A small diagram helps:


In the figure, you see (180 - 2x) + y = 180 (straight angle) so 2x = y

Also, the moment you see the circle and its two radii, mark them equal and the corresponding angles equal. (the red angle = blue angle)
x + 180 - 2y = y
5x = 180 (from above, y = 2x)
x = 36
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Since OC = AC, ∆AOC is an isosceles triangle, which means ∠OAC is also x°


Since all 3 angles in ∆AOC must add to 180°, we can conclude that ∠OCA = (180-2x)°


Since angles on a LINE must add to 180°, we can conclude that ∠ACB = 2x°


Since AC = AB, ∆ACB is an isosceles triangle, which means ∠CBA is also 2x°


Finally, since OA and OB are radii of the same circle, we know that ∆OAB is an isosceles triangle, which means ∠OABis also 2x°


At this point, we can see that the 3 angles ∆OAB are x°, 2x° and 2x°
Since the angles in a triangle must add to 180°, we can write: x° + 2x° + 2x° = 180°
Simplify: 5x = 180
Solve: x = 36

Answer: B

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It goes like this -

In triangle OAB , Angle O + Angle A + Angle B = 180 -------1
OA = OB (Radius) => Angle A = Angle B

In triangle ACB, Angle C = Angle O + Angle OAC (Sum of interior opposite angles)
=> Angle ACB = 2x;
Also, AC = AB => Angle ACB = Angle ABC = 2x each

Thus Angle A = Angle B = 2x each.

So, substituting in 1

5x = 180 => x = 36 deg.

This will help.

As \(OC=AC\)
So, \(∠OAC=x\)
Then, \(∠ACB=2x\) [As the external angle of a triangle is equal to the sum opposite two angles]

As, \(AC=AB\)
So, \(∠ACB=∠ABC=2x\)

\(OA=AB\) as they are two radii
\(So, ∠OAB=2X\)

Now for the \(∆OAB\)
\(x+2x+2x=180\)
\(5x=180\)
\(X=36\)

The answer is \(B\)
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In the figure attached, point O is the center of the circle and OC = AC = AB. What is the value of x (in degrees)?

OC = AC = AB tells us that triangle OAC and BAC are isosceles. Also, as with any triangle on a straight line with an exterior angle, the exterior angle (or in this case, angle ACB) can be found by subtracting the interior angle from 180.

We know that in triangle OAC (because it is an isosceles triangle) that angle o and angle a are equal to one another. Therefore, angle c = 180 - x - x --> angle c = 180-2x. Angle C (of the smaller isosceles triangle) also happens to share the same angle measurement as angle B. Furthermore, because OA and OB are radii, we know that they equal one another and that angle A = angle B.

We know that angle c in the smaller isosceles triangle = 180 - the measure of the obtuse angle C. As shown above the obtuse angle C = (180-2x) So, the small angle C = 180 - (180-2x) --> angle C = 2x. So, angle C = B = 2x. If angle A = B then A = 2x and becaause the obtuse triangle has two x measurements, we know that the measure of A in the small isosceles triangle = x. Therefore, in the small isosceles triangle we have 2x+2x+x = 180. 5x = 180. x = 36.

b. 36

Hi Karishma,

Thanks for the detailed explanation but I still have a nagging issue. I was having a hard time setting this up with the variables. For example, <OAC, you have it broken up as 180-2y and x, which i see why you did. When I started working on this problem, I labeled <ACB and <ABC as "x" because of the congruent lines and I labeled OAC and COA as "x" too because all of those lines were same according to the question stem. Doing so, conflicted with the fact that <OAB and <ABO are equal and that's when my whole setup went kaboom Why is that wrong?

I guess I'm not following why you gave some variables "x" vs. some "y"?

Thanks!

When two sides of a triangle are equal, the two opposite angles are equal. But can you say what the two angles are? No. Say a triangle has two sides of length 5 cm each. Do we know the measure of equal angles? No. They could be 40-40 or 50-50 or 80-80 etc. So if you have two different triangles with 2 sides of length 5 cm each, the equal angle could have different measures - in one triangle the equal angles could be 50-50, in the other triangle, the equal angles could be 70-70.

