In the figure above, QRS is a straight line and QR = PR. Is

In the figure, QRS is a straight line. QR=PR. Are TR and PQ parallel?
1) Length PQ = Length PR
2) Line TR bisects angle PRS

From the prompt, we know that triangle QPR is isosceles, with QR = PR. By the Isosceles Triangle theorem, we know that angle Q = angle P.

Statement #1 PQ = PR.
This is enough to guarantee that triangle QPR is equilateral, but we don't know anything about ray RT, so we have no idea whether that is parallel to anything else.
This statement, alone and by itself, is not sufficient.

Statement #2 Line TR bisects angle PRS
A fascinating statement. Let's think about this. We already know angle Q = angle P. Call the measure of that angle M. Look angle QRP --- call the measure of that angle K. Clearly, within triangle QPR, M + M + K = 180, but Euclid's famous theorem.
Now, look at angle PRS. This is what is known as the "exterior angle" of a triangle, and there's a special theorem about this.
The Remote Interior Angle Theorem:
If the exterior angle of a triangle is adjacent to the angle of the triangle, then the measure of the exterior angle is equal to the sum of the two "remote" interior angles of the triangle --- that is, the two angles of the triangle which the exterior angle is not touching.
If you think about this, it has to be true, because
(angle Q) + (angle P) + (angle PRQ) = 180, because they're the three angles in a triangle
(angle PRQ) + (exterior angle PRS) = 180, because they make a straight line
Subtract (angle PRQ) from both sides of both equations:
(angle Q) + (angle P) = 180 - (angle PRQ)
(exterior angle PRS) = 180 - (angle PRQ)
Since the two things on the left are equal to the same thing, they are equal to each other.
(angle Q) + (angle P) = (exterior angle PRS)
Now, going back to the letters we were using ---- if (angle Q) = (angle P) = M, this means (exterior angle PRS) = 2M. If we bisect exterior angle PRS, each piece will have a measure of M. Thus, (angle PRT) = (angle TRS) = M
Well, now we know that (angle Q) = (angle TRS) = M. If corresponding angles are congruent, then the lines must be parallel. PQ must be parallel to TR.
This statement, alone and by itself, is sufficient to answer the prompt question.

Answer = B

Does all this make sense?

Mike
_________________

Hi Mike well explained,

I do have a doubt !

now we know that (angle Q) = (angle TRS) = M. If corresponding angles are congruent, then the lines must be parallel. PQ must be parallel to TR.
This statement, alone and by itself, is sufficient to answer the prompt question.


Correspoding angle means the angle made on its side?
angle Q ==> side PQ
angle TRS ==> side TR

very basic question but i understood complex things

No. "Corresponding angles" is a technical term from Euclidean Geometry. It's the name of a particular pair of angles formed when a transversal crosses a pair of parallel lines.

The following pairs are corresponding angles
1 & 5
2 & 6
3 & 7
4 & 8
Corresponding angles are congruent if and only if the lines are parallel.

The following pairs are alternate interior angles
3 & 6
4 & 5
Alternate interior angles are congruent if and only if the lines are parallel

The following pairs are alternate exterior angles
1 & 8
2 & 7
Alternate exterior angles are congruent if and only if the lines are parallel

The following pairs are same side interior angles
3 & 5
4 & 6
Same side interior angles are supplementary if and only if the lines are parallel

The following pairs are same side exterior angles
1 & 7
2 & 8
Same side exterior angles are supplementary if and only if the lines are parallel

Those are all the names relating pairing an angle at one vertex with an angle at the other vertex, when a transversal intersects a pair of parallel lines.

Does all this make sense?

Mike
_________________

Ans: For lines TR and PQ to be parallel angle PQR= angle TRS. From statement 1 we get angle PQR=x=60 but nothing about angle TRS.
From statement 2 we get PRQ=180-2X , therefore PRS=180-(180-2x)=2x and TR bisects it so angle TRS=x which is equal to PQR. Therefore the answer is (B).

Let's solve. So we need to know if TR is parallel to PQ.

Now then, let's hit the first statement. We are told that PQ=QR. Now we know that PQR is an equilateral triangle but still no info on TR. Therefore, insufficient.

Statement 2, we have that TR bisects PRS. Now let's see. So we know that QR=PR from the question stem. Hence angle QRP is 180-2x, 'x' being the angles P and Q respecively. Therefore angle PRS will be 2x since QRS is a straight line with total measure of 180 degrees. Now if TRS bisects then angle TRS is x only. Which means that the angles Q and R are equal and thus PQ // TR.

B stands

Cheers!
J

(1) this tells us that all 3 sides are equal = Equilateral Triangle. Not sufficient, as there is no info about TR
(2) Now look at the attachment, I'll use the property of the "Exterior Angles of a Triangle" Angle Y is the sum of the angles on the both sides of TR

Exterior angle of a triangle is equal to the sum of the opposite interior angles

Angle Y=2X and if TR bisects angle Y it means that Angle TRS = \(\frac{Y}{2}=\frac{2X}{2}=X\) and when this two angles of to different triangles are equal then QP and TR are parallel.
Answer B

(1) PQ = PR = QR, it is an equilateral triangle; each angle is 60 degrees. Insufficient

(2) angle PRS = 2 * angle QPS [exterior angle property]
Line TR bisects angle PRS
therefore, angle PRT = angle QPR
If alternate interior angles are equal, then the lines are parallel. Sufficient

B is correct
_________________

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________