Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let  = the vertex angle

and  = base angle

So the equation to solve becomes 

or

Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.

This triangle has an angle of . We also know it has another angle of  at  because the two sides are equal. Adding those two angles together gives us  total. Since a triangle has total, we subtract 130 from 180 and get 50.

Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let  = vertex angle and  = base angle.

Then the equation to solve becomes

or

.

Solving for  gives a vertex angle of 24 degrees and a base angle of 78 degrees.

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let  be the vertex angle and  be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.

Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.

Example:

If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.

Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180. 

We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:

(1/2)y = 2x.

Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.

(1/2)(180-x) = 2x.

Multiply both sides by 2 to get rid of the fraction.

(180 – x) = 4x.

Add x to both sides.

180 = 5x.

Divide both sides by 5.

x = 36.

The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:

x + z = 90.

Now, we can substitute 36 as the value of x and then solve for z.

36 + z = 90.

Subtract 36 from both sides.

z = 54.

The answer is 54. 

Refer to the following diagram while reading the explanation:

We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection).  Therefore, it is 20°. 

Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°.  Based on the rule for supplements, we know that a + 40° = 180°.  Solving for a, we get a = 140°.

Therefore, a + b = 140° + 20° = 160°