SAT II Math II : Finding Angles

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

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Example Question #1 : Finding Angles

Solve for and .

Question_3

(Figure not drawn to scale).

Possible Answers:

Correct answer:

Explanation:

The angles containing the variable  all reside along one line, therefore, their sum must be .

Because  and  are opposite angles, they must be equal.

Example Question #1 : Finding Angles

What angle do the minute and hour hands of a clock form at 6:15?

Possible Answers:

Correct answer:

Explanation:

There are twelve numbers on a clock; from one to the next, a hand rotates . At 6:15, the minute hand is exactly on the "3" - that is, on the  position. The hour hand is one-fourth of the way from the "6" to the "7" - that is, on the  position. Therefore, the difference is the angle they make:

.

Example Question #3 : Finding Angles

In triangle  and . Which of the following describes the triangle?

Possible Answers:

 is obtuse and isosceles.

 is acute and isosceles.

 is obtuse and scalene.

None of the other responses is correct.

 is acute and scalene.

Correct answer:

 is acute and isosceles.

Explanation:

Since the measures of the three interior angles of a triangle must total 

All three angles have measure less than , making the triangle acute. Also, by the Isosceles Triangle Theorem, since ; the triangle has two congruent sides and is isosceles. 

Example Question #4 : Finding Angles

In  and  are complementary, and . Which of the following is true of  ?

Possible Answers:

None of the other responses is correct.

 is right and scalene.

 is acute and isosceles.

 is acute and scalene.

 is right and isosceles.

Correct answer:

 is right and scalene.

Explanation:

 and  are complementary, so, by definition, 

Since the measures of the three interior angles of a triangle must total 

 is a right angle, so  is a right triangle. 

 and  must be acute, so neither is congruent to ; also,  and   are not congruent to each other. Therefore, all three angles have different measure. Consequently, all three sides have different measure, and  is scalene.

Example Question #2 : Finding Angles

Decagon

The above figure is a regular decagon. Evaluate .

Possible Answers:

Correct answer:

Explanation:

As an interior angle of a regular decagon,  measures

.

Since  and  are two sides of a regular polygon, they are congruent. Therefore, by the Isosceles Triangle Theorem,

The sum of the measures of a triangle is , so

Example Question #3 : Finding Angles

Hexagon

The above hexagon is regular. What is ?

Possible Answers:

None of the other responses is correct.

Correct answer:

Explanation:

Two of the angles of the quadrilateral formed are angles of a regular hexagon, so each measures

.

The four angles of the quadrilateral are . Their sum is , so we can set up, and solve for  in, the equation:

Example Question #4 : Finding Angles

What angle do the minute and hour hands of a clock form at 4:15?

Possible Answers:

Correct answer:

Explanation:

There are twelve numbers on a clock; from one to the next, a hand rotates . At 4:15, the minute hand is exactly on the "3" - that is, on the  position. The hour hand is one-fourth of the way from the "4" to the "5" - that is, on the  position. Therefore, the difference is the angle they make:

.

Example Question #5 : Finding Angles

If the vertical angles of intersecting lines are:   and , what must be the value of ?

Possible Answers:

Correct answer:

Explanation:

Vertical angles of intersecting lines are always equal.

Set the two expressions equal to each other and solve for .

Subtract  from both sides.

Subtract 6 from both sides.

The answer is:  

Example Question #6 : Finding Angles

If the angles in degrees are  and  which are complementary to each other, what is three times the value of the smallest angle?

Possible Answers:

Correct answer:

Explanation:

Complementary angles add up to 90 degrees.

Set up an equation such that the sum of both angles equal to 90.

Subtract 10 from both sides.

Divide by 2 on both sides.

The angles are:

Three times the value of the smallest angle is:

The answer is:  

Example Question #7 : Finding Angles

If the angles  and  are supplementary, what must be the value of ?

Possible Answers:

Correct answer:

Explanation:

Supplementary angles sum up to 180 degrees.

Add five on both sides.

Divide by negative five on both sides to determine .

The answer is:  

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