From the picture below, what is the measure of GDF?

The measure of arc GDF is 304°

Solution:

Given data:

m∠CHD = 90°, m(ar EF) = 34°

The angle measure of the central angle is equal to the measure of the intercepted arc.

m∠CHD = m(ar CD) = 90°

Sum of the adjacent angles in a straight line = 180°

⇒ m∠GHC + m∠CHD = 180°

⇒ m∠GHC + 90° = 180°

Subtract 90° from both sides, we get

⇒ m∠GHC = 90°

⇒ m(ar GC) = m∠GHC

⇒ m(ar GC) = 90°

Sum of the adjacent angles in a straight line = 180°

⇒ m∠EHD + m∠CHD = 180°

⇒ m∠EHD + 90° = 180°

Subtract 90° from both sides, we get

⇒ m∠EHD = 90°

⇒ m(ar ED) = m∠EHD

⇒ m(ar ED) = 90°

m(ar GDF) = m(ar GC) + m(ar CD) + m(ar DE) + m( EF)

                 = 90° + 90° + 90° + 34°

                 = 304°

The measure of arc GDF is 304°.

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.