Tricky Isosceles triangle problem!!!

[JA]

In isosceles triangle RST (figure not shown), 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.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Why is choice 2 not sufficient? Could someone please explain.

[Guest]

In an isosceles triangle, two sides and two angles are equal.

With the first case: angle T =100. There cannot be two obtuse angles within a triangle. So it can be found that the other two angles are equal to 40 and so the angle R = 40.

With the second case: angle S = 40, angle R can be 40 or 100.

So the answer is Choice (1)

[RonPurewal]

In an isosceles triangle, two sides and two angles are equal.

With the first case: angle T =100. There cannot be two obtuse angles within a triangle. So it can be found that the other two angles are equal to 40 and so the angle R = 40.

With the second case: angle S = 40, angle R can be 40 or 100.

So the answer is Choice (1)

there are actually THREE possible answers with statement (2).

first off, a fact.
FACT: there are two different isosceles triangles with a 40° angle.
in particular, there's 40°-70°-70°, in which the 40° is the vertex angle, and then there's 40°-40°-100°, in which the 40° is one of the base angles.

you can generalize this:
GENERAL FACT: if 0 < n < 90, then there are two different shapes of isosceles triangle containing an n° angle, one of which has n° at the vertex angle and one of which has n° at a base angle. the only exception to this is n = 60, in which case the triangle is equilateral either way.
(by contrast, if 90 < n < 180, then there's only one isosceles triangle containing an n° angle, and that's the one that has n° at the vertex.)

--

using the FACT above, we can see that, under statement 2, angle R could be 40°, 70°, or 100°.

[JA]

Great explanation. Thanks Ron!!!

[RonPurewal]

Great explanation. Thanks Ron!!!

no problem.

and now, if you ever need a degree symbol, you can copy and paste one from this post. awesome.