1.In the given figure l and m are two lines which intersect at  O forming four angles a, b, c, d.

       a. Name two pairs of equal angles
       b. Name four pairs of adjacent angles
       c. Name four pairs of angles forming a linear pair
       d. If a = 105^o then find the measure of b,c and d

a. ∠a and ∠c, ∠b and ∠d (vertically opposite angles)
b. ∠a and ∠b, ∠b and ∠c, ∠c and ∠d, ∠d and ∠a.
c. ∠a and ∠b, ∠b and ∠c, ∠c and ∠d, ∠d and ∠a.
d. ∠b = 180 - 105^o (linear pair)
          = 75^o
    ∠d = ∠b = 75^o (vertically opposite angles)
    ∠c = ∠a = 105^o (Vertically opposite angles)

2. An angle is 10^o more than its complement. Find the angle.

Let x be the required angle. Its complement = 90 - x^o. Then 

                            x = 90 - x^o + 10
                    ⇒  2x^o  = 100
                 

3. Find the complement of    a. 56^o      b. 38^o.

a. Complement of  56^o    = 90 - 56
                                             = 34^o
b. Complement of 38^o     = 90^o - 38^o
                                             = 52^o.

4. Find the supplement of      a. 105^o         b. 198^o

a. Supplement of 105^o   =  180^o - 105^o
                                            =  75^o
b. Since 198^o is greater than 180^o, it has no supplement.

5. Two supplementary angles are such that one is two third of the other. Find them.

Let x and 180^o - x are the supplementary angles. Then according to the given condition.

                 x  = ²/₃ (180^o-x)

                ⇒  3 x = 360^o - 2x
                ⇒  5x = 360^o
                 
∴ The angles are 72^o and 180 - 72^o.

        ie = 72^o and 108⁰

6. Find the angle whose measure is equal to the measure of     a. its complement     b. its supplement.

Let x be the required angle.

Its complement   = 90^o - x. Then

            x = 90^o - x    ⇒  2 x   = 90^o
                                 

b. Let y be the required angle.

Its supplement   = 180 - x^o . Then
x  = 180 - x^o   ⇒  2 x = 180^o
                      

7.    Find the value of x.

∠AOC  + ∠COD   = 180^o (linear pair)
that is, ∠ AOB + ∠BOC + ∠COD  = 180^o
⇒   x^o + 2x^o  + 3x  = 180^o
 ⇒      6x^o  = 180^o

8. Find the value of x if p||q.

Since p||q, x and 110^o are interior angles of the same side.

Therefore they are supplementary.

     ∴  110^o + x  = 180^o
   ⇒  x  = 180^o  - 110^o
            = 70^o

^9. Find the measure of ∠A, ∠B, ∠C, ∠D, ∠E in the given figure if x||y and y||z.

∠A  = 130^o (vertically opposite angles)

∠B = 130^o (Alternate angles)

∠C = 180^o - 130^o (Linear pair)
      = 50^o
∠D = ∠C = 50^o (Alternate angles)
∠E = ∠D = 50^o (Vertically opposite angles).

10  . In the given figure P||q. Find measure of ∠X, ∠Y, ∠Z, ∠W.

Since R is the transversal, ∠X and ∠Z are corresponding angles.

               ∠X   = 120^o  (Vertically opposite angles)
               ∠Z   = ∠X  = 120^o (Corresponding angles)
               ∠Y  = 55^o  (VOA)
               ∠W = ∠Y = 55^o (Alternate angles)