Question

Can someone walk me through it

Answer 1

Answer: x = 123
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Here are the basic steps:

Step 1) Find the measure of angle 7 (near the 35 degree angle)
Step 2) Find angle 6 (in the center; bottom angle)
Step 3) Use angle 6 to find the value of x
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Let's go through those steps mentioned
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Step 1) Finding the measure of angle 7

We see that the 35 degree angle and angle 7 combine to form a 90 right angle. So they must add to 90 degrees

(angle 7) + 35 = 90
(angle 7) + 35 - 35 = 90 - 35
angle 7 = 55

So angle 7 is 55 degrees. We'll use it on the next step
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Step 2) Finding the measure of angle 6

We'll use the result from step 1. The triangle with angle 7, angle 6, and the 68 degree angle will be focused on here. Recall that for any triangle, the three angles must add to 180 degrees. 

So,
(angle 7) + (angle 6) + 68 = 180
(55) + (angle 6) + 68 = 180
(angle 6) + 55 + 68 = 180
(angle 6) + 123 = 180
(angle 6) + 123 - 123 = 180 - 123
angle 6 = 57

So we now know that angle 6 is 57 degrees
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Step 3) Find x

We now use the fact that angle 6 and angle x are a linear pair. They combine to form a straight line, or straight angle. In other words they add to 180 degrees (they are supplementary angles)

So,
(angle 6) + x = 180
(57) + x = 180
x + 57 = 180
x + 57 - 57 = 180 - 57
x + 57 - 57 = 123
x = 123
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Side note: we can use the exterior angle theorem to skip over step 2. To do this, we add up angle 7 (which was 55 degrees) to 68 to get 123 degrees which is the same answer.