Intersecting Angles and Lines - GMAT Quantitative

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 .

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.

If two chords of a circle intersect inside it, then the measure of an angle formed is equal to the arithmetic mean of the measures of arc it intercepts and the arc its vertical angle intercepts. In other words, in this diagram,

Both arc measures are needed to find the measure of the angle. Neither statement alone give both; the two together do.

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, .

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 .

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.

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.

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 .

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.

If two chords of a circle intersect inside it, then the measure of an angle formed is equal to the arithmetic mean of the measures of arc it intercepts and the arc its vertical angle intercepts. In other words, in this diagram,

Both arc measures are needed to find the measure of the angle. Neither statement alone give both; the two together do.

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, .

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 .

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.

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.

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 .

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.

If two chords of a circle intersect inside it, then the measure of an angle formed is equal to the arithmetic mean of the measures of arc it intercepts and the arc its vertical angle intercepts. In other words, in this diagram,

Both arc measures are needed to find the measure of the angle. Neither statement alone give both; the two together do.

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, .

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 .

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.