9类RD Sharma解决方案–第14章四边形–练习14.2

问题1.平行四边形的两个相反角度是(3x – 2)°和(50 – x) ° 。找到平行四边形的每个角度的度量。

解决方案：

Given: Two opposite angles of a parallelogram are (3x – 2)° and (50 – x)°.

(3x – 2)°= (50 – x)° [Opposite sides of a parallelogram are equal]

$\to$3x + x = 50 + 2

$\to$4x = 52

$\to$x = 13

Angle x is 13°

$\therefore$(3x – 2) = (3*13 – 2) = 37°

(50 – x)° = (50 – 13)°= 37°

$\therefore$x + 37°= 180° [Adjacent angles of a parallelogram are supplementary]

$\therefore$x = 180°− 37°= 143°

Hence, required angles are : 37°, 143°, 37°, 143°.

问题2.如果平行四边形的角度是其相邻角度的三分之二，请找出平行四边形的角度。

解决方案：

Let the measure of the angle be x.

Therefore, the measure of the angle adjacent is 2x/3

Hence, x + 2x/3 = 180° [Consecutive angles of a parallelogram are supplementary]

2x + 3x = 540°

$\to$5x = 540°

$\to$x = 108°

Now,

⟹ x + 108°= 180° [Consecutive angles of a parallelogram are supplementary]

⟹ x + 108°= 180°

⟹ x = 180°- 108°= 72°

⟹ x = 72°

$\therefore$required angles are180°, 72°, 180°, 72°

问题3.如果一个角度比最小角度的两倍小24° ，则求出平行四边形所有角度的度量。

解决方案：

x + 2x – 24°= 180° [Consecutive angles of a parallelogram are supplementary]

$\to$3x – 24°= 180°

$\to$3x = 108° + 24°

$\to$3x = 204°

$\to$x = 204/3 = 68°

$\to$x = 68°

Other angle of parallelogram=$\to$2x – 24°= 2*68°- 24°= 112°

$\therefore$required angles are 68°,112°,68°,112°

问题4.平行四边形的周长是22厘米。如果较长的边长为6.5厘米，较短的边长是多少?

解决方案：

Given: perimeter of a parallelogram is 22 cm

Let us consider the shorter side be ‘y’.

$\therefore$perimeter = y + 6.5 + 6.5 + x [Sum of all sides]

22 = 2(y + 6.5)

11 = y + 6.5

$\to$y = 11 – 6.5 = 4.5 cm

Hence, the measure of the shorter side = 4.5 cm

问题5.在平行四边形ABCD中，, D = 135 ° 。确定∠A和∠B的小节。

解决方案：

In a parallelogram ABCD

Given: ∠D=135°

So, ∠D + ∠C = 180° [Consecutive angles of a parallelogram are supplementary]

∠C = 180°− 135°

∠C = 45°

In a parallelogram opposite sides are equal.

∠A = ∠C = 45° [opposite sides of parallelogram are equal]

∠B = ∠D = 135°

hence, the measures of ∠A and ∠B are 45°,135°respectively.

问题6. ABCD是一个平行四边形，其中∠A= 70 ° 。计算∠B，∠C和∠D。

解决方案：

In a parallelogram ABCD

Given: ∠A = 70°

∠A + ∠B = 180° [Consecutive angles of a parallelogram are supplementary]

70°+ ∠B = 180° [given ∠A = 70°]

∠B = 180°− 70°

∠B = 110°

Now,

∠A = ∠C = 70° [opposite sides of parallelogram are equal]

∠B = ∠D = 110°

hence, the measures of ∠A and ∠B are 70°,110°respectively.

问题7.在图中，ABCD是一个平行四边形，其中∠A= 60 ° 。如果∠A和∠B的等分线在P处相遇，则证明AD = DP，PC = BC和DC = 2AD。

解决方案：

Given: ∠A = 60°

To prove: AD = DP, PC = BC and DC = 2AD

AP bisects ∠A

so, ∠DAP = ∠PAB = 30°

Now,

∠A + ∠B = 180° [Consecutive angles of a parallelogram are supplementary]

∠B + 60°= 180°

∠B = 180°− 60°

∠B = 120°

BP bisects ∠B

so, ∠PBA = ∠PBC = 60°

∠PAB = ∠APD = 30°[Alternate interior angles]

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

Similarly

∠PBA = ∠BPC = 60° [Alternate interior angles]

Therefore, PC = BC

DC = DP + PC

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

DC = 2AD [Since, AD = BC, The opposite sides of parallelogram are parallel and congruent]

hence proved.

问题8.在图中，ABCD是一个平行四边形，其中∠DAB= 75 °和∠DBC= 60 ° 。计算∠CDB和∠ADB。

解决方案：

Given: ∠DAB = 75°and ∠DBC = 60°

To find ∠CDB and ∠ADB

∠CBD = ∠ADB = 60° [Alternate interior angle. AD∥ BC and BD is the transversal]

In$\triangle$BDA

∠DAB + ∠ADB + ∠ABD = 180° [Angle sum property]

$\to$75°+ 60°+ ∠CDB = 180°

$\to$∠ABD = 180°− (135°)

$\to$∠ABD = 45°

∠ABD = ∠CDB = 45° [Alternate interior angle. AD∥ BC and BD is the transversal]

Hence, ∠CDB = 45°, ∠ADB = 60°

问题9.在图中，ABCD是平行四边形，E是BC边的中点。如果生产时的DE和AB在F处相遇，则证明AF = 2AB。

解决方案：

Given: ABCD is a parallelogram and E is the mid-point of side BC.

To prove: AF = 2AB.

Now,

In ΔBEF and ΔCED

∠BEF = ∠CED [Verified opposite angle]

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

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

$\therefore$ΔBEF ≅ ΔCED [ASA congruence]

$\therefore$BF = CD [Corresponding Parts of Congruent Triangle]

AF = AB + AF

AF = AB + CD [BF = CD by Corresponding Parts of Congruent Triangle ]

AF = AB + AB [CD=AB, The opposite sides of parallelogram are parallel and congruent]

AF = 2AB.

Hence proved.

问题10.以下哪个陈述是正确的(T)，哪些是错误的(F)?

(i)在平行四边形中，对角线是相等的。

(ii)在平行四边形中，对角线一分为二。

(iii)在平行四边形中，对角线彼此成直角相交。

(iv)在任何四边形中，如果一对相对的边相等，则为平行四边形。

(v)如果四边形的所有角度均相等，则为平行四边形。

(vi)如果四边形的三个边相等，则为平行四边形。

(vii)如果四边形的三个角度相等，则为平行四边形。

(viii)如果四边形的所有边均相等，则为平行四边形。

解决方案：

(i) False

(ii) True

(iii) False

(iv) False

(v) True

(vi) False

(vii) False

(viii) True