Q.1 Given a parallelogram ABCD. Complete each statement along with the definition or property used.(i) AD =......
(ii) âˆ DCB =......
(iii) OC =......
(iv) m âˆ DAB + m âˆ CDA = ......
Sol. (i) AD = BC
In a parallelogram, opposite sides are equal.
(ii) âˆ DCB = âˆ DAB
In a parallelogram, opposite angles are equal.
(iii) OC = OA
In a parallelogram, diagonals bisect each other.
(iv) m âˆ DAB + m âˆ CDA = 180áµ’.
In a parallelogram, adjacent angles are supplementary to each other.
Q.2 Consider the following parallelograms. Find the values of the unknowns x, y, z.Sol. (i) In a parallelogram, adjacent angles are supplementary to each other.
Therefore, âˆ B + âˆ C = 180áµ’
100áµ’ + x = 180áµ’
x = 180áµ’ - 100áµ’
x = 80áµ’
In a parallelogram, opposite angles are equal.
Therefore, z = x = 80áµ’
In a parallelogram, opposite angles are equal.
Therefore, y = âˆ B = 100áµ’
(ii) In a parallelogram, adjacent angles are supplementary to each other.
x + 50áµ’ = 180áµ’
x = 180áµ’ - 50áµ’
x = 130áµ’
In a parallelogram, opposite angles are equal.
Therefore, y = x = 130áµ’
In a parallelogram, corresponding angles are equal.
Therefore, z = x = 130áµ’
(iii) In a parallelogram, vertically opposite angles are equal.
Therefore, x = 90áµ’
We know that, sum of interior angles of triangle is 180áµ’
y + x + 30áµ’ = 180áµ’
y + 90áµ’ + 30áµ’ = 180áµ’
y + 120áµ’ = 180áµ’
y = 180áµ’ - 120áµ’
y = 60áµ’
In a parallelogram, alternate angles are equal.
Therefore, z = y = 60áµ’
(iv) In a parallelogram, corresponding angles are equal.
Therefore, z = 80áµ’
In a parallelogram, adjacent angles are supplementary to each other.
x + 80áµ’ = 180áµ’
x = 180áµ’ - 80áµ’
x = 100áµ’
In a parallelogram, opposite angles are equal.
Therefore, y = 80áµ’
(v) In a parallelogram, opposite angles are equal.
Therefore, y = 112áµ’
We know that, sum of interior angles of triangle is 180áµ’
40áµ’ + y + x = 180áµ’
40áµ’ + 112áµ’ + x = 180áµ’
152áµ’ + x = 180áµ’
x = 180áµ’ - 152áµ’
x = 28áµ’
In a parallelogram, alternate angles are equal.
Therefore, z = x = 28áµ’
Q.3 Can a quadrilateral ABCD be a parallelogram if
(i) âˆ D + âˆ B = 180Â°?
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) âˆ A = 70Â° and âˆ C = 65Â°?
Sol. (i) For âˆ D + âˆ B = 180Â°, quadrilateral ABCD might be a parallelogram if (a) sum of the measures of adjacent angles is 180áµ’ (b) Opposite angles are equal.
(ii) For AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm, quadrilateral ABCD cannot be a parallelogram since, opposite sides AD and BC are of different lengths.
(iii) For âˆ A = 70Â° and âˆ C = 65Â°, quadrilateral ABCD cannot be a parallelogram since, opposite angles A and C are not equal.
Q.4 Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Sol. For the quadrilateral ABCD which is a kite, there are two interior angles âˆ A and âˆ C of same measure. But, still quadrilateral ABCD is not a parallelogram as the measure of the remaining pair of opposite angles âˆ D and âˆ B are not equal.
Q.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.
Sol. Let the two adjacent angles be 3a and 2a.
We know that in a parallelogram adjacent angles are supplementary to each other.
Therefore, 3a + 2a = 180áµ’
5a = 180áµ’
a = 36áµ’
Therefore, one angle = 3a = 3 x 36áµ’ = 108áµ’
Other angle = 2a = 2 x 36áµ’ = 72áµ’
Q.6 Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Sol. Let each adjacent angle be a.
We know that in a parallelogram adjacent angles are supplementary to each other.
Therefore, a + a = 180áµ’
2a = 180áµ’
a = 90áµ’
Thus, each adjacent angle is 90áµ’
We know that, sum of interior angles of triangle is 180áµ’
Therefore, a + a + a = 180áµ’
3a = 180áµ’
a = 60áµ’
Hence, the measure of each angle of parallelogram is 90áµ’, 90áµ’ and 60áµ’.
Q.7 The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.Sol. Here, âˆ EHP and âˆ y forms alternate interior angles.
Therefore, âˆ y = 40áµ’
In a parallelogram, corresponding angles are equal.
Therefore, 70áµ’ = z + 40áµ’
70áµ’ - 40áµ’ = z
z = 30áµ’
In a parallelogram, adjacent pair of angles are equal.
x + (z + 40áµ’) = 180áµ’
x + 70áµ’ = 180áµ’
x = 110áµ’
Q.8 The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm).
Sol. (i) In parallelogram GUNS,
In a parallelogram, opposite sides are equal.
Therefore, GS = UN
3x = 18
x = 6 cm
Similarly, GU = SN
3y -1 = 26
3y = 26 + 1
3y = 27
y = 9 cm
Hence, the measures of x and y are 6 cm and 9 cm respectively.
(ii) In a parallelogram, diagonals bisect each other.
Therefore, y + 7 = 20
y = 13 cm
Similarly, x + y = 16
x + 13 = 16
x = 3 cm
Hence, the measures of x and y are 3 cm and 13 cm respectively.
Q.9
In the above figure both RISK and CLUE are parallelograms. Find the value of x.
Sol. In a parallelogram, adjacent angles are supplementary.
For parallelogram RISK, âˆ RSK + âˆ ISK = 180áµ’
120áµ’ + âˆ ISK = 180áµ’
âˆ ISK = 60áµ’
In a parallelogram, opposite angles are equal.
For parallelogram CLUE, âˆ ULC = âˆ CEU = 70áµ’
We know that the sum of measures of all the interior angles of a triangle is 180áµ’
x + 60áµ’ + 70áµ’= 180áµ’
x = 50áµ’
Q.10 Explain how this figure is a trapezium. Which of its two sides are parallel?
Sol. We know that if a transversal line is intersecting two given lines such that the sum of the measures of the angles on the same side of transversal is 180áµ’, then the given two lines will be parallel to each other.
For the given trapezium,
âˆ M + âˆ L =100áµ’ + 80áµ’ = 180áµ’ (Since sum of interior opposite angles is 180áµ’)
Therefore, NM and KL are parallel.
This proves that KLMN is a trapezium.
Q.11 Find mâˆ C in Fig if || .
Sol. Given, || .
âˆ B + âˆ C = 180áµ’ (Since they angles on same side of transversal)
120áµ’ + âˆ C = 180áµ’
âˆ C = 180áµ’ - 120áµ’
âˆ C = 60áµ’
Q.12 Find the measure of âˆ P and âˆ S if || in Fig. (If you find mâˆ R, is there more than one method to find mâˆ P?)
Sol. From the figure, we can see that âˆ P and âˆ Q are angles on the same side of transversal
Therefore, âˆ P + âˆ Q = 180áµ’
âˆ P = 50áµ’
From the figure, we can see that âˆ R and âˆ S are angles on the same side of transversal
Therefore, âˆ R + âˆ S = 180áµ’
90áµ’ + âˆ S = 180áµ’
âˆ S = 90áµ’
Alternative method to find âˆ P:
Using the angle sum property of quadrilateral, we can write,
âˆ P + âˆ Q + âˆ R + âˆ s = 360áµ’
âˆ P + 130áµ’ + 90áµ’ + 90áµ’ = 360áµ’
âˆ P = 360áµ’ - 310áµ’
âˆ P = 50áµ’
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what about question number 6
Message *good answers well done