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**Question 1.** Can you construct the following quadrilateral MIST if we have 100° at M instead of 75°?

**Solution:** Yes, the quadrilateral MIST can be constructed with ∠M = 100° instead of ∠75°.**Question 2.** Can you construct the quadrilateral PLAN if PL = 6 cm, LA = 9.5 cm, ∠P = 75°, ∠L = 150° and ∠A = 140°?

(**Hint:** Recall angle-sum property)

**Solution**: Here, ∠P + ∠L + ∠A + ∠N = 75° + 150° + 140° + ∠N

= 365° + ∠N

But the sum of all the angles of a quadrilateral is 360°.

∴ Construction of quadrilateral PLAN is not possible.**Question 3.** In a parallelogram, the lengths of adjacent sides are known. Do we still need measures of the angles to construct as in the Q-1 above?**Solution: **No, the measures of three angles are not necessary in case of a parallelogram as its opposite sides are parallel.**EXERCISE 4.1(Question 1.** Construct the following quadrilaterals:

**(i) Quadrilateral MORE **

MO = 6 cm

OR = 4.5 cm

∠M = 60°

∠O = 105°

∠R = 105° **(ii) Quadrilateral PLAN**

PL = 4 cm

LA = 6.5 cm

∠P = 90°

∠A = 110°

∠N = 85°**(iii) Parallelogram HEAR **

HE = 5 cm

EA = 6 cm

∠R = 85°**(iv) Rectangle OKAY**

OK = 7 cm

KA = 5 cm**Solution: **(i) Steps of construction:

I . Draw a line segment MO = 6 cm.

II. At M, draw , such that ∠OMX = 60°.

III. At O, draw , such that ∠MOY = 105°.

IV. From , cut off OR = 4.5 cm.

V. At R, draw , such that ∠ORZ = 105°.

Let intersects at E.

Thus, MORE is the required quadrilateral.

**(ii) Steps of construction: **

I. Draw a line segment AL = 6.5 cm.

II. At A, draw such that ∠LAX = 110°.

III. At L, draw such that ∠ALY = 75°.**Note:** ∠L = 75° is not given, but we can determine it using angle sum property

∵ Sum of the three given angles = 110° + 90° + 85° = 285°

∴ The fourth angle ∠L = 360° – 285° = 75°.

IV. From, cut-off LP = 4 cm.

V. At P, draw such that ∠LPZ = 90°

Let PZ and intersect at N.

Thus, PLAN is the required quadrilateral.**(iii) Steps of construction: **

I. Draw a line segment = 5 cm.

II. At E, draw such that ∠HEA = 85°.

III. From , cut-off EA = 6 cm.

IV. With centre at A and radius = 5 cm, draw an arc towards H.

V. With centre at H and radius = 6 cm, draw an arc such that it intersects the previous arc at R.

VI. Join RA and RH.

Thus, HEAR is the required quadrilateral.

**(iv) Steps of constructions: **

I. Draw a line segment OK = 7 cm.

II. At O, draw such that ∠KOP = 90°.

III. From , cut-off = 5 cm.

IV. At K, draw such that ∠OKQ = 90°.

V. From cut-off = 5 cm.

VI. Join A and Y.

Thus, OKAY is the required quadrilateral.

**CONSTRUCTION OF A QUADRILATERAL WHEN THREE SIDES AND TWO INCLUDED ANGLES ARE GIVEN**

**Question**. We used some five measurements to draw quadrilateral so far. Can there be different sets of five measurements (other than seen so far) to draw a quadrilateral? The following problems may help you in answering the question.

(i) Quadrilateral ABCD with AB = 5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and ∠B = 80°.

(ii) Quadrilateral PQRS with PQ = 4.5 cm, ∠P = 70°,∠Q = 100°, ∠ R = 80° and ∠S = 110°.

Construct a few more examples of your own to find sufficiency/insufficiency of the data for construction of a quadrilateral**Solution:** (i) It is possible to construct a quadrilateral ABCD with the given measurements.

(ii) Not possible, because we cannot locate the points R and S with the help of given measurements.**EXERCISE 4.4 ****Question 1 **Construct the following quadrilaterals.

(i) Quadrilateral DEAR

DE = 4 cm

EA = 5 cm

AR = 4.5 cm

∠E = 60°

∠A = 90°**(ii) Quadrilateral TRUE**

TR = 3.5 cm

RU = 3 cm

UE = 4 cm

∠R = 75°

∠U = 120°**Solution:** **(i) Steps of construction:**

I. Draw a line segment DE = 4 cm.

II. At E, draw such that ∠DEX = 60°.

III. From , cut-off EA = 5 cm.

IV. At A, draw ray such that ∠EAY = 90°.

V. From , cut-off AR = 4.5 cm.

VI. Join R and D.

Thus, DEAR is the required quadrilateral.

**(ii) Steps of construction: **

I. Draw a line segment = 3.5 cm.

II. At R, draw a ray such that ∠TRX = 75°.

III. From , cut-off RU = 3 cm.

IV. At U, draw such that ∠RUY = 120°.

V. From UY, cut-off= 4 cm.

VI. Join E and T.

Thus, TRUE is the required quadrilateral.

**SOME SPECIAL CASES**

Using the properties of a square, rectangle, kite and rhombus, etc., we can construct some special quadrilaterals even with less than 5 measurements.

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