GMAT Math : DSQ: Calculating the angle of an intersection
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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:
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.
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 gives insufficient information. It only gives a relationship between 
Statement 2 alone gives insufficient information; 
Now assume both statements are true. Again, from Statement 1, 
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:
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.
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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Assume Statement 1 alone. From the diagram, the three angles of measure 
From Statement 1,
so by the subtraction property of equality,
Assume Statement 2 alone. 
Example Question #2261 : Gmat Quantitative Reasoning
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.
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.
Assume Statement 1 alone. 
Assume Statement 2 alone. 
Example Question #2262 : Gmat Quantitative Reasoning
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.
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 insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Assume Statement 1 alone. 
Assume Statement 2 alone. Again, 
Since the angles of measures 
Example Question #1 : Dsq: Calculating The Angle Of An Intersection
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.
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.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
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 
The angle of measure 
From Statement 1,
Since 
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:
Again, all values are consistent with the diagram and both statements.
Since at least two different values of 
Example Question #6 : 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 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.
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.
Statement 1 alone gives us that 
Assume both statements are true. From Statement 1, 
Example Question #5 : Dsq: Calculating The Angle Of An Intersection
Note: Figure NOT drawn to scale.
Give the measure of 
Statement 1: 
Statement 2: 
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.
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.
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.
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.
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