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TOPICS
Exercise - 3.3

Question-1 :-  Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = ......   (ii) ∠ DCB = ......   (iii) OC = ......   (iv) m ∠DAB + m ∠CDA = ......

Solution :-
```(i) AD = BC [Since opposite sides of a parallelogram are equal]
(ii) ∠ DCB = ∠ DAB [Since opposite angles of a parallelogram are equal]
(iii) OC = OA	[Since diagonals of a parallelogram bisect each other]
(iv)  m ∠DAB + m ∠CDA = = 180 [Adjacent angles in a parallelogram are supplementary]

```

Question-2 :-  Consider the following parallelograms. Find the values of the unknowns x, y, z.

Solution :-
```(i) ∠ B + ∠ C = 180° [Adjacent angles in a parallelogram are supplementary]
100° + x = 180°
And Also,
x = 180° - 100° = 80°
z = x = 80° [Since opposite angles of a parallelogram are equal]
y = 100° [Since opposite angles of a parallelogram are equal]
```
```(ii) x + 50° = 180° [Adjacent angles in a //gm are supplementary]
x = 180° - 50° = 130°
z = x = 130° [Corresponding angles]
```
```(iii) x = 90° [Vertically opposite angles]
y + x + 30° = 180° [Angle sum property of triangle]
y + 90° + 30° = 180°
y + 120° = 180°
y = 180° - 120°
y = 60°
z = y = 60° [Alternate angles]
```
```(iv) z = 80° [Corresponding angles]
x + 80° = 180° [Adjacent angles in a //gm are supplementary]
x = 180° - 80°
x = 100°
y = x = 100° [opposite angles are equal in a //gm]
```
```(v) y = 112° [opposite angles]
x + y + 40° = 180° [Angle sum property of triangle]
x + 112° + 40° = 180°
x + 152° = 180°
x = 180° - 152°
x = 28°
z = x = 28° [Alternate angles]
```

Question-3 :-  Can a quadrilateral ABCD be a parallelogram if
(i) ∠D + ∠B = 180°?   (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?   (iii) ∠A = 70° and ∠C = 65°?

Solution :-
```(i)It can be , but not always as you need to look for other criteria as well.

(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram.

(iii) Here opposite angles are not equal, so it is not a parallelogram.
```

Question-4 :-  Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Solution :-
```  ABCD is a quadrilateral in which angles ∠ A = ∠ C = 110°.
Therefore, it could be a kite.

```

Question-5 :-  The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Solution :-
```
Let two adjacent angles be  3x  and 2x.
Since the adjacent angles in a parallelogram are supplementary.
3x + 2x = 180°
5x = 180°
x = 180°/5
x = 36°
one angle = 3x = 3 x 36° = 108°
second angle = 2x = 2 x 36° = 72°
```

Question-6 :-  Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Solution :-
```  Let each adjacent angle be x.
Since the adjacent angles in a parallelogram are supplementary.
x + x = 180°
2x = 180°
x = 180°/2
x = 90°
So, each adjacent angle is 90°.
```

Question-7 :-  The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

Solution :-
```  Angle opposite to y = 180° - 70° = 110°
Hence, y = 40° [alternate angle of ∠ PHE]
x = 180° - (110° + 40°) = 30°, [triangle’s angle sum]
z = 30° [Alternate angle of a transversal]
```

Question-8 :-  The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

Solution :-
```(i) As opposite sides are equal in a parallelogram.
so, In //gm GUNS
3y - 1 = 26
3y = 26 + 1
3y = 27
y = 27/3
y = 9 cm
similarly,
3x = 18
x = 18/3
x = 6 cm
```
```(ii) As you know diagonals bisect each other in a parallelogram.
so, In //gm RUNS
y + 7 = 20   [Diagonals of //gm bisects each other]
y = 20 - 7
y = 13 cm
similarly.
x + y = 16
x + 13 = 16
x = 16 - 13
x = 3 cm
```

Question-9 :-
In the above figure both RISK and CLUE are parallelograms. Find the value of x.

Solution :-
```  In //gm RISK,
∠ RIS = ∠ K = 120°   [opposite angles of //gm are equal]
∠ m + 120° = 180°      [Linear Pair]
∠ m = 180° - 120°
∠ m = 60°

Now, ∠ ECI = ∠ L = 70°  [Corresponding Angles]
m + n + ∠ ECI = 180° [Angle sum property of triangle]
60° + n + 70° = 180°
130° + n = 180°
n = 180° - 130°
n = 50°
Also, x = n = 50° [Vertically opposite angles]
```

Question-10 :-  Explain how this figure is a trapezium. Which of its two sides are parallel?

Solution :-
```  Here, ∠M + ∠L = 100° + 80° = 180°  [sum of interior opposite angles]
NM and KL are parallel.
So, KLMN is a trapezium.
```

Question-11 :-  Find m∠C in Figure if AB||DC .

Solution :-
```  Here, ∠B + ∠C = 180°  [AB||DC]
120° + m∠C = 180°
m∠C = 180° - 120°
m∠C = 60°
```

Question-12 :-  Find the measure of ∠P and ∠S if SP||RQ in Figure. (If you find m∠R, is there more than one method to find m∠P?)

Solution :-
```  Here, ∠P + ∠Q = 180°  [sum of co-interior angles]
∠P + 130° = 180°
∠P = 180° - 130°
∠P = 50°
∠R = 90°
∠S + 90° = 180°  [Given]
∠S = 180° - 90°
∠S = 90°
```
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