# SAT II Math II : Finding Angles

## Example Questions

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### Example Question #1 : Angles

Solve for and .

(Figure not drawn to scale).

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 #55 : 2 Dimensional Geometry

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

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 #56 : 2 Dimensional Geometry

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

is obtuse and scalene.

is obtuse and isosceles.

None of the other responses is correct.

is acute and scalene.

is acute and isosceles.

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 #57 : 2 Dimensional Geometry

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

is acute and isosceles.

None of the other responses is correct.

is right and isosceles.

is right and scalene.

is acute and scalene.

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 #58 : 2 Dimensional Geometry

The above figure is a regular decagon. Evaluate .

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 #1 : Finding Angles

The above hexagon is regular. What is ?

None of the other responses is correct.

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 #2 : Finding Angles

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

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 #3 : Finding Angles

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

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.

### Example Question #4 : 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?

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:

### Example Question #5 : Finding Angles

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

Explanation:

Supplementary angles sum up to 180 degrees.