Examples with solutions for Sum and Difference of Angles: Using quadrilaterals

Exercise #1

ABCD rhombus.

B=80 ∢B=80

Calculate the size A ∢A

AAABBBDDDCCC80

Video Solution

Step-by-Step Solution

It is known that according to the properties of a quadrilateral, each pair of opposite angles are equal to each other.

That is:

B=C=80 B=C=80

A=D A=D

Additionally, we know that the sum of angles in a quadrilateral is equal to 360 degrees.

Therefore, we can calculate angles A and D as follows:

3608080=200 360-80-80=200

200:2=100 200:2=100

In other words, angle A is equal to 100

Answer

100

Exercise #2

ABCD is a quadrilateral.

A=80 ∢A=80

C=95 ∢C=95

D=45 ∢D=45

Calculate the size of B ∢B .

AAABBBDDDCCC809545

Video Solution

Step-by-Step Solution

We know that the sum of the angles of a quadrilateral is 360°, that is:

A+B+C+D=360 A+B+C+D=360

We replace the known data within the following formula:

80+B+95+45=360 80+B+95+45=360

B+220=360 B+220=360

We move the integers to one side, making sure to keep the appropriate sign:

B=360220 B=360-220

B=140 B=140

Answer

140°

Exercise #3

ABCD is a trapezoid.

A=110 ∢A=110

B=130 ∢B=130

C=70 ∢C=70

Calculate the size of angle D ∢D .

AAABBBDDDCCC13011070

Video Solution

Step-by-Step Solution

As known, the sum of angles in a trapezoid is 360 degrees.

Therefore:

360=A+B+C+D 360=A+B+C+D

Let's substitute the known data into the above formula:

360=110+130+70+D 360=110+130+70+D

360=310+D 360=310+D

We'll move terms and maintain the appropriate sign:

360310=D 360-310=D

50=D 50=D

Answer

50

Exercise #4

ABCD is a rectangle.

ABC=? ∢\text{ABC}=?

AAABBBDDDCCC30

Video Solution

Step-by-Step Solution

Since we are given that ABCD is a rectangle, we know that AC is parallel to BD

Therefore, angles ACB and CBD are equal to each other at 30 degrees.

In a rectangle, we know that all angles are equal to 90 degrees, meaning angle ABD is equal to 90.

Now we can calculate angle ABC as follows:

9030=60 90-30=60

Answer

60

Exercise #5

ABCD is a quadrilateral.

According to the data, calculate the size of B ∢B .

AAABBBDDDCCC80140

Video Solution

Step-by-Step Solution

As we know, the sum of the angles in a square is equal to 360 degrees, therefore:

360=A+B+C+D 360=A+B+C+D

We replace the data we have in the previous formula:

360=140+B+80+90 360=140+B+80+90

360=310+B 360=310+B

Rearrange the sides and use the appropriate sign:

360310=B 360-310=B

50=B 50=B

Answer

50

Exercise #6

ABCD Deltoid.

Calculate the size D ∢D

AAABBBDDDCCC3040

Video Solution

Step-by-Step Solution

We know that in a kite, the side angles are equal to each other, meaning:

B=C=70 B=C=70

And therefore also:

ACB=30 ACB=30

BCD=40 BCD=40

Now we can calculate angle A. As we know, the sum of angles in a triangle is 180, so:

1803030=120 180-30-30=120

Now we can calculate angle D. As we know, the sum of angles in a kite is 360, so:

3601207070=100 360-120-70-70=100

D=100 D=100

Answer

100

Exercise #7

ABCD is a quadrilateral.

AB||CD
AC||BD

Calculate angle A ∢A .

90°90°90°AAABBBDDDCCC45°45°

Video Solution

Step-by-Step Solution

Angles ABC and DCB are alternate angles and equal to 45.

Angles ACB and DBC are alternate angles and equal to 45.

That is, angles B and C together equal 90 degrees.

Now we can calculate angle A, since we know that the sum of the angles of a square is 360:

360909090=90 360-90-90-90=90

Answer

90°

Exercise #8

The deltoid ABCD is shown below.

C=100 ∢C=100

Calculate the size of D ∢D .

858585AAABBBDDDCCC100

Video Solution

Answer

75°