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**Q.1. Two opposite angles of a parallelogram are (3x-2) ^{0} and (50-x)^{0}. Find the measure of each angle of the parallelogram.**

**Solution:**

We know that,

Opposite sides of a parallelogram are equal.

(3x-2)^{0} = (50-x)^{0}

⇒ 3x + x = 50 + 2

⇒ 4x = 52

⇒ x = 13^{0}

Therefore, (3x-2)^{0} = (3*13-2) = 37^{0}

(50-x)^{0} = (50-13) = 37^{0}

Adjacent angles of a parallelogram are supplementary.

∴ x+37 = 1800

∴ x = 1800−370 = 1430

Hence, four angles are : 37^{0}, 143^{0}, 37^{0}, 143^{0}.

**Q.2. If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.**

**Solution:**

Let the measure of the angle be x.

Therefore, the measure of the angle adjacent is

We know that the adjacent angle of a parallelogram is supplementary.

Hence

2x + 3x = 540^{0}

⇒ 5x = 540^{0}

⇒ x = 108^{0}

Adjacent angles are supplementary

⇒ x + 108^{0} = 180^{0}

⇒ x = 180^{0} – 108^{0} = 72^{0}

⇒ x = 72^{0}

Hence, four angles are 180^{0}, 72^{0}, 180^{0}, 72^{0}

**Q.3. Find the measure of all the angles of a parallelogram, if one angle is 24 ^{0} less than twice the smallest angle.**

**Solution:**

x + 2x – 24 = 180^{0}

⇒ 3x – 24 = 180^{0}

⇒ 3x = 108^{0} + 24

⇒ 3x = 204^{0}

⇒ x = 68^{0}

⇒ 2x – 24^{0} = 2*68^{0} – 24^{0} = 112^{0}

Hence, four angles are 68^{0}, 112^{0}, 68^{0}, 112^{0}.

**Q.4. The perimeter of a parallelogram is 22cm. If the longer side measures 6.5cm what is the measure of the shorter side?**

**Solution:**

Let the shorter side be ‘x’.

Therefore, perimeter = x + 6.5 + 6.5 + x [Sum of all sides]

22 = 2 (x + 6.5)

11 = x + 6.5

⇒ x = 11 – 6.5 = 4.5cm

Therefore, shorter side = 4.5cm

**Q.5. In a parallelogram ABCD, ∠D = 1350. Determine the measures of ∠Aand∠B.**

**Solution:**

In a parallelogram ABCD

Adjacent angles are supplementary

So, ∠D+∠C = 1800

∠C = 1800−1350

∠C = 450

In a parallelogram opposite sides are equal.

∠A =∠C = 450

∠B = ∠D = 1350

**Q. 6. ABCD is a parallelogram in which ∠A = 700. Compute ∠B,∠Cand∠D.**

**Solution:**

In a parallelogram ABCD

∠A = 700

∠A+∠B = 1800 [ Since, adjacent angles are supplementary ]

700+∠B = 1800 [∵∠ A = 70°]

∠B = 1800−700

∠B =1100

In a parallelogram opposite sides are equal.

∠A = ∠C = 700

∠B = ∠D = 1100

**Q.7. In Figure 14.34, ABCD is a parallelogram in which ∠A = 600. If the bisectors of ∠A,and∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.**

**Solution:**

AP bisects ∠A

Then, ∠DAP = ∠PAB = 300

Adjacent angles are supplementary

Then, ∠A+∠B = 1800

∠B+600 = 1800

∠B = 1800−600

∠B = 1200

BP bisects ∠B

Then, ∠PBA = ∠PBC = 300

∠PAB = ∠APD = 300 [Alternate interior angles]

Therefore, AD = DP [Sides opposite to equal angles are in equal length]

Similarly

∠PBA = ∠BPC = 600 [Alternate interior angles]

Therefore, PC = BC

DC = DP + PC

DC = AD + BC [Since, DP = AD and PC = BC]

DC = 2AD [Since, AD = BC, opposite sides of a parallelogram are equal]

**Q.8. In figure 14.35, ABCD is a parallelogram in which ∠DAB = 750 and ∠DBC = 600. Compute ∠CDB, and ∠ADB.**

**Solution:**

To find ∠CDB and ∠ADB

∠CBD = ∠ABD = 600 [Alternate interior angle. AD║BC and BD is the transversal]

In ∠BDC

∠CBD+∠C+∠CDB = 1800 [Angle sum property]

⇒ 600+750+∠CDB =1800

⇒ ∠CDB = 1800−(600+750)

⇒ ∠CDB = 450

Hence, ∠CDB = 450,∠ADB = 600

**Q. 9. In figure 14.36, ABCD is a parallelogram and E is the mid-point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.**

**Solution:**

In ΔBEFandΔCED

∠BEF = ∠CED [Verified opposite angle]

BE = CE [Since, E is the mid-point of BC]

∠EBF = ∠ECD [Since, Alternate interior angles are equal]

∴ ΔBEF ≅ ΔCED [ASA congruence]

∴ BF = CD[CPCT]

AF = AB + AF

AF = AB + AB

AF = 2AB.

Hence proved.

**Q.10. Which of the following statements are true (T) and which are false (F)?**

**(i) In a parallelogram, the diagonals are equal.**

**(ii) In a parallelogram, the diagonals bisect each other.**

**(iii) In a parallelogram, the diagonals intersect each other at right angles.**

**(iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.**

**(v) If all the angles of a quadrilateral are equal, it is a parallelogram.**

**(vi) If three sides of a quadrilateral are equal, it is a parallelogram.**

**(vii) If three angles of a quadrilateral are equal, it is a parallelogram.**

**(viii) If all the sides of a quadrilateral are equal, it is a parallelogram.**

**Solution:**

(i) False

(ii) True

(iii) False

(iv) False

(v) True

(vi) False

(vii) False

(viii) True

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