In triangle OAC, since OC = AC, you have two equal angles as x each. The third angle here is 180 - 2x.

In triangle ACB, since AC = AB, angle ACB = angle ABC but what makes you say that they must be x each too? This is a different triangle. Even if the sides have the same length as the sides of triangle OAC, there is no reason to believe that the equal angles need to be x each. So you call the angles y. The third angle here is 180 - 2y.
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Answer = B = 36

Please refer diagram below:

Hi Karishma,

This is news to me -- you can't assume that because length AC is shared, that implies that the corresponding angle in Triangle ACO will equal the corresponding angle in ACB?

Hmm -- learn something new everyday! Thanks!

Corresponding angles are equal only when you have parallel lines with a common transversal.
Make two triangles with a common side. Can you make the angles made on the common side such that the angles have very different measures? Sure!



Here one angle is 90 degrees and the other is acute.
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Karishma - why is it that you did not have x+ x + (180 - 2y)+ y = 180 to include the entire triangle? I solved this equation by adding an additional line to form a diameter and then constructed another angle(titled z) and approached the solution that way but I am attempting to understand this method that you used to solve this problem.

You need to find the relation between x and y. You can do it in any way you like; you will get the same result.

Doing it your way:
x+ x + (180 - 2y)+ y = 180
2x = y
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Thank you.

I must have overlooked something because I get:

x+x+(180-2y)+y= 180
2x-y=0
2x-2x= 0

Ignore the highlighted step.
after 2x - y = 0, when you take y to the other side, you get
2x = y

Without doing this step, how did you substitute 2x for y in the highlighted step?
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How are we getting different variables x and y when the sides are equal. Can you explain Krishna. Cause three sides are equal shouldn't their angles be noted with the same variable?

I must have overlooked something because I get:

x+x+(180-2y)+y= 180
2x-y=0
2x-2x= 0[/quote]

Ignore the highlighted step.
after 2x - y = 0, when you take y to the other side, you get
2x = y

Without doing this step, how did you substitute 2x for y in the highlighted step?[/quote]
I see what I did. Below was my thought process:

x+x+(180-2y)+y= 180
2x-y=0 I then looked at the graph and applied the sum of two interior angles is equal to the opposite exterior which you labeled y. Once I got 2x= y I then went back to x+x+(180-2y)+y= 180 and thought I would end up with 5x= 180 but instead kept getting 2x-2x= 0

Note the sides that are equal
OC = AC = AB

OC and AC are sides if a triangle and the angles opposite to them are marked as x each (i.e. they are equal)

AC and AB are equal sides of another triangle and angles opposite to them are marked as y each. Note that you cannot mark them as x too because they are equal angles in a different triangle. Their measure could be different from x. I have explained this in detail in a post given below. Giving the explanation here:

"When two sides of a triangle are equal, the two opposite angles are equal. But can you say what the two angles are? No. Say a triangle has two sides of length 5 cm each. Do we know the measure of equal angles? No. They could be 40-40 or 50-50 or 80-80 etc. So if you have two different triangles with 2 sides of length 5 cm each, the equal angle could have different measures - in one triangle the equal angles could be 50-50, in the other triangle, the equal angles could be 70-70.

In triangle OAC, since OC = AC, you have two equal angles as x each. The third angle here is 180 - 2x.

In triangle ACB, since AC = AB, angle ACB = angle ABC but what makes you say that they must be x each too? This is a different triangle. Even if the sides have the same length as the sides of triangle OAC, there is no reason to believe that the equal angles need to be x each. So you call the angles y. The third angle here is 180 - 2y."
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From this equation: x+x+(180-2y)+y= 180,
you derive that 2x = y. If you try to substitute 2x = y in this equation itself, you will just get 2x - 2x = 0 which implies 0 = 0.
This equation has 2 variables and you need two distinct equations to get the value of the two variables. If both equations are just 2x = y, you cannot get the value of x. You need another equation to get the value of x.
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What step am I doing wrong here? I can't figure it out. I don't understand how to go to 5x = 180

I just get to 4x + 180 - 4x = 180 aka 4x - 4x = 0..

edit: that last "8" kinda looks like a 5, but its supposed to be an 8