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In isosceles triangle RST what is the measure of angle R?  [#permalink]

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In isosceles triangle RST what is the measure of angle R?

(1) The measure of ∠T is 100°.

(2) The measure of ∠S is 40°.
Math Expert V
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In isosceles triangle RST what is the measure of angle R?

Triangle RST is isosceles means that two of its angles are equal.

(1) The measure of angle T is 100 degrees --> since no other angle can be equal to 100 degrees (because in this case the sum of the angles will be more than 180 degrees) then the other two angles, R and S, are equal: R=(180-100)/2=40. Sufficient.

(2) The measure of angle S is 40 degrees --> consider two cases: S=R=40 and T=100 AND S=40 and R=T=70. Not sufficient.

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gmatgg,

This is a fairly one simple one that you should spend no more than 30 seconds on.

The keyword is "isosceles"---which means 2 sides are the same and also...2 angles are the same

With statement (1), you are told measure of angle T is 100 degrees. You already know an entire triangle = 180 degrees.

So either one of the other angles also equals 100 degrees, or the other two angles not mentioned are equivalent.

Well, it's impossible for one of the other angles to also be 100 degrees, because that would exceed that total of 180 degrees possible in a triangle.

So the only choice that makes sense is to have the other two angles equal and everything adds to 180.

In other words, the other 2 angles combined must equal the difference between 180 and 100.
Mathematically, 180 - 100 = 80 and then divide that by 2 to get 40 degrees.

So (1) by itself is good enough.

Statement (2) is just telling us that one angle is 40 degrees. Well, what possibilities are there? You could have one 40 degrees and then the other two angles are equivalent and all 3 add up to 180. OR you can have another angle that is also 40 degrees and the third angle would add up to 180.
In other words, you'd have two possibilities:

1) 40 - 70 - 70 (adds up to 180)
or 2) 40 - 40 - 100 (also adds up to 180)

Since you have MULTIPLE possibilities with statement (2), then you do not have enough information to definitively say what the answer is. So Statement (2) is not good enough.

With (1) good and (2) no good, answer is (A).

Hope that helps.
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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enigma123 wrote:
In isosceles triangle RST what is the measure of angle R?

(1) The measure of angle T is 100 degrees.

(2) The measure of angle S is 40 degrees.

How come statemen1 is correct and not C?

Is it because of if one angle is 100 then others have to be 40 and 40 as the triangle is isoceles.

But then if that's the case why statement 2 is not sufficient?

IN ISOCELES TRIANGLE ANGLES ARE ALWAYS. X,X,(180-2X)

(1) The measure of angle T is 100 degrees.
CLEARLY X CANT BE equal to 100 because in that case sum of angle will exceed 180 degrees
therefore 180-2X = 100. CLEARLY X=40
SUFFICIENT.

(2) The measure of angle S is 40 degrees.
IF X CAN BE 40 OR 180-2X CAN BE 40
WE ARE GETTING 2 DIFFERENT VALUES OF X.
HENCE INSUFFICIENT.

HENCE A
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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HI,

Thanks a lot for the reply! That is very useful.
So I suppose for 2, we cannot assume that S is the adjacent angle to two isosceles angles? As this is what I did and I thought that it was sufficient.

Thanks a lot again,

Really appreciate it!

Bunuel wrote:
In isosceles triangle RST what is the measure of angle R?

Triangle RST is isosceles means that two of its angles are equal.

(1) The measure of angle T is 100 degrees --> since no other angle can be equal to 100 degrees (because in this case the sum of the angles will be more than 180 degrees) then the other two angles, R and S, are equal: R=(180-100)/2=40. Sufficient.

(2) The measure of angle S is 40 degrees --> consider two cases: S=R=40 and T=100 AND S=40 and R=T=70. Not sufficient.

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Yela wrote:
HI,

Thanks a lot for the reply! That is very useful.
So I suppose for 2, we cannot assume that S is the adjacent angle to two isosceles angles? As this is what I did and I thought that it was sufficient.

Thanks a lot again,

Really appreciate it!

Bunuel wrote:
In isosceles triangle RST what is the measure of angle R?

Triangle RST is isosceles means that two of its angles are equal.

(1) The measure of angle T is 100 degrees --> since no other angle can be equal to 100 degrees (because in this case the sum of the angles will be more than 180 degrees) then the other two angles, R and S, are equal: R=(180-100)/2=40. Sufficient.

(2) The measure of angle S is 40 degrees --> consider two cases: S=R=40 and T=100 AND S=40 and R=T=70. Not sufficient.

Yes, from (2) we cannot assume that S is one of the two angles that are equal.
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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I have a general question: The order of the letters after the sign of the triangle, here <RST, does not give any indication of the order of the angles?
Sorry I don't know how to insert the proper angle sign.
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lou34 wrote:
I have a general question: The order of the letters after the sign of the triangle, here <RST, does not give any indication of the order of the angles?
Sorry I don't know how to insert the proper angle sign.

