Question

O is the centre of the circle, AM is perpendicular to BC and AO is perpendicular to OD, if
∠BOC = 30֯. Find x and y
please answer correctly, I am counting on you guys

Answer 1

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Answer 2

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CBSE

Mathematics

Grade 8

Mensuration

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Prove that if two lines intersecting each other, then the vertically opposite angles are equal.

Answer

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Hint: We start solving the problem by drawing the two lines which were intersecting each other at a fixed point. We then use the fact that the sum of the angles lying on a straight line is equal to 180∘ (which is also known as linear pair axiom) for the angles present on the both lines. We then make the necessary calculations using the relations obtained between the angles present on the two lines to complete the required proof.

Complete step-by-step answer:

According to the problem, we need to prove that if two lines intersect each other, then the vertically opposite angles are equal.

Let us draw the two lines AB and CD intersecting at point O.

We know that the sum of angles lying on a straight line is 180∘.

Let us consider the angles on the line CD. So, we get ∠COB+∠BOD=180∘.

⇒∠COB=180∘−∠BOD ---(1).

Now, let us consider the angles on the line AB. So, we get ∠COB+∠AOC=180∘ ---(2).

Let us substitute equation (2) in equation (1).

So, we get 180∘−∠BOD+∠AOC=180∘.

⇒∠AOC=180∘−180∘+∠BOD.

⇒∠AOC=∠BOD.

From the figure, we can see that the angles ∠AOC and ∠BOD are vertically opposite angles.

So, we have proved that if two lines intersect each other, then the vertically opposite angles are equal.

Note: We can also prove that the other pair of vertically opposite angles ∠COB and ∠DOA equal in the similar way as shown below:

Let us consider the angles on the line CD. So, we get ∠COB+∠BOD=180∘.

⇒∠BOD=180∘−∠COB ---(3).

Now, let us consider the angles on the line AB. So, we get ∠BOD+∠DOA=180∘ ---(4).

Let us substitute equation (3) in equation (4).

So, we get 180∘−∠COB+∠DOA=180∘.

⇒∠DOA=180∘−180∘+∠COB.

⇒∠DOA=∠COB.

We use this result to get the required answers.

Step-by-step explanation:

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