In the figure above, ΔPQR has angle measures as shown. Is x < y ?

Is x < y?

1. PQ = QR

The greater the angle is the greater is the side opposite to it. If the triangle has two equal sides, it is an isosceles triangle with two equal angles opposite to those sides. Since PQ = QR, x = 58. Thus y = 180 - 58 - 58 = 64. x < y. Sufficient


2. PR > QR

Since the side opposite to y is greater than the side opposite to x, y must be greater than x. Accordingly, x < y. Sufficient

In this case, Statement (1) tells us that triangle PQR is an isosceles triangle, with sides PQ=QR, thus corresponding angles PRQ and QPR are also equal. So, x must also be 58 degrees, and since the sum of the angles of a triangle must be 180 degrees, angle y must be 180-58-58, or 64 degrees, answering the question yes. (Sufficient) Keep in mind, on test day, as soon as we know that statement one will give us precisely one situation for the angle measures, we don't actually need to make the calculation, regardless of whether the situation answers the question "yes" or "no," if we have one distinct possibility for the angle measures, the statement will be sufficient to answer our question.

Statement (2) tells us that the length of PR is greater than the length of QR, thus, the corresponding opposite angle from PR, angle PQR, must be greater than the angle opposite QR, angle QPR - so, yes, y is once again greater than x. (Sufficient)

This question really just comes down to understanding the relationships between side lengths and their corresponding angle measures, and utilizing a bit of data sufficiency strategy to recognize that when we are given enough information to solve for precisely one situation, we'll be able to answer the question, often without having to actually make the calculation!

We are given that angle QRP = 58 and need to determine whether x is less than y.

Statement One Alone:

PQ = QR

Since PQ = QR, x = 58.

Thus, y = 180 - 58 - 58 = 64.

So we see that x is less than y.

Statement one alone is sufficient to answer the question.

Statement Two Alone:

PR > QR

Since PR is greater than QR, the angle opposite PR is greater than the angle opposite QR.

Thus, y > x, so we see that statement two alone is also sufficient to answer the question.

Answer: D

Hi All,

We're told that the triangle PQR has angle measures as shown. We're asked if X is less than Y. This is a YES/NO question and is based around a specific Triangle rule (re: in simple terms, the bigger the angle, the bigger the side that's across from it). Since Angle R is 58 degrees, the sum of Angles X and Y is 122 degrees. If X and Y are EQUAL, then the sides across from them will also be EQUAL.

(1) PQ = QR

With the information in Fact 1, we know that two of the sides are EQUAL, so the angles across from them will also be EQUAL. Thus, Angle X = 58 degrees. By extension, Angle Y = 122 - 58 = 64 degrees and the answer to the question is YES.
Fact 1 is SUFFICIENT

(2) PR > QR

Fact 2 tells us that the side across from Angle Y is GREATER than the side across from Angle X, so Y is greater than X and the answer to the question is YES.
Fact 2 is SUFFICIENT

Final Answer:

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Rich

Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are given

• The figure of a triangle PQR, where angle P = x°, angle Q = y° and angle R = 58°

From this information, we need to determine

• If x < y or not.

As PQR is a triangle, the sum of the three angles must be equal to 180°

• x° + y° + 58° = 180°

Or, x° + y° = 180° - 58° = 122°

Hence, to determine whether x < y, we need to know the value of either x or y.
With this information, let us now analyse the individual statements.

Step 3: Analyse Statement 1

As per the information given in statement 1, PQ = QR.

• Therefore, we can say angle QPR = angle QRP

Or, x° = 58°
As we can determine the value of x, we can also determine the value of y and find out whether x < y or not.
Hence, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 2, PR > QR.

• Therefore, the angle opposite to PR > the angle opposite to QR

Or, y > x
As we can determine whether x < y or not, statement 2 is sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)

Since we could determine the answer from either of the statements individually, this step is not required.

Hence, the correct answer is option D.

Foe me definitely D.

OG 21 explanation

Given that PQ = QR, it follows that the angle at vertex R has the same measure as the angle at vertex P, or 58 = x. Therefore, using x + y + 58 = 180 gives 58 + y + 58 = 180, or y = 64. Since the values of both x and y are now known, it can be determined whether x < y; SUFFICIENT.
Given that PR > QR, it follows that the measure of the angle at vertex Q is greater than the measure of the angle at vertex P, or y > x; SUFFICIENT.

(1) so x 58°, y will greater than x; Sufficient.

(2) A side represents it's opposite angle. so y>x. Sufficient.

The answer is D

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Answer: Option D

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Statement 1: lets us find all three angles as two equal sides will subtent equal angles with the common base.
thus sufficient.

Statement 2: Lets us find a relation betweeen x and y as larger side will be opposite to bigger angle; hence sufficient.
Answer: D