Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

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

\[
  \left( A \right)\quad {\left( {p + r} \right)^\circ } \\
  \left( B \right)\quad {\left( {p + t} \right)^\circ } \\
  \left( C \right)\quad {\left( {q + s} \right)^\circ } \\
  \left( D \right)\quad {\left( {r + s + t} \right)^\circ } \\
  \left( E \right)\quad {\left( {t + u} \right)^\circ } \\
 \]

seo-qna
SearchIcon
Answer
VerifiedVerified
444.6k+ views
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 \[\left( A \right)\quad {\left( {p + r} \right)^\circ }\]
As we can observe that
$\angle p + \angle m = {180^\circ }$ (Supplementary angles)
And $\angle m = \angle r$ (Corresponding angles)
$ \Rightarrow \angle p + \angle m = \angle p + \angle r = {180^\circ }$ (Since they are equal)

Checking option \[\left( B \right)\quad {\left( {p + t} \right)^\circ }\]
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 \[\left( C \right)\quad {\left( {q + s} \right)^\circ }\]
As we can observe that
$\angle q + \angle n = {180^\circ }$ (Supplementary angles)
And $\angle n = \angle s$ (Corresponding angles)
$ \Rightarrow \angle q + \angle n = \angle q + \angle s = {180^\circ }$ (Since they are equal)
Checking option \[\left( D \right)\quad {\left( {r + s + t} \right)^\circ }\]
As we can observe from the figure
$
  \angle r = \angle x \\
  \angle s = \angle y \\
  \angle t = \angle z \\
 $(Vertical angles)
In addition, we know that sum of all interior angles of a triangle $ = {180^\circ }$.Therefore,
$
   \Rightarrow \angle x + \angle y + \angle z = {180^\circ } \\
   \Rightarrow \angle r + \angle s + \angle t = {180^\circ } \\
 $
Therefore \[{\left( {r + s + t} \right)^\circ } = 180^\circ \]
Checking option \[\left( E \right)\quad {\left( {t + u} \right)^\circ }\]
We can see that $\angle t$ and $\angle u$ are supplementary angles. Therefore,
$ \Rightarrow \angle t + \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 $r,s\;and\;t$ 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.