Question
Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line.
✓
Answer
Draw two lines $AB$ and $CD$ intersecting each other at $O.$
We know that the vertically opposite angles are equal.
Therefore, $\angle\text{BOC}=\angle\text{AOD}$ and
$\angle\text{AOC}=\angle\text{BOD}.$
We bisect angle $AOC$ and draw the bisecting ray as $OX.$
Similarly, we bisect angle $BOD$ and draw the bisecting ray as $OY.$
Now, $\angle\text{XOA}+\angle\text{AOD}+\angle\text{DOY}$
$=\frac{1}{2}\angle\text{AOC}+\angle\text{AOD}+\frac{1}{2}\angle\text{BOD}$
$=\frac{1}{2}\angle\text{BOD}+\angle\text{AOD}+\frac{1}{2}\angle\text{BOD}$
$[\text{As,}\angle\text{AOC}=\angle\text{BOD}]$
$=\angle\text{AOD}+\angle\text{BOD}$
Since, $AB $ is a line.
Therefore, $\angle\text{AOD}$ and $\angle\text{BOD}$ are supplementary angles and the sum of these two angles will be $180^\circ .$
Therefore, $\angle\text{XOA}+\angle\text{AOD}+\angle\text{DOY}=180^{\circ}$
We know that the angles on one side of a straight line will always add to $180^\circ .$
Also, the sum of the angles is $180^\circ .$
Therefore, $XY$ is a straight line.
Thus, $OX$ and $OY$ are in the same line.
We know that the vertically opposite angles are equal.
Therefore, $\angle\text{BOC}=\angle\text{AOD}$ and
$\angle\text{AOC}=\angle\text{BOD}.$
We bisect angle $AOC$ and draw the bisecting ray as $OX.$
Similarly, we bisect angle $BOD$ and draw the bisecting ray as $OY.$
Now, $\angle\text{XOA}+\angle\text{AOD}+\angle\text{DOY}$
$=\frac{1}{2}\angle\text{AOC}+\angle\text{AOD}+\frac{1}{2}\angle\text{BOD}$
$=\frac{1}{2}\angle\text{BOD}+\angle\text{AOD}+\frac{1}{2}\angle\text{BOD}$
$[\text{As,}\angle\text{AOC}=\angle\text{BOD}]$
$=\angle\text{AOD}+\angle\text{BOD}$
Since, $AB $ is a line.
Therefore, $\angle\text{AOD}$ and $\angle\text{BOD}$ are supplementary angles and the sum of these two angles will be $180^\circ .$
Therefore, $\angle\text{XOA}+\angle\text{AOD}+\angle\text{DOY}=180^{\circ}$
We know that the angles on one side of a straight line will always add to $180^\circ .$
Also, the sum of the angles is $180^\circ .$
Therefore, $XY$ is a straight line.
Thus, $OX$ and $OY$ are in the same line.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Alok purchased $1\ kg\ 200\ g$ potatoes, $250\ g$ dhania, $5\ kg\ 300\ g$ onion, $500\ g$ palak and $2\ kg\ 600\ g$ tomatoes. Find the total weight of his purchases in kilograms.
→Compare $\frac{4}{5}$ and $\frac{5}{6}$.
→Use number line and add the integers: $(-1) + (-7).$
→Construct with ruler and compasses, angles of measure: $45^\circ $
→A room is $8m$ long and $6m$ wide. Its floor is to be covered with rectangular tiles of size $25\ cm$ by $20\ cm$. How many tiles will be required? Find the cost of these tiles at $Rs. 25$ per tile.
→Find the $LCM$ of $12$ and $18.$
→Use a pair of compasses and construct the following angles:$15^\circ $
→Construct a right angle $A B C$ with the help of a protractor. Mark a point $D$ in the interior of $\angle A B C$. Draw ray $B D$. Measure $\angle A B D, \angle D B C$. Verify that they are complementary.
→Draw number line and locate the points on them: $\frac18,\;\frac28,\frac38,\;\frac78$.
→If $a, b, c,$ are in proportion, then
→