HOW TO FIND MISSING ANGLES OF A RHOMBUS WITH DIAGONALS

Find the all numbered angles in the following rhombus.

Problem 1 :

Solution :

Here ∠2 = ∠1, because the diagonal is the angle bisector.

∠2 = ∠4, because the equal sides will create equal angles.

∠4 = ∠3

73 + ∠2 + ∠4 = 180

73 + ∠2 + ∠2 = 180

73 + 2∠2 = 180

2∠2 = 180 - 73

2∠2 = 107

∠2 = 107/2 ==> 53.5

∠2 = 53.5

∠1 = 53.5

∠4 = 53.5

∠3 = 53.5

Problem 2 :

Solution :

∠2 = ∠4

In the triangle below,

∠2 + ∠4 + 104 = 180

∠4 + ∠4 = 180 - 104

2∠4 = 76

∠4 = 76/2

∠4 = 38

∠3 = 38 (angle bisector)

∠2 = 38 (Equal sides, will have same angles)

∠1 = 38 (Angle bisector)

Problem 3 :

Solution :

In the triangle above, 

∠1 + ∠2 + 26 = 180

∠1 = 26

26 + ∠2 + 26  = 180

52 + ∠2 = 180

∠2 = 180 - 52

∠2 = 128

∠3 = 128 (Opposite angles)

Problem 4 :

Solution :

∠1 = 118

∠1 + ∠2 + ∠2 = 180

118 + 2∠2 = 180

2∠2 = 180 - 118

2∠2 = 62

∠2 = 62/2

∠2 = 31

∠3 = 31

Problem 5 :

Solution :

From the figure given above, ∠3 = 58

∠2 = 90

∠1 + ∠2 + ∠3 = 180

∠1 + 90 + 58 = 180

∠1 + 148 = 180

∠1 = 180 - 148

∠1 = 32

∠4 = 32

Problem 6 :

Solution :

∠2 = ∠3

Here ∠1 = 90

∠1 + 30 + ∠2 = 180

90 + 30 + ∠2 = 180

120 + ∠2 = 180

∠2 = 180 - 120

∠2 = 60

∠3 = 60

∠4 = 30

Problem 7 :

Solution :

∠4 = 90

∠2 = 35

∠3 + ∠4 + 35 = 180

∠3 + 90 + 35 = 180

∠3 + 125 = 180

∠3 = 180 - 125

∠3 = 55

∠1 = 55

Problem 8 :

Solution :

∠2 = 90, ∠1 = 60

∠1 + ∠2 + ∠3 = 90

Problem 9 :

Solution :

∠1 = ∠3 = 90

In a triangle,

∠1 + ∠2 + 35 = 180

90 + ∠2 + 35 = 180

∠2 + 125 = 180

∠2 = 180 - 125

∠2 = 55

Problem 10 :

Solution :

∠1 = ∠4

∠1 + 52 + 52 + ∠4 = 180

∠1 + ∠1 + 104 = 180

2∠1 = 180 - 104

2∠1 = 76

∠1 = 76/2

∠1 = 38

∠2 = 90

∠3 = 90 (vertically opposite angles)

Problem 11 :

for any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning.

a) ∠L ≅ ∠M

b)∠K ≅ ∠M

c) JM ≅  KL

d) JK ≅ KL

e) JL ≅ KM

f) ∠JKM ≅ ∠LKM

Solution :

a) In the rhombus JKLM,  ∠L and ∠M are co-interior angles they add upto 180 degree. They are not congruent to each other. So, it is sometimes true.

b)∠K and ∠M are opposite angles and they are congruent. So, it is always true.

c) JM and KL are opposite sides, they must be equal. So, it is true always.

d) In rhombus all four sides will be equal then JK and KL are adjacent sides and they are equal. So, it is true always.

e) JL and KM are diagonals, so they are not equal always. Then it is sometimes true.

f) ∠JKM ≅ ∠LKM

The diagonals are angle bisector, then they must be equal. True always.

Problem 12 :

Name each quadrilateral parallelogram, rectangle, rhombus, or square for which the statement is always true.

a) It is equiangular.

b) It is equiangular and equilateral.

c) The diagonals are perpendicular.

d) Opposite sides are congruent.

e) The diagonals bisect each other.

f) The diagonals bisect opposite angles.

Solution :

a) It is equiangular.

In square, rectangle all angles measures must be 90 degree.

b) It is equiangular and equilateral.

All four angle measures will be equal and all side lengths will be equal in the shape of square.

c) The diagonals are perpendicular.

Diagonals are perpendicular in the shape of rhombus.

d) Opposite sides are congruent.

Opposite sides will be equal in the shapes parallelogram, square, rectangle and rhombus.

e) The diagonals bisect each other.

Diagonals will bisect each other in all the shapes parallelogram, square, rectangle and rhombus.

f) The diagonals bisect opposite angles.

In the quadrilaterals like parallelogram, square, rectangle and rhombus, the diagonals will bisect opposite angles.

Problem 13 :

Given a quadrilateral ABCD, and diagonals AC and BD bisect each other at P such that AP = CP and BP = DP. Also ∠APD = 90°, then quadrilateral is a

(a) rhombus      (b) trapezium     (c) parallelogram     (d) rectangle

Solution :

It must be the rhombus.