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12. Find the measure of $\angle P$and $\angle S$ if $\overline {SP} || \overline {RQ}$ in Fig 3.34.
(If you find $m\angle R$, is there more than one method to find $m\angle P$?)

Given,                       (angles on same side of transversal)                                                 (angles on same side of transversal) Yes, to find there is more than one method.  PQRS is quadrilateral so sum of all angles is 360 and we know so put valuse of  and we get measurement of

11. Find $m\angle C$ in Fig 3.33 if $\overline{AB} || \overline{DC}$.

Given ,          (Angles on same side of tranversal) Hence,.

10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)

Given, = . A transverse line is intersecting two lines such that sum of angles on same side of transversal line is . And hence, lines KL and MN are parallel to each other. Quadrilateral KLMN has a pair of parallel lines so it is a trapezium.

9.

In the above figure both RISK and CLUE are parallelograms. Find the value of $x$.

( adjacent angles are supplemantary)                (oppsite angles are equal)      (sum of angles of a triangle is )

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

Diagonals of parallelogram intersect each other.   Hence,cm  and   cm.

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

(i)

GUNS is a parallelogram, so opposite sides are equal in length                      Hence, cm  and   cm.

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

The adjacent figure HOPE is a parallelogram.            (linear pairs)               (opposite angles of parallelogram are equal)    ( adjacent angles are supplemantary )   y=                        (Alternate interior angles are equal)

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

Given : Two adjacent angles of a parallelogram have equal measure = .                   ( adjacent angles of a parallelogram are suplemantary)                            and      ( Opposite angles of prallelogram are equal) Hence,

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.

The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Sum of adjacent angles is .                                                                                                                     Hence, angles are  and . Let  there be a parallelogram ABCD then,        and . (Opposite angles are equal)

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

The above shown figure shows two opposite angles equal.. But, its not a parallelogram because other two angles are different  i.e. .

3. Can a quadrilateral ABCD be a parallelogram if

(iii) $\angle A = 70 \degree$ and $\angle C = 65 \degree$?

Opposite angles of parallelogram are equal Since, here   and  are different . So, it is not a parallelogram.

3. Can a quadrilateral ABCD be a parallelogram if

(ii) $AB = DC = 8 cm, AD = 4 cm$ and $BC = 4.4 cm$?

Opposite sides of a parallelogram are equal in length. Since,   and  are opposite sides and have different length. No, it is not a parallelogram.

3. Can a quadrilateral ABCD be a parallelogram if

(i) $\angle D + \angle B = 180\degree$

(i) Opposite angles should be equal and adjacent angles should be supplemantary to each other.  are opposite angles. Hence,a quadrilateral ABCD can be parallelogram but it is not conform.

2. Consider the following parallelograms. Find the values of the unknowns $x, y, z$.

y =                                       (opposite angles are equal)       (adjacent angles are supplemantary) x = z =            (alternate angles are equal)

2. Consider the following parallelograms. Find the values of the unknowns $x, y, z$.

(adjacent angles are supplementary) y =                    ( opposite angles are equal) z=                     (corresponding angles are equal)

2. Consider the following parallelograms. Find the values of the unknowns $x, y, z$.

x=                 (vertically opposite angles)       (sum of angles of a triangle is ) y=z=                (alternate interior angles)

2. Consider the following parallelograms. Find the values of the unknowns $x, y, z$.

( Two adjacent angles are supplemetary to each other) x=y=                 (opposite angles are equal) z=x=              ( corresponding angles are equal)

2. Consider the following parallelograms. Find the values of the unknowns$x, y, z$.

In a parallelogram, adjacent angles are supplementary to each other.  Opposite angles are equal . Hence, z =  and y = 100

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(iv) $m \angle DAB + m\angle CDA =$ .....

In a parallelogram, adjacent angles are supplementary to each other.    (iv)

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(iii) $OC =$......

In a parallelogram, both diagonals bisect each other.  (iii)
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