\angle }d$$. We know that vertical angles are opposite to each other and when the two angles are across from each other are called vertical angles which are congruent and they are equal to each other. So we can write $$\mathrm{\angle }b=\mathrm{\angle }d$$. We can now use the supplementary angles to find the remaining angles.

Complete step by step answer:
Here we are given a diagram in which $$\mathrm{\angle }b={70}^{\circ }$$and we have to find $$\mathrm{\angle }a,\mathrm{\angle }c,\mathrm{\angle }d$$.

 Here we can see that we have $$\mathrm{\angle }b={70}^{\circ }$$ which is opposite to $$\mathrm{\angle }d$$.
We know that vertical angles are opposite to each other. When the two angles are across from each other they are called vertical angles, which are congruent so they are equal to each other. So, we can write as,
 $$\Rightarrow \mathrm{\angle }b=\mathrm{\angle }d={70}^{\circ }$$.
We can now use the supplement angels to find angle a and c.
We know that the sum of two angles is equal to $${180}^{\circ }$$ as supplementary angles.
We can see in the diagram that angle d and c form a straight line whose angle is $${180}^{\circ }$$.
We can now write it as
$$\begin{array}{rl}& \Rightarrow \mathrm{\angle }c+\mathrm{\angle }d={180}^{\circ }\\ & \Rightarrow \mathrm{\angle }c={180}^{\circ }-{70}^{\circ }={110}^{\circ }\end{array}$$
Similarly, angle a and c form a straight line whose angle is $${180}^{\circ }$$.
$$\begin{array}{rl}& \Rightarrow \mathrm{\angle }a+\mathrm{\angle }d={180}^{\circ }\\ & \Rightarrow \mathrm{\angle }c={180}^{\circ }-{70}^{\circ }={110}^{\circ }\end{array}$$
 Therefore, $$\mathrm{\angle }a=\mathrm{\angle }c={110}^{\circ }\text{ and }\mathrm{\angle }d={70}^{\circ }$$.

Note: We should always remember that vertical angles are opposite to each other. When the two angles are across from each other they are called vertical angles, which are congruent so they are equal to each other. We should also know that the sum of two angles is equal to $${180}^{\circ }$$ as supplementary angles.