Order of letters for a triangle does not matter. Triangles RST, RTS, STR, ... are all the same.

As for other polygons (quadrilaterals, pentagons, ...): here order of the letters is important meaning that rectangle ABCD (diagonals here are AC and BD) is different from rectangle ACBD (diagonals here are AB and CD). Usually OG uses sequential (ABCDE...) labeling. Also, most of the times the labeling is clockwise but I wouldn't worry about it because if a diagram is not given then this won't be important (meaning that you can consider it clockwise as well as counter-clockwise).
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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Bunuel wrote:
In isosceles triangle RST what is the measure of angle R?

Triangle RST is isosceles means that two of its angles are equal.

(1) The measure of angle T is 100 degrees --> since no other angle can be equal to 100 degrees (because in this case the sum of the angles will be more than 180 degrees) then the other two angles, R and S, are equal: R=(180-100)/2=40. Sufficient.

(2) The measure of angle S is 40 degrees --> consider two cases: S=R=40 and T=100 AND S=40 and R=T=70. Not sufficient.

Hi Bunuel

How can S=R=40 because that would take it out of the triangle definition isnt it? Sum of two smaller sides wont be greater than the longest side so can it be a valid scenario? Does this rule only apply to sides measure and not angles in a triangle?
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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sinhap07 wrote:
Bunuel wrote:
In isosceles triangle RST what is the measure of angle R?

Triangle RST is isosceles means that two of its angles are equal.

(1) The measure of angle T is 100 degrees --> since no other angle can be equal to 100 degrees (because in this case the sum of the angles will be more than 180 degrees) then the other two angles, R and S, are equal: R=(180-100)/2=40. Sufficient.

(2) The measure of angle S is 40 degrees --> consider two cases: S=R=40 and T=100 AND S=40 and R=T=70. Not sufficient.

Hi Bunuel

How can S=R=40 because that would take it out of the triangle definition isnt it? Sum of two smaller sides wont be greater than the longest side so can it be a valid scenario? Does this rule only apply to sides measure and not angles in a triangle?

Yes, the rule is about the lengths of the sides.

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

For the angles of a triangle we have that their sum must be 180 degrees.

Check for more here: math-triangles-87197.html

Hope it helps.
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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gmatgg wrote:
In isosceles triangle RST what is the measure of angle R?

(1) The measure of angle T is 100 degrees.

(2) The measure of angle S is 40 degrees.

IMPORTANT: In an isosceles triangle there are 2 IDENTICAL angles, and 1 LONE angle.

Target question: What is measure of ∠R?

Statement 1: ∠T = 100 degrees
We should recognize that ∠T CANNOT be one of the identical angles. If this were the case, we'd have two angles with measures of 100 degrees each, which would result in a triangle in which the sum of the angles is GREATER than 180 degree (which is IMPOSSIBLE)
So, we can conclude that ∠T must be the LONE angle, which means ∠R and ∠S are the two IDENTICAL angles.
Since the sum of the 3 angles must be 180, we can conclude that ∠R = 40, ∠S = 40, and ∠T = 100
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: ∠S = 40 degrees
Here are two possible cases to consider:
Case a: ∠S is the LONE angle, in which case the ∠R = 70, ∠S = 40, and ∠T = 70
Case b: ∠S is one of the IDENTICAL angles, in which case we could have ∠R = 40, ∠S = 40, and ∠T = 100
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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Stmt 1: R is 100 deg, so remaining two angles have a sum of 80 deg. Since it is an isosceles triangle, 2 angles are equal. Now, another angle cannot be equal to R as sum of just two angles will be more than 180 deg. Hence, the other two angles have to be half of 80 each, or 40 deg. Hence suff. Eliminate B,C,E

Stmt 2: T is 40 deg, so remaining two angles have a sum of 140 deg. If one of the other two angles is 40, then third is 100, but it can be possible than the other two are equal to 70 deg each as well. Hence insuff. Eliminate D.

Hence Option A
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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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isosceles ∆ RST, what is the measure of ∠ S?
St:1
∠ R is 100 ◦
R+S+T=180
R=100 so S+T has to 80
Isosceles so S=T=40
Sufficient.
St:-2
The measure of ∠ T is 40 ◦ .
S+R=140
So s=r=70 ( if side correspond to angle s and r are equal)
But,
If Sides correspond to angle T and angle S is equal then s=40

Two values \ Not sufficient.

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Re: In isosceles triangle RST what is the measure of angle R?  [#permalink]

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gmatgg wrote:
In isosceles triangle RST what is the measure of angle R?

(1) The measure of ∠T is 100°.

(2) The measure of ∠S is 40°.

I read the options as angle T is less than 100 degreee!!

All my analysis of this question is wrong  Re: In isosceles triangle RST what is the measure of angle R?   [#permalink] 07 Jun 2019, 02:23
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