# GMAT Math : DSQ: Calculating the angle of an intersection

## Example Questions

### Example Question #1 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2:

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

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.

EITHER statement ALONE is sufficient to answer the question.

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

Explanation:

Statement 1 alone gives insufficient information. It only gives a relationship between  and , but no further clues about the measures of any angle are given.

Statement 2 alone gives insufficient information; , since the angles with those measures are vertical; since no measures are known,  cannot be calculated.

Now assume both statements are true. Again, from Statement 1, ; from Statement 2, . Again, from the diagram, . Three angles with measures  together form a straight angle, so

Therefore, both statements together are sufficient to answer the question.

### Example Question #2 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2:

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 sufficient to answer the question, but NEITHER statement ALONE is 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.

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

Explanation:

Assume Statement 1 alone. From the diagram, the three angles of measure  together form a straight angle, so

From Statement 1,

,

so by the subtraction property of equality,

Assume Statement 2 alone. , but there is no clue about the value of  or any other angle measure, so the value of  cannot be computed.

### Example Question #3 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale. Do not assume lines are parallel or perpendicular simply by appearance.

Evaluate .

Statement 1:

Statement 2:

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.

EITHER statement ALONE is 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.

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

Explanation:

Assume Statement 1 alone.  and  are vertical angles, so they must have the same measure; .

Assume Statement 2 alone.  and  are alternating exterior angles, which are congruent if and only if ; however, we do not know whether , so no conclusions can be made about .

### Example Question #4 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

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

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. , since the three angles of these measures together form a straight angle. Also, from Statement 1, . Therefore:

Assume Statement 2 alone. Again, , and from Statement 2, . Therefore,

Since the angles of measures  and  form a linear pair, they are supplementary, and  .

### Example Question #5 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2:

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

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

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show that both statements together give insufficient information. Assume both statements together. Consider the following two cases:

Case 1:

Since the angles of measures  and  form linear pairs with the angle of measure , each is supplementary to that angle and, subsequently,  .

The angle of measure  is vertical to the angle of measure , so the two must be congruent; .

From Statement 1,

Since  and , then .

These values are therefore consistent with the diagram and with both statements.

Case 2:

We can find the values of the other variables as before:

, so .

Again, all values are consistent with the diagram and both statements.

Since at least two different values of  satify the conditions, the two statements are insufficient.

### Example Question #1 : Dsq: Calculating The Angle Of An Intersection

Note: Figure NOT drawn to scale. Do not assume lines are parallel or perpendicular simply by appearance.

Evaluate .

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

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

Explanation:

Statement 1 alone gives us that , but reveals no clues about any of the eight angle measures. From Statement 2 alone, that , we can assume that , and  all have measure , but no clues are given about any of the other four angles—in particular, .

Assume both statements are true. From Statement 1, , and by way of the Parallel Postulate, corresponding angles have the same measure—in particular, . From Statement 2, we know that . From these two statements, .

### Example Question #7 : Intersecting Angles And Lines

Note: Figure NOT drawn to scale.

Give the measure of  in the above diagram.

Statement 1:  is an arc of measure .

Statement 2:  is an arc of measure .

BOTH statements TOGETHER are insufficient 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 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.