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B
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Difficulty:
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52% (02:05) correct
48%
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Attachment:
Screen shot 2012-12-09 at 1.40.06 PM.png [ 19.42 KiB | Viewed 27639 times ]
(1) Length PQ = Length PR
(2) Line TR bisects angle PRS
09 Dec 2012, 15:58
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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
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
General Discussion
10 Dec 2012, 17:53
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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
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
10 Dec 2012, 18:09
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Bookmarks
shanmugamgsn
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.
Attachment:
parallel line diagram.JPG [ 22.96 KiB | Viewed 26665 times ]
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
