Q1. Proof the opposite angles formed by 2 intersecting lines are of equal measures
Q2. Coordinates of some pairs are given below. Hence find the distance between each pair
a. y+2, y-2 b.31, -42
Q3. On a number line, points X, Y and Z are such that d( X,Z) = 11.5,d(Z,Y)= 7.5. Find d (X,Y) considering all possibilities.
Q4. Write the following statement in conditional form and also its Converse statement
Answers
STEP BY STEP EXPLANATON :-
When two lines intersect each other, then the opposite angles, formed due to intersection are called vertical angles or vertically opposite angles. A pair of vertically opposite angles are always equal to each other. Also, a vertical angle and its adjacent angle are supplementary angles, i.e., they add up to 180 degrees. For example, if two lines intersect and make an angle, say X=45°, then its opposite angle is also equal to 45°. And the angle adjacent to angle X will be equal to 180 – 45 = 135°.
When two lines meet at a point in a plane, they are known as intersecting lines. When the lines do not meet at any point in a plane, they are called parallel lines. Learn about Intersecting Lines And Non-intersecting Lines here.
Definition
As we have discussed already in the introduction, the vertical angles are formed when two lines intersect each other at a point. After the intersection of two lines, there are a pair of two vertical angles, which are opposite to each other.
The given figure shows intersecting lines and parallel lines.
Intersecting and parallel lines
In the figure given above, the line 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°
HOPE IT HELPS