Answer:
160° (But you should input the answer as 160 as the instructions say)
Explanation:
We know that the interior angles of a triangle add up to 180°. In this case, (interior) angles A, B, and C. For those three angles, we are given only A and B. Let's solve for C.
First, let's set up an equation with our given values:
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m∠A + m∠B + m∠C = 180°
Plug in our known values:
↓
(5x - 10)° + (12x)° + m∠C = 180°
Let's call m∠C just InC (to stand for interior C) for now and solve the equation for InC:
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Combine the x's
(5x + 12x) -10 + InC = 180
↓
17x - 10 + InC = 180
(-17x + 10) (-17x + 10)
↓
InC = 190 - 17x
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Okay, so what we've got for InC looks pretty confusing at first. You're probably wondering what the heck we're supposed to do with this weird value. Well, this is where the exterior C angle the problem gives us comes in handy:
We can see that line AC is a straight line, which means InC and the exterior C angle are supplementary angles, or that they add up to 180°.
Looks like it's time to solve for another equation now that we know this!
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We'll call the exterior C angle ExC. So:
InC + ExC = 180°
Well, look at that! The good news is, we've already got a value for InC (from the previous equation we just solved) AND ExC (the question tells us).
That means all we need to do is plug this in and solve for x! Let's go:
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InC + ExC = 180°
Plug in our known values:
↓
(190 - 17x) + 16x = 180°
Combine the x's:
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(-17x +16x) + 190 = 180°
↓
Isolate x:
-x + 190 = 180
(-190) (-190)
↓
-x = -10
Multiply both sides by -1 to cancel:
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x = 10
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Yay, we've got x!
Now, be careful not to fall for the trick of putting the value of x for your answer. Remember, they're asking for the value of the exterior angle at C.
Well, we know that the exterior angle at C equals 16x, AND we've figured out the value of x. All we have to do is plug in our value:
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16 * 10 = 160°
↓
There it is! 160° is our value of the exterior angle at C
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From here on, we're just going to double-check our answer to be sure! You can skip it if you would like.
So, what we're going to do to make sure that we got the right value for x is plug it into the values of m∠A, m∠B, and m∠C. If the three numbers add up to 180° (and if m∠C and the exterior angle at C equal 180°), then we're 100% correct :)
Let's see if the triangle adds up to 180° first:
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m∠A = 5x - 10
Let's plug in 10 into x (since that's the value we calculated for x):
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m∠A = 5(10) - 10
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m∠A = 50 - 10 = 40
↓
m∠A = 40°
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m∠B = 12x
↓
m∠B = 12(10) = 120
↓
m∠B = 120°
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Let's use the value of InC we calculated in the very first equation we did as our value of m∠C:
m∠C = 190 - 17x
↓
m∠C = 190 - 17(10)
↓
m∠C = 190 - 170 = 20
↓
m∠C = 20°
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Now:
m∠A + m∠B + m∠C = 40° + 120° + 20°
↓
= 180°
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Woo! Look's like we're right after all. While we're at it, why don't we do the "m∠C and the exterior angle at C equal 180°" check I mentioned too:
↓
Since we just solved for the value of x, we can already see that 20° + 160° = 180°, so this double-check method also tells us we're right, but let's pretend that we haven't done the first double-check method and therefore don't know what m∠C equals:
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m∠C + Exterior Angle at C = 180° (because they're supplementary angles)
↓
We'll plug in the original value of InC for m∠C, and of course our answer for the questions for "Exterior Angle at C":
(190 - 17x) + 160 = 180°
Plug in our known value of x:
↓
(190 - 17(10)) + 160 = 180
↓
(190 - 170) + 160 = 180
↓
20 + 160 = 180
↓
180 = 180
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So, now that we've double-checked our answer twice, we can be sure that 160° is the correct answer! You don't have to double-check your answer all the time, but these are the double-check methods if you're ever unsure about your answer.
And we're finally finished! If you have any more questions about this problem (I didn't completely explain things that I felt you might already know), please comment, and I will answer as soon as possible!