ISEE Upper Level Quantitative : Lines

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #80 : Plane Geometry

Lines

Examine the above diagram. If , give  in terms of .

Possible Answers:

Correct answer:

Explanation:

The two marked angles are same-side exterior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for  in this equation:

Example Question #81 : Geometry

Which is the greater quantity? 

(a) The measure of an angle complementary to a  angle

(b) The measure of an angle supplementary to a  angle

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Supplementary angles and complementary angles have measures totaling  and , respectively.

(a) The measure of an angle complementary to a  angle is 

(b) The measure of an angle supplementary to a  angle is 

This makes (b) greater.

Example Question #81 : Geometry

 and  are complementary; .

Which is the greater quantity?

(A) 

(B) 

Possible Answers:

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

(B) is greater 

Correct answer:

(B) is greater 

Explanation:

Two angles are complementary if their degree measures total 90. Therefore, 

Since , we can substitute, and we can solve for :

, making (B) the greater quantity.

Example Question #83 : Geometry

Untitled

Note: diagram is not drawn to scale

Refer to the above diagram. If  , what is  ?

Possible Answers:

Correct answer:

Explanation:

 and  form a linear pair, so 

Since , this can be rewritten as

, and the first equation can be rewritten as:

Example Question #1 : Lines

Untitled

Note: Figure NOT drawn to scale.

Which of the following pairs of numbers could give the measures of  and  ?

Possible Answers:

None of the pairs given in the other choices is correct.

Correct answer:

Explanation:

The two angles form a linear pair and therefore their measures total . We check all of the pairs for this sum.

The correct pair is .

Example Question #85 : Geometry

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following triples could refer to the measures of , and  ?

Possible Answers:

All of the other responses are correct.

Correct answer:

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are   and . Therefore, we are looking for the sum of the first two angle measures to be equal to the third.

All four triples satisfy this condition.

Example Question #86 : Geometry

Untitled

Note: figure NOT drawn to scale

The degree measure of  is five degrees greater than twice that of . Which is the greater quantity?

(A) 

(B) 

Possible Answers:

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

The degree measure of  is five degrees greater than twice that of  - that is, 

Since  and  form a linear pair, 

 , and by substitution, 

This makes (A) greater.

Example Question #87 : Geometry

Untitled

Note: Figure NOT drawn to scale

 . What is  ?

Possible Answers:

Correct answer:

Explanation:

 and  form a linear pair and therfore, the the sum of their degree measures is 

Example Question #88 : Geometry

Triangle

Note: Figure NOT drawn to scale.

Evaluate .

Possible Answers:

Correct answer:

Explanation:

, so 

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are   and .

 

Example Question #89 : Geometry

What is the slope of a line that passes through points  and ?

 

Possible Answers:

Correct answer:

Explanation:

The equation for solving for the slope of a line is 

Thus, if  and , then:

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