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theorem of chap.6 |
| Answer» Theorem 6. 1 :\xa0If two lines intersect each other, then the vertically opposite angles are equalGiven :-\xa0two lines intersect each otherTo prove :- ∠AOC = ∠BOD and ∠AOD = ∠BOCProof :- AB and CD be two lines intersecting at O as shown in Fig 6 8 They lead to two pairs ofvertically opposite angles, namely, (i) ∠AOC and ∠ BOD(ii) ∠ AOD and ∠ BOCWe need to prove that ∠AOC = ∠BODand ∠AOD = ∠BOCNow, ray OA stands on line CDFrom (Linear pair axiom)∠AOC + ∠AOD = 180° … (1)∠AOD + ∠BOD = 180° … (2)From (1) and (2), we can write∠AOC + ∠AOD = ∠AOD + ∠BODThis implies that ∠AOC = ∠BOD\xa0Similarly, it can be proved that ∠AOD = ∠BOC\xa0Parallel Lines and a Transversal∠1, ∠ 2, ∠7 and ∠ 8 are called\xa0exterior angles,∠ 3, ∠ 4, ∠ 5 and ∠6 are called\xa0interior angles(a)\xa0Corresponding angles :(i) ∠1 = ∠ 5 (ii) ∠ 2 = ∠ 6 (iii) ∠ 4 = ∠ 8 (iv) ∠ 3 = ∠ 7(b)\xa0Alternate interior angles\xa0:(i) ∠ 4 = ∠ 6 (ii) ∠ 3 = ∠ 5(c)\xa0Alternate exterior angles:(i) ∠1 = ∠ 7 (ii) ∠ 2 = ∠8(d)\xa0Interior angles on the same side of the transversal:(i) ∠4 + ∠5 =180°(ii) ∠3 + ∠ 6 =180° | |