In a triangle, the measure of the first angle is four times the measure of the second angle. The measure of the third angle is 84 degrees more than the second angle. What is the measure of the first angle?

2nd angle --- x

1st angle ---- 4x
3rd angle ---- x+84

x + 4x + x+84 = 180
6x=96
x = 16

1st angle = 4x = 4(16) = 64°

To find the measure of the first angle, we need to apply some algebraic reasoning. Let's assume that the measure of the second angle is represented by the variable "x".

According to the problem, the measure of the first angle is four times the measure of the second angle, so we can write it as 4x.

The measure of the third angle is described as 84 degrees more than the second angle. Therefore, it can be written as x + 84.

The sum of the three angles in a triangle is always 180 degrees. So, we can set up an equation to solve for x:

4x + x + (x + 84) = 180

Simplifying the equation, we combine the terms with x:

6x + 84 = 180

Next, we isolate x by subtracting 84 from both sides:

6x = 180 - 84
6x = 96

Now, we solve for x by dividing both sides by 6:

x = 96 / 6
x = 16

Therefore, the measure of the second angle is 16 degrees.

To find the measure of the first angle, we substitute x back into the expression 4x:

4x = 4 * 16
4x = 64

So, the measure of the first angle is 64 degrees.