Sum and Difference of Angles: Using quadrilaterals

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

Therefore, angles ACB and CBD are equal (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:

$90-30=60$

The side angles in a kite are equal, therefore:

$B=C=70$

Also, therefore:

$ACB=30$

$BCD=40$

Now we can calculate angle A. Since the sum of angles in a triangle is 180, this is done as follows:

$180-30-30=120$

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:

$360-90-90-90=90$