Mathematics

# Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.

##### SOLUTION
AB and CD are straight lines intersecting at O. OX the bisector of angles $\angle$AOC and OY is the OY is the bisector of $\angle$BOD.

OY is the bisector of $\angle$BOD.

$\therefore \angle 1=\angle 6$ ……..$(1)$

OX is the bisector of $\angle$AOC

$\therefore \angle 3=\angle 4$ …….$(2)$

$\angle 2=\angle 5$ ……….$(3)$ (vertically opposite angles)

We know that, the sum of the angles formed at a point is $360^o$.

$\therefore \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=360^o$

$\Rightarrow \angle 1+\angle 2+\angle 3+\angle 3+\angle 2+\angle 1=360^o$
(using $1, 2$ and $3$)

$\Rightarrow 2\angle 1+2\angle 2+2\angle 3=360^o$

$2(\angle 1+\angle 2+\angle 3)=360^o$

$\Rightarrow \angle DOY+\angle AOD+\angle AOX=180^o$

$\angle XOY=180^o$

$\therefore$ The bisectors of pair of vertically opposite angles are on the same straight line.

You're just one step away

Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
In $\Delta ABC$ fig, $\angle x + \angle y + \angle z$ is equals :
• A. $120^{\circ}$
• B. $180^{\circ}$
• C. $240^{\circ}$
• D. $360^{\circ}$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q2 Single Correct Medium
Which of the given statements are correct when two straight lines intersect?
(iii) Opposite angles are equal.
(iv) Opposite angles are supplementary.
• A. (i) and (iii) only
• B. (i) and (iv) only
• C. (ii) and (iv) only
• D. (ii) and (iii) only

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q3 Single Correct Medium
Mark the correct alternative of the following.
Two complementary angles are in the ratio $2:3$. The measure of the larger angle is?
• A. $60^o$
• B. $66^o$
• C. $48^o$
• D. $54^o$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q4 Single Correct Medium
Given AB, CD and EF straight   lines intersecting at the point O.  If $a = c < 45$, then:
• A. CD bisects $\angle AOF$
• B. AB bisects $\angle COD$
• C. CD bisects $\angle EOB$
• D. EF bisects $\angle BOD$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020

Q5 Subjective Medium
Classify the angles whose magnitudes are given below:
$328^{o}$

Asked in: Mathematics - Lines and Angles

1 Verified Answer | Published on 09th 09, 2020