Answer:
160° (But you should input the answer as 160 as the instructions say)
Step-by-step 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:
↓
Combine the x's
(5x + 12x) -10 + InC = 180
↓
17x - 10 + InC = 180
(-17x + 10) (-17x + 10)
↓
<u>InC</u><u> = 190 - 17x</u>
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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 <u>straight line,</u> which means InC and the exterior C angle are <u>supplementary angles</u>, 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, <u><em>we've already got a value for </em></u><u><em>InC </em></u><u>(from the previous equation we just solved) </u><u><em>AND </em></u><u><em>ExC </em></u><u>(the question tells us).</u>
<u />
That means all we need to do is plug this in and solve for x! Let's go:
↓
InC + ExC = 180°
Plug in our known values:
↓
(190 - 17x) + 16x = 180°
Combine the x's:
↓
(-17x +16x) + 190 = 180°
↓
Isolate x:
-x + 190 = 180
(-190) (-190)
↓
-x = -10
Multiply both sides by -1 to cancel:
↓
<u>x = 10</u>
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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 <u>16x</u>, AND we've figured out the value of x. All we have to do is plug in our value:
↓
16 * 10 = 160°
↓
There it is! <u>160° is our value of the exterior angle at C</u>
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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
↓
m∠A = 50 - 10 = 40
↓
<u>m∠A = 40°</u>
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m∠B = 12x
↓
m∠B = 12(10) = 120
↓
<u>m∠B = 120°</u>
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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
↓
<u>m∠C = 20°</u>
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Now:
m∠A + m∠B + m∠C = 40° + 120° + 20°
↓
<u>= 180°</u>
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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 <em>just</em> solved for the value of x, we can already see that 20° + 160° = 180°, so this double-check method <em>also</em> 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
↓
<u>180 = 180</u>
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So, now that we've double-checked our answer <u>twice</u>, 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!
