Question

The two horizontal lines shown in the above figure are parallel to each other. Which of the following does NOT equal ${180}^{\circ }$.

$$\begin{gathered} \left(A\right){\left(p+r\right)}^{\circ } \\ \left(B\right){\left(p+t\right)}^{\circ } \\ \left(C\right){\left(q+s\right)}^{\circ } \\ \left(D\right){\left(r+s+t\right)}^{\circ } \\ \left(E\right){\left(t+u\right)}^{\circ } \end{gathered}$$

Answer 1

Hint: We will be solving the question by individually checking the options provided to us. We will use the properties of angles such as
$\left(1\right)$ Sum of Supplementary angles is ${180}^{\circ }$.
$\left(2\right)$ Sum of all interior angles of a triangle is ${180}^{\circ }$.
$\left(3\right)$ Vertical angles are equal.
$\left(4\right)$ Corresponding angles are equal.

Complete step-by-step answer:
Let us add some more angles in the figure in order to understand better
Checking option $$\left(A\right){\left(p+r\right)}^{\circ }$$
As we can observe that
$\mathrm{\angle }p+\mathrm{\angle }m={180}^{\circ }$ (Supplementary angles)
And $\mathrm{\angle }m=\mathrm{\angle }r$ (Corresponding angles)
$\Rightarrow \mathrm{\angle }p+\mathrm{\angle }m=\mathrm{\angle }p+\mathrm{\angle }r={180}^{\circ }$ (Since they are equal)

Checking option $$\left(B\right){\left(p+t\right)}^{\circ }$$
We can see that from the data given we cannot conclude that the value of $${\left(p+t\right)}^{\circ }={180}^{\circ }$$.
Therefore, we will check other options.
Checking option $$\left(C\right){\left(q+s\right)}^{\circ }$$
As we can observe that
$\mathrm{\angle }q+\mathrm{\angle }n={180}^{\circ }$ (Supplementary angles)
And $\mathrm{\angle }n=\mathrm{\angle }s$ (Corresponding angles)
$\Rightarrow \mathrm{\angle }q+\mathrm{\angle }n=\mathrm{\angle }q+\mathrm{\angle }s={180}^{\circ }$ (Since they are equal)
Checking option $$\left(D\right){\left(r+s+t\right)}^{\circ }$$
As we can observe from the figure
$$\begin{gathered} \mathrm{\angle }r=\mathrm{\angle }x \\ \mathrm{\angle }s=\mathrm{\angle }y \\ \mathrm{\angle }t=\mathrm{\angle }z \end{gathered}$$(Vertical angles)
In addition, we know that sum of all interior angles of a triangle $={180}^{\circ }$.Therefore,
$$\begin{gathered} \Rightarrow \mathrm{\angle }x+\mathrm{\angle }y+\mathrm{\angle }z={180}^{\circ } \\ \Rightarrow \mathrm{\angle }r+\mathrm{\angle }s+\mathrm{\angle }t={180}^{\circ } \end{gathered}$$
Therefore $${\left(r+s+t\right)}^{\circ }={180}^{\circ }$$
Checking option $$\left(E\right){\left(t+u\right)}^{\circ }$$
We can see that $\mathrm{\angle }t$ and $\mathrm{\angle }u$ are supplementary angles. Therefore,
$\Rightarrow \mathrm{\angle }t+\mathrm{\angle }u={180}^{\circ }$
After checking all the options, we can Conclude that the options $\left(A\right),\left(C\right),\left(D\right),\left(E\right)$ are all equal to ${180}^{\circ }$. So by eliminating these options, we are only left with option $\left(B\right)$
Hence, the correct answer is $\left(B\right)$.

Note: It should be noted that the angles $r,sandt$ are not the exterior angles of the triangle formed. Therefore, you cannot apply “the sum of exterior angles of a convex polygon is ${360}^0$” property.