Question

how to proof a vertically opposite angle theorem

Answer 1

friend here is your right answer



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. The given figure shows intersecting and parallel lines.



In the figure given above, the line segment  and  meet at the point  and these represent two intersecting lines. The line segment  and represent two parallel lines as they have no common intersection point in the given plane. Infinite lines can pass through a single point and hence through , multiple intersecting lines can be drawn. Similarly, infinite parallel lines can be drawn parallel to  and  . Also, it is worth noting that the perpendicular distance between two parallel lines is constant.

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. These angles are also known as vertical angles or opposite angles.

For a pair of opposite angles the following theorem, known as vertical angle theorem holds true:

Theorem: In a pair of intersecting lines the vertically opposite angles are equal.

Proof: Consider two lines  and which intersect each other at . The two pairs of vertical angles are: i) ∠AOD and ∠COB ii) ∠AOC and ∠BOD as shown.



It can be seen that ray  stands on the line  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,  stands on the line .

Therefore, ∠AOC + ∠BOC = 180° —(2) (Linear pair of angles)

From (1) and (2),

∠AOD + ∠AOC = ∠AOC + ∠BOC

⇒ ∠AOD = ∠BOC —(3)

Also,  stands on the line ..

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.

Consider the figure given below to understand this concept.



In the given figure ∠AOC = ∠BOD and ∠COB = ∠AOD(Vertical Angles)

⇒ ∠BOD = 105° and ∠AOD = 75°

Answer 2

Given two lines AB and CD intersect each other at the point O.
To prove: ∠1 = ∠3 and ∠2 = ∠4
Proof:
From the figure, ∠1 + ∠2 = 180°  [Linear pair]  → (1)
∠2 + ∠3 = 180°  [Linear pair]  → (2)
From (1) and (2), we get
∠1 + ∠2 = ∠2 + ∠3
∴ ∠1 = ∠3
Similarly, we can prove ∠2 = ∠4 also.