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## Example Questions

### Example Question #2 : Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Even with both statements, cannot be determined because the length of is missing.

For example, we can have and , making ; or, we can have and , making . Neither scenario violates the conditions given.

### Example Question #3 : Lines

, , and are distinct points.

True or false: and are opposite rays.

Statement 1: is on

Statement 2: is on

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

Both statements are equivalent, as both are equivalent to stating that , , and are collinear. Therefore, it suffices to determine whether the fact that the points are collinear is sufficient to answer the question.

In both of the above figures, , , and are collinear, so the conditions of both statements are met. But in the top figure, and are the same ray, since is on ; in the bottom figure, since and are on opposite sides of , and are opposite rays.

### Example Question #4 : Lines

, , and are distinct points.

True or false: and are opposite rays.

Statement 1: .

Statement 2: is the midpoint of .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below. In the first figure, is the midpoint of .

In both figures, . But only in the second figure, and are on the opposite side of the line from , so only in the second figure, and are opposite rays.

Assume Statement 2 alone. If is the midpoint of , then, as seen in the top figure, is on . Therefore, and are the *same *ray, not opposite rays.

### Example Question #1 : Dsq: Understanding Rays

, , and are distinct points.

True or false: and are opposite rays.

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 alone does not answer the question.

Case 1: Examine the figure below.

,

thereby meeting the condition of Statement 1.

Also, and are opposite rays, since and are on opposite sides of the same line from .

Case 2: Suppose , , and are noncollinear.

The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

.

Furthermore, and are not part of the same line and are not opposite rays.

Now assume Statement 2 alone. As can be seen in the diagram above, if and are opposite rays, then by segment addition, , making Statement 2 false. Contrapositively, if Statement 2 holds, and , then and are not opposite rays.

### Example Question #2 : Dsq: Understanding Rays

, , and are distinct points.

True or false: and are the same ray.

Statement 1: .

Statement 2: is the midpoint of .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below:

In both figures, , but only in the first figure, and are the same ray.

Assume Statement 2 alone. If is the midpoint of , must be on , as in the top figure, so and are one and the same.

### Example Question #3 : Dsq: Understanding Rays

, , and are distinct points.

True or false: and are the same ray.

Statement 1:

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

We show that both statements together provide insufficient information by giving two scenarios in which both statements are true.

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

and

.

Also, since the three points are not on a single line, and are parts of different lines and cannot be the same ray.

Case 2: with length 2 and midpoint .

and , so ; similarly, . Also, and are the same ray, since they have the same endpoint and is on .

### Example Question #4 : Dsq: Understanding Rays

, , and are distinct points.

True or false: and are the same ray.

Statement 1: , , and are collinear.

Statement 2: .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

Statement 1 alone does not prove the rays to be the same or different, as seen in these diagrams:

In both figures, , , and are collinear, satisfying the condition of Statement 1. But In the top figure, and are the same ray, since is on ; in the bottom figure, since is not on , and are distinct rays.

Assume Statement 2 alone. Suppose and are *not* the same ray. Then one of two things happens:

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the triangle inequality,

,

contradicting Statement 2.

Case 2: , , and are collinear. must be between and , as in the bottom figure, since if it were not, and would be the same ray. By segment addition,

,

contradicting Statement 2.

By contradiction, and are the same ray.

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