Sum and Difference of Angles: Using quadrilaterals

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

$A+B+C+D=360$

We replace the known data within the following formula:

$80+B+95+45=360$

$B+220=360$

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

$B=360-220$

$B=140$

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

Therefore:

$360=A+B+C+D$

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

$360=110+130+70+D$

$360=310+D$

We'll move terms and maintain the appropriate sign:

$360-310=D$

$50=D$

According to the properties of a quadrilateral, each pair of opposite angles are equal to each other.

Therefore:

$B=C=80$

$A=D$

Additionally, we know that the sum of the angles in a quadrilateral equals 360 degrees.

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

$360-80-80=200$

$200:2=100$

Angle A is equal to 100.

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

$360=A+B+C+D$

We replace the data we have in the previous formula:

$360=140+B+80+90$

$360=310+B$

Rearrange the sides and use the appropriate sign:

$360-310=B$

$50=B$

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:

$90-30=60$

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

$B=C=70$

And therefore also:

$ACB=30$

$BCD=40$

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

$180-30-30=120$

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

$360-120-70-70=100$

$D=100$

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:

$360-90-90-90=90$