Problem Solving

Mike@Magoosh wrote:Dear phoenix 99^2,

I'm happy to help with this.

In #1, you need a few geometric fact.

Alternate Interior Angles: Because KL || JM, that necessarily means a = b.

Angles a & b are inscribed angles --- angles that have their vertex on the circumference of a circle.

Congruent inscribed angles subtend congruent arcs. Therefore, arc KJ just equal arc LM. Statement (b) must be true.

Congruent arcs have congruent chords. Therefore, if we drew them, KJ and LM would be equal, which means the JKLM is an isosceles trapezoid.

KL and JM are the bases of an isosceles trapezoid, which don't have to be equal. We could imagine a very special case in which J, K, L, and M were chosen so that they were the vertices of a perfect rectangle, and in that special case KL would equal JM, but in the general case the are not equal. Statement (a) does not have to be true.

In a general isosceles trapezoid, one leg (LM) would not equal the upper base (KL). Again, in a very special case --- an isosceles trapezoid with angles of exactly 60, 120, 120, and 60, then in that very elite special case, the upper base would equal each leg, but here we are in a general case, and in the general case, there's no reason to assume they have to be equal. If the segments are not equal, then the arcs wouldn't be equal either. Statement (c) doesn't have to be true.

BTW, another thing that would have to be true, about which the problem doesn't ask --- if we drew segment KM, and compared it to segment JL, those two would have to be equal, because isosceles trapezoids have congruent diagonals. But this question doesn't ask about that.

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In the second problem, we know that each small angle at R is x degrees, and each small angle at P is y degrees, because those two angles have been bisect.

In this problem, given this information, there is absolutely no way to solve for the individual value of x, or for the individual value of y. I 100% guarantee that, from the information given, those would be impossible to answer.

What you could be asked to find in this question is either (a) the sum (x + y), or (b) the measure of angle RSP.

In the big triangle, at R we have an angle of (2x), at P we have angle of (2y), and at Q we have an angle of 88 degrees.

2x + 2y + 88 = 180

2x + 2y = 92

2(x + y) = 92

x + y = 46

That's how you solve for the sum (x + y). Now, if you were asked for the measure of angle RSP, we would then consider the smaller triangle, triangle RSP. In that triangle, the sum of the angles is also 180 ---

x + y + (angle RSP) = 180

(x + y) + (angle RSP) = 180

46 + (angle RSP) = 180

(angle RSP) = 180 - 46 = 134 degrees

Those are the numbers you can find, and with only this given information, it's impossible to find anything else. Does that make sense?

Let me know if you have any further questions.

Mike

gmat_and_me wrote:Yes, it is explicitly mentioned that segment RS bisects PRQ and
that's why angle PRQ is 2x. Same is the case for RPQ too...

HTH