#### Topics

1.In the adjoining figure l and m are a pair of coplanar lines and ‘ p ‘ is the transversal intersecting them. If ∠A = 60o and ∠G = 85o, then is l || m ? State reasons. l and m are a pair of coplanar lines.

P is the transversal intersecting them.

∠A = 60o

If ∠A =∠ E = ∠G , then l and m will be parallel.

But ∠G is given as 85o.

∴ l and m are not parallel.

2. In the adjoining figure l and m are a pair of coplanar lines and p is the transversal intersecting them. If ∠A = ∠E = 65o, then 1) ∠B = ?    2) ∠C = ?   3) ∠G= ?   4) ∠H= ? l and m are a pair of coplanar line.

p is the transversal intersecting them.

Given ∠A = ∠E = 65o

1. ∠A and ∠B are a linear pair

∴  ∠A + ∠B = 180o

⇒ B = 180o – ∠A

= 180o – 65o = 115o

2. ∠C= 65o  [ A and C are vertically opposite angles]

3. ∠G = ∠E (VOA)

But ∠A = ∠E  (corresponding angles)

But given ∠A = 65

⇒ G = 65o

4. ∠A = ∠E (corresponding angles)

But ∠A = 65o

∠E = 65o

∠E + ∠H = 180o

⇒ ∠H = 180o – 65o = 115o

3. In the adjoining figure l and m are two lines intersected by a transversal P. If ∠1 = 130o and ∠8 = 50o, then l || m? State  reasons. l and m are two lines intersected by a transversal P.

Given ∠1 = 130o and ∠8 = 50o

∠1 = 5 (corresponding angles)

∠5 = 130o

∠8 and ∠5 are a linear pair.

∠5 + ∠8 = 180o

∠8 = 180o – 130o = 50o

It is given that ∠8 = 50o

∴ l is parallel to m.

4. In the below figure lines m and n are parallel and l is the transversal intersecting them. If ∠A = 100o, then 1. Find all the other angles.
2. If m is not parallel to n, then we can find all the other angles?

m || n and l is the transversal.

Given ∠A = 100o

1. ∠A + ∠B = 180o

⇒∠B = 180o – 100o = 80o

∠A = ∠C

⇒  C = 100o

∠B = ∠D = 80o

∠E = ∠A (corresponding angles)

∴ ∠E = 100o

∠B = ∠F = 80o

∠G = 100o  ( ∠C = ∠G)

∠H = 80o ( ∠F = ∠H)

2. No, we cannot find other angles.

5. In the adjoining figure p||q and r is the transversal intersecting the lines p and q. If ∠A : ∠B = 2 : 3, then find all the other angles. Given ∠A : ∠B = 2 : 3

But ∠A + ∠B = 180o

∠A = 180o × 2/5 = 72o

∠B = 180o × 3/5 = 108o

∠C = 72 (∠A and ∠C are vertically opposite angles)

∠D = 108o ( ∠B = ∠D)

∠E = 72o ( ∠A = ∠E)

∠F = 108o ( ∠B and ∠F are corresponding angles)

∠G = 720o ( ∠G = ∠C)

∠H = 108o

6. In the adjoining figure p and q are intersecting by a transversal l. If ∠A = 120o and ∠B = 45o then , is p||q? Give reason. p and q are intersecting lines and l is the transversal.

Given ∠A = 120o and ∠B = 45o

∠A + ∠B = 120o + 45o = 165o

If the sum of the interior angles on the same side of the transversal is 180o , then they are parallel. But here it is 165o only.

∴ p and q are not parallel lines.

7. In the adjoining figure l and m are two coplanar line intersected by a transversal 'n'. If ∠A = ∠B, then, is l || m? State reason. Given ∠A = ∠B  → (1)

But ∠A = ∠K     → (2)

(Vertically opposite angles)

From (1) and (2)

∠B = ∠K

But ∠B and ∠K are corresponding angles.

∴ l || m

8. In the adjoining figure is the transversal intersecting them. Then is ∠x = ∠y? State reasons. Given  is the transversal intersecting them. are not parallel lines.

∴ ∠x ≠ ∠y

If are parallel then only ∠x = ∠y.

9. In the adjoining figure then show that ∠x = ∠y. Hypothesis : Conclusion : We have to prove ∠x = ∠y

Construction : Produce Proof : is the transversal.

∠y = ∠k  → (1)

(Corresponding angle) is the transversal.

∠x = ∠k  → (2)

From (1) and (2) we infer that ∠x = ∠y.

10. In the adjoining figure l || m. A is a point on l and B is a point on m. C is a point not on l or m, show that ∠z = ∠x + ∠y. Hypothesis : l || m ; A is a point on l; and B is a point on m. C is not a point either on l or m.

Conclusion : ∠z = ∠x + ∠y

Construction : Through C draw a line parallel to l and m.

Proof : m || p BC is the transversal.

∠x = ∠BCR (Alternate angles)

∠y = ∠ACR  (Alternate angles.

Now ∠x + ∠y = ∠BCD + ∠ACR

= ∠ACB = ∠z

∴ ∠x + ∠y = ∠z

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