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

l and m are a pair of coplanar lines.

P is the transversal intersecting them.

∠A = 60^o

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

But ∠G is given as 85^o.

∴ 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 = 65^o, 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 = 65^o

1. ∠A and ∠B are a linear pair

            ∴  ∠A + ∠B = 180^o 

            ⇒ B = 180^o – ∠A

                    = 180^o – 65^o = 115^o

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

     3. ∠G = ∠E (VOA)

         But ∠A = ∠E  (corresponding angles)

        But given ∠A = 65^o  

       ⇒ G = 65^o

      4. ∠A = ∠E (corresponding angles)

           But ∠A = 65^o

                 ∠E = 65^o

         ∠E + ∠H = 180^o

      ⇒ ∠H = 180^o – 65^o = 115^o

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

l and m are two lines intersected by a transversal P.

Given ∠1 = 130^o and ∠8 = 50^o

∠1 = 5 (corresponding angles)

∠5 = 130^o

∠8 and ∠5 are a linear pair.

∠5 + ∠8 = 180^o

∠8 = 180^o – 130^o = 50^o

It is given that ∠8 = 50^o

∴ 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 = 100^o, 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 = 100^o

      1. ∠A + ∠B = 180^o

       ⇒∠B = 180^o – 100^o = 80^o                                                                         

          ∠A = ∠C                                                                              

      ⇒  C = 100^o                                                                                                      

          ∠B = ∠D = 80^o                                                                                           

         ∠E = ∠A (corresponding angles)

     ∴ ∠E = 100^o

         ∠B = ∠F = 80^o

        ∠G = 100^o  ( ∠C = ∠G)

        ∠H = 80^o ( ∠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 = 180^o

∠A = 180^o × ²/5 = 72^o

∠B = 180^o × ³/5 = 108^o

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

∠D = 108^o ( ∠B = ∠D)

∠E = 72^o ( ∠A = ∠E)

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

∠G = 720^o (∠G = ∠C)

∠H = 108^o

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

p and q are intersecting lines and l is the transversal.

Given ∠A = 120^o and ∠B = 45^o

∠A + ∠B = 120^o + 45^o = 165^o

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