Question

2) prove that if two lines intersect each other, then the
vertically opposite angles are equal​

Answer 1

Answer:

Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.

Proof

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC.

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. 

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)Can we write ∠ AOD + ∠ BOD = 180°?  (Linear pair axiom)……………(2)

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)Can we write ∠ AOD + ∠ BOD = 180°?  (Linear pair axiom)……………(2)From (1) and (2), we can write

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)Can we write ∠ AOD + ∠ BOD = 180°?  (Linear pair axiom)……………(2)From (1) and (2), we can write∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)Can we write ∠ AOD + ∠ BOD = 180°?  (Linear pair axiom)……………(2)From (1) and (2), we can write∠ AOC + ∠ AOD = ∠ AOD + ∠ BODThis implies that ∠ AOC = ∠ BOD

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC. We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.Now, ray OA stands on line CD.Therefore, ∠ AOC + ∠ AOD = 180°  (Linear pair axiom) ………..(1)Can we write ∠ AOD + ∠ BOD = 180°?  (Linear pair axiom)……………(2)From (1) and (2), we can write∠ AOC + ∠ AOD = ∠ AOD + ∠ BODThis implies that ∠ AOC = ∠ BODSimilarly, it can be proved that ∠AOD = ∠BOC

Step-by-step explanation:

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