PROBLEMS ON GEOMETRY PROPERTIES OF PARALLELOGRAM WITH DIAGONALS

Problem 1 :

Find angle measure ∠YWX.

Solution :

Opposite angles are equal.

∠X = 140

∠V = 140

∠V + ∠WYV + ∠VWY = 180

140 + ∠WYV + 25 = 180 

165 + ∠WYV = 180 

∠WYV = 180 - 165

∠WYV = 15

∠YWX = 15 (alternate interior angles).

Problem 2 :

Find ∠KLJ.

Solution :

In parallelogram, diagonal is a transversal of parallel lines.

∠KLJ = ∠LJM

Alternate interior angles.

∠KLJ = 30

Problem 3 :

Solution :

∠MLJ = ∠LJK

In the triangle JLK,

∠JLK + ∠LKJ + ∠KJL = 180

50 + 10x + 14 + 36 = 180

10x + 14 + 86 = 180

10x + 100 = 180

10x = 180 - 100

10x = 80

x = 80/10

x = 8

Problem 4 :

FH = 14, QH = 2x - 13

Solution :

In a parallelogram, the diagonals will be equal. Diagonals will bisect each other.

FH = 2QH

14 = 2(2x - 13)

14/2 = 2x - 13

7 = 2x - 13

2x = 7 + 13

2x = 20

x = 20/2

x = 10

Problem 5 :

CJ = 5 + 3x, JE = 2x + 11 and find CJ.

Solution :

CJ = JE

5 + 3x = 2x + 11

5 - 11 = 2x - 3x

-6 = -x

x = 6

Applying the value of x in CJ = 5 + 3x

CJ = 5 + 3(6)

= 5 + 18

CJ = 23

Problem 6 :

Solution :

∠EDB = ∠DBC

∠DBC = 4x + 1

∠DCB + ∠CBD + ∠CDB = 180

21x - 5 + 4x + 1 + 34 = 180

21x + 4x - 4 + 34 = 180

25x + 30 = 180

25x = 180 - 30

25x = 150

x = 150/25

x = 6

Problem 7 :

Find the indicated measure in ▱LMNQ. Explain your reasoning.

Solution :

a) LM = 13 (In parallelogram, opposite sides will be equal)

b) LP = 7 (Diagonals will bisect each other)

c) LQ = 8(opposite sides are equal)

d) MQ = 2(8.2) ==> 16.4

e) m∠LMN

m∠QLM + m∠LMN = 180 (cointerior angles)

100 + m∠LMN = 180

m∠LMN = 180 - 100

m∠LMN = 80

f) m∠NQL = 80 (Opposite angles are equal)

g) m∠MNQ = 100 (Opposite angles are equal)

h) m∠LMQ = 29 (Alternate interior angles)

Problem 8 :

For what value of x is quadrilateral CDEF a parallelogram?

Solution :

Since it is parallelogram, FD and EC are equal. The diagonals will bisect each other. Then,

FN = ND

CN = NE

5x - 8 = 3x

5x - 3x = 8

2x = 8

x = 8/2

x = 4

So, the value of x is 4.

Problem 9 :

For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.

Solution :

2x = 10 - 3x

2x + 3x = 10

5x = 10

x = 10/5

x = 2

So, the value of x is 2.

Problem 10 :

Find the value of x that makes the quadrilateral a parallelogram

Solution :

Opposite sides will be equal

4x + 2 = 5x - 6

4x - 5x = -6 - 2

-x = -8

x = 8

So, the value of x is 8.

Problem 11 :

What value of x makes the quadrilateral a parallelogram? Explain how you found your answer.

Solution :

4x - 2 = 3x + 3

4x - 3x = 3 + 2

x = 5

So, the value of x is 5.

Problem 12 :

Prove the Parallelogram Diagonals Converse. Given Diagonals JL and KM bisect each other. Prove JKLM is a parallelogram

Solution :

JP = PL ------(1)

KP = PM -----(2)

In triangle JPM, KPL

M = KP

JP = PL

<JPM = <KPL (vertical angles)

So, triangle JPM and KPL are congruent, So, JM and KL are equal. Since the opposite sides are equal, then the given shape is a parallelogram.

Problem 13 :

Two adjacent angles of a parallelogram are as 2 : 3. Find the measure of each of its angles. 

Solution :

Let the angle measure be 2x and 3x

Since these two are co-interior angles they add upto 180 degree.

2x + 3x = 180

5x = 180

x = 180/5

x = 36

2(36) ==> 72

3(36) ==> 108

So, the adjacent angles are 72 and 108 respectively.