Deck 16: Polygons

Question 1

Draw a tree diagram showing the hierarchy of the following quadrilaterals: kites, trapezoids, parallelograms, squares, rhombuses, isosceles trapezoids, and rectangles. (Note: Use "quadrilaterals" at the top of your diagram.)

Answer 1

Question 2

Sketch an example, if it is possible, of each shape described. If any are not possible tosketch, explain why.

A) an isosceles triangle that is not an acute triangle
B) a quadrilateral with two 90° angles that is not a rectangle (Be sure to mark the 90° angles.)
C) a kite that is also a rectangle
D) a triangular right prism (Show every hidden edge as a dashed segment.)

Answer 2

A) a sketch of a right isosceles triangle or an obtuse isosceles triangle
B) a sketch of a trapezoid that is not isosceles (two right angles)
C) a sketch of a square
D) a sketch of a right prism with triangular bases or a right prism with right triangular bases (The latter is probably acceptable, unless you have emphasized the former.)

Question 3

For each shape, sketch an example if it is possible. If it is not possible, say so and explain why.

A) a trapezoid with exactly one right angle
B) a parallelogram that is an isosceles trapezoid
C) a pentagon with exactly one right angle
D) an isosceles obtuse triangle
E) a rhombus that is not a kite

Answer 3

A) This is not possible because the parallel sides will force at least two right angles if there is one.
B) a sketch of any rectangle/square
C) a sketch with exactly one right angle and five sides
D) One example is a triangle with angles of 120°, 30°, and 30°.
E) This is not possible because every rhombus is a special kite.

Question 4

What is the sum of the measures of the exterior angles, one at each vertex, of every convex polygon? Explain your reasoning.

Question 5

Tell whether each statement is always true, sometimes true, or never true. Justify your choice.

A) The diagonals of a parallelogram bisect each other.
B) The diagonals of a kite bisect each other.

Question 6

Indicate whether each statement is always true, sometimes true, or never true. Justify your choice.

A) A parallelogram is an isosceles trapezoid.
B) A square is a rhombus.
C) A scalene triangle is an acute triangle.

Question 7

Consider the following definitions of a trapezoid.
Definition A: A quadrilateral with at least one pair of opposite sides parallel.
Definition B: A quadrilateral with exactly one pair of opposite sides parallel.
Is it possible to draw a figure that is a trapezoid according to definition A but not a trapezoid according to definition B? If yes, draw a figure that satisfies the conditions and explain why your figure satisfies those conditions. If no, explain why not.

Question 8

Every square is a special quadrilateral.

Question 9

Every rectangle is a parallelogram.

Question 10

Every rhombus is a parallelogram.

Question 11

Every rectangle is