The sum of the three interior angles of a triangle is \(\displaystyle 180\) degrees.

The interior angles of a triangle must have measures whose sum is \(\displaystyle 180^{\circ }\), so the measure of the third angle must be \(\displaystyle 180 - (50+45) = 85^{\circ }\).

By the Triangle Exterior-Angle Theorem, an exterior angle of a triangle measures the sum of its remote interior angles; therefore, to get the greatest measure of any exterior angle, we add the two greatest interior angle measures: \(\displaystyle 50+85= 135^{\circ }\)

Since the angles are supplementary, their sum is 180 degrees.  Because they are in a ratio of 1:4, the following expression could be written:

\(\displaystyle x+4x=180\)

\(\displaystyle 5x=180\)

\(\displaystyle x=36^{\circ}\)

Since the sum of the angles of a triangle is \(\displaystyle 180^{\circ}\), and given that the angles are in a ratio of 1:3:5, let the measure of the smallest angle be \(\displaystyle x\), then the following expression could be written:

\(\displaystyle x+3x+5x=180\)

\(\displaystyle 9x=180\)

\(\displaystyle x=20\)

If the smallest angle is 20 degrees, then given that the middle angle is in ratio of 1:3, the middle angle would be 3 times as large, or 60 degrees.

The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.

The measure of \(\displaystyle \angle2\) is twenty degrees greater than the measure \(\displaystyle x\) of \(\displaystyle \angle 1\), so its measure is 20 added to that of \(\displaystyle \angle 1\) - that is, \(\displaystyle x + 2 0\).

The measure of \(\displaystyle \angle 3\) is thirty degrees less than twice that of \(\displaystyle \angle 1\). Twice the measure of \(\displaystyle \angle 1\) is \(\displaystyle 2x\), and thirty degrees less than this is 30 subtracted from \(\displaystyle 2x\) - that is, \(\displaystyle 2x-30\).

The sum of the measures of the three angles of a triangle is 180, so, to solve for \(\displaystyle x\) - thereby allowing us to calulate all three angle measures - we add these three expressions and set the sum equal to 180. This yields the equation:

\(\displaystyle x + (x+20) + (2x-30) = 180\)

The measure of \(\displaystyle \angle2\) is forty degrees less than the measure \(\displaystyle x\) of \(\displaystyle \angle 1\), so its measure is 40 subtracted from that of \(\displaystyle \angle 1\) - that is, \(\displaystyle x -40\).

The measure of \(\displaystyle \angle 3\) is ten degrees less than twice that of \(\displaystyle \angle 1\). Twice the measure of \(\displaystyle \angle 1\) is \(\displaystyle 2x\), and ten degrees less than this is 10 subtracted from \(\displaystyle 2x\) - that is, \(\displaystyle 2x-10\).

The sum of the measures of the three angles of a triangle is 180, so, to solve for \(\displaystyle x\) - thereby allowing us to calulate all three angle measures - we add these three expressions and set the sum equal to 180. This yields the equation:

\(\displaystyle x + (x-40) + (2x - 10) = 180\)

The sum of the three angles of a triangle add up to 180 degrees.  Subtract 64 degrees to determine the third angle.

\(\displaystyle 180-64= 116\)

A right angle has a measure of \(\displaystyle 90$^{\circ}$\).  One fifth of the angle is:

\(\displaystyle \angle A=\frac{90}{5}= 18$^{\circ}$\)

To find the other angle, subtract the given angle from \(\displaystyle 90$^{\circ}$\) since complementary angles add up to \(\displaystyle 90$^{\circ}$\).

The complementary is:

\(\displaystyle \angle 90-\angle65 = \angle25\)