First Problem:
angle R and angle S are complementary which means they add to 90 degrees. Think "C" for "Complementary" and "Corner" as in a sharp 90 degree corner. That's how I remember it.
Let's solve for x
(angle R) + (angle S) = 90
(12x-3) + (7x-2) = 90
12x-3 + 7x-2 = 90
(12x+7x)+(-3-2) = 90
19x-5 = 90
19x-5+5 = 90+5
19x = 95
19x/19 = 95/19
x = 5
Since x = 5, we can say
angle R = 12x - 3
angle R = 12*5 - 3
angle R = 60 - 3
angle R = 57 degrees
Final Answer: 57 degrees
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Second Problem:
Angle 1 and angle 2 "form a linear pair". This means they are adjacent and supplementary (add to 180 degrees). Put another way, they combine to form a single straight angle or line.
Let's make
x = measure of angle 1
y = measure of angle 2
We're also told that "the measure of angle 2 is six more than twice the measure of angle 1", which basically translates to
y = 2*x + 6
Because the angles (x and y) are supplementary, we can also say
x+y = 180
So we have this system of equations
y = 2x+6
x+y = 180
Start with equation (2). Plug in the equation (1) and solve for x
x+y = 180
x+2x+6 = 180 ... note how y has been replaced with 2x+6
3x+6 = 180
3x+6-6 = 180-6
3x = 174
3x/3 = 174/3
x = 58
Now that we know x, we can find y
y = 2x+6
y = 2*58+6
y = 116+6
y = 122
Final Answer: 122 degrees