Prove that opposite angles formed by two intersecting lines are of equal measures
Answers
Answer:
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Step-by-step explanation:
segment AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯ meet at the point O and these represent two intersecting lines. The line segment PQ¯¯¯¯¯¯¯¯ and RS¯¯¯¯¯¯¯ represent two parallel lines as they have no common intersection point in the given plane.
In a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. In the figure given above, ∠AOD and ∠COB form a pair of vertically opposite angle and similarly ∠AOC and ∠BOD form such a pair. Therefore,
∠AOD = ∠COB
∠AOC = ∠BOD
For a pair of opposite angles the following theorem, known as vertical angle theorem holds true.
Note: A vertical angle and its adjacent angle is supplementary to each other. It means they add up to 180 degrees
Vertical Angles: Theorem and Proof
Theorem: In a pair of intersecting lines the vertically opposite angles are equal.
Proof: Consider two lines AB←→ and CD←→ which intersect each other at O. The two pairs of vertical angles are:
i) ∠AOD and ∠COB
ii) ∠AOC and ∠BOD
Vertically opposite angles
It can be seen that ray OA¯¯¯¯¯¯¯¯ stands on the line CD←→ and according to Linear Pair Axiom, if a ray stands on a line, then the adjacent angles form a linear pair of angles.
Therefore, ∠AOD + ∠AOC = 180° —(1) (Linear pair of angles)
Similarly, OC¯¯¯¯¯¯¯¯ stands on the line AB←→.
Therefore, ∠AOC + ∠BOC = 180° —(2) (Linear pair of angles)
From (1) and (2),
∠AOD + ∠AOC = ∠AOC + ∠BOC
⇒ ∠AOD = ∠BOC —(3)
Also, OD¯¯¯¯¯¯¯¯ stands on the line AB←→.
Therefore, ∠AOD + ∠BOD = 180° —(4) (Linear pair of angles)
From (1) and (4),
∠AOD + ∠AOC = ∠AOD + ∠BOD
⇒ ∠AOC = ∠BOD —(5)
Thus, the pair of opposite angles are equal.
Hence, proved.
Solved Example
Consider the figure given below to understand this concept.
Vertically Opposite Angles - Example
In the given figure ∠AOC = ∠BOD and ∠COB = ∠AOD(Vertical Angles)
⇒ ∠BOD = 105° and ∠AOD = 75°
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Frequently Asked Questions – FAQs
What is vertical angles?
When two lines intersect each other, then the angles opposite to each other are called vertical angles.
How to measure vertical angles?
If the angle next to the vertical angle is given to us, then we can subtract it from 180 degrees to get the measure of vertical angle, because vertical angle and its adjacent angle are supplementary to each other.
If x=30 degrees is a vertical angle, when two lines intersect, then find all the angles?
Given, vertical angle, x = 30
Let y is the angle vertically opposite to x, then y = 30 degrees
Now, as we know, vertical angle and its adjacent angle add up to 180 degrees, therefore,
The other two angles are: 180 – 30 = 150 degrees
What are complementary angles with example?
The angles which are adjacent to each other and their sum is equal to 90 degrees, are called complementary angles. For example, x = 45 degrees, then its complement angle is: 90 – 45 = 45 degree
