Properties of parallel lines cut by a transversal
Introduction When two parallel lines are cut by a transversal, Transversal Lines Parallel lines When two parallel lines are cut by a transversal, Each pair of interior alternate angles are equal Each pair of exterior alternate angles are equal Each pair of corresponding angles are equal Each pair of interior angles on the same side of the transversal are supplementary (i.e 180°) Let us see each rule one by one
Each pair of corresponding angles are equal Rule 1 Each pair of corresponding angles are equal 2 1 6 5 ∠2 = ∠6 ∠1 = ∠5 4 3 8 7 ∠4 = ∠8 ∠3 = ∠7
Each pair of interior alternate angles are equal Rule 2 Each pair of interior alternate angles are equal 3 4 5 6 ∠3= ∠5 ∠4 = ∠6
Each pair of exterior alternate angles are equal Rule 3 Each pair of exterior alternate angles are equal 1 2 8 7 ∠2= ∠8 ∠1 = ∠7
Rule 4 Each pair of interior angles on the same side of the transversal are supplementary (i.e 180°) 4 3 5 6 ∠4 + ∠5 = 180° ∠3 + ∠6 = 180°
Rule 5 Each pair of exterior angles on the same side of the transversal are supplementary (i.e 180°) 1 2 7 8 ∠2 + ∠7 = 180° ∠1 + ∠8 = 180°
Each pair of vertically opposite angles are equal 2 1 4 3 ∠2 = ∠4 ∠1 = ∠3 5 6 7 8 ∠5 = ∠7 ∠6 = ∠8
Sum of pair of adjacent supplementary angles are supplementary (i Sum of pair of adjacent supplementary angles are supplementary (i.e 1800) 1 2 3 2 3 4 1 4 ∠1 + ∠2 = 180 ∠2 + ∠3 = 180 ∠3 + ∠4 = 180 ∠1 + ∠4 = 180 5 6 7 6 7 8 5 8 ∠5 + ∠6 = 180 ∠6 + ∠7 = 180 ∠7 + ∠8 = 180 ∠5 + ∠8 = 180
Pair of angles are equal Sum of pair of angles are 1800 Properties of parallel lines cut by a transversal 3 5 2 1 4 6 7 8 Pair of angles are equal Sum of pair of angles are 1800 Each pair of corresponding angles are equal Each pair of interior angles on the same side of the transversal are supplementary (i.e 1800) Each pair of interior alternate angles are equal Each pair of exterior angles on the same side of the transversal are supplementary (i.e 1800) Each pair of exterior alternate angles are equal Sum of pair of adjacent supplementary angles are supplementary (i.e 1800) Each pair of vertically opposite angles are equal
l || m Example 1: Solution: Lines l || m, t is a transversal, ∠x? Given: l || m t is a transversal To find: ∠x ∠70° and ∠x are the pair of interior alternate angles Ans: ∠ x = 70° Rule2: Each pair of interior alternate angles are equal
Example 2: Solution: Lines l || m, t is a transversal, ∠z? Given: To find: ∠z 98° and ∠ z are the pair of interior angles lying on the same side of the transversal. 98° + ∠ z = 180° 98° + ∠ z - 98° = 180° - 98° (Subtract 98° from both sides of the equation) ∠ z = 82° Rule4: Each pair of interior angles on the same side of the transversal are supplementary (i.e 180°)
Rule1: Each pair of corresponding Example 3: Lines l || m, t is a transversal, ∠x? Solution: Given: l || m t is a transversal To find: ∠x ∠140° and ∠ x are the pair of corresponding angles ∠x = 140° Rule1: Each pair of corresponding angles are equal
Example 4: l 1, l2 be two lines and t is a transversal. Is ∠a = ∠b? Solution: Given: l1, l2 be two lines t is a transversal To show: ∠a = ∠b We know that when a parallel line cut by a transversal, each pair of alternate angles are equal. But l1 and l2 are not parallel lines, so none of the properties holds good. Therefore, ∠a ≠ ∠b
Try These Lines l || m, t is a transversal, ∠x? 2. Lines l || m, t is a transversal, ∠h?