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## Example Questions

### Example Question #1 : Calculating An Angle In A Quadrilateral

Note: Diagram is NOT drawn to scale.

Refer to the above diagram.

Any of the following facts *alone* would be enough to prove that is *not* a parallelogram, EXCEPT:

**Possible Answers:**

*Any one* of these facts *alone* would prove that is *not* a parallelogram.

**Correct answer:**

Opposite sides of a parallelogram are congruent; if , then , violating this condition.

Consecutive angles of a parallelogram are supplementary; if , then , violating this condition.

Opposite angles of a parallelogram are congruent; if , then , violating this condition.

Adjacent sides of a parallelogram, however, may or may not be congruent, so the condition that would not by itself prove that the quadrilateral is not a parallelogram.

### Example Question #2 : Calculating An Angle In A Quadrilateral

Which of the following can not be the measures of the four interior angles of a quadrilateral?

**Possible Answers:**

All four of the other choices fit the conditions.

**Correct answer:**

The four interior angles of a quadrilateral measure a total of , so we test each group of numbers to see if they have this sum.

This last group does not have the correct sum, so it is the correct choice.

### Example Question #3 : Calculating An Angle In A Quadrilateral

A circle can be circumscribed about each of the following figures except:

**Possible Answers:**

An isosceles trapezoid with one base three times as long as the other

A rectangle twice as long as it is wide

An isosceles triangle with its base one-half as long as either leg

A rhombus with a angle

A right scalene triangle

**Correct answer:**

A rhombus with a angle

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure ; the rectangle fits this description, but the rhombus does not.

### Example Question #4 : Calculating An Angle In A Quadrilateral

Two angles of a parallelogram measure and . What are the possible values of ?

**Possible Answers:**

**Correct answer:**

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and

### Example Question #5 : Calculating An Angle In A Quadrilateral

Rhombus has two diagonals that intersect at point ; .

What is ?

**Possible Answers:**

**Correct answer:**

The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.

### Example Question #6 : Calculating An Angle In A Quadrilateral

Quadrilateral is inscribed in circle . . What is ?

**Possible Answers:**

**Correct answer:**

Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so

### Example Question #51 : Other Quadrilaterals

Note: Figure NOT drawn to scale.

The above figure is of a rhombus and one of its diagonals. What is equal to?

**Possible Answers:**

Not enough information is given to answer the question.

**Correct answer:**

The four sides of a rhombus are congruent, so a diagonal of the rhombus cuts it into two isosceles triangles. The two angles adjacent to the diagonal are congruent, so the third angle, the one marked, measures:

### Example Question #7 : Calculating An Angle In A Quadrilateral

Refer to the above figure. You are given that Polygon is a parallelogram, but NOT that it is a rectangle.

Which of the following statements is *not* enough to prove that Polygon is also a rectangle?

**Possible Answers:**

and are complementary angles

**Correct answer:**

To prove that Polygon is also a rectangle, we need to prove that any one of its angles is a right angle.

If , then by definition of perpendicular lines, is right.

If , then, since and form a linear pair, is right.

If , then, by the Converse of the Pythagorean Theorem, is a right triangle with right angle .

If and are complementary angles, then, since

, making right.

However, since, by definition of a parallelogram, , by the Alternate Interior Angles Theorem, regardless of whether the parallelogram is a rectangle or not.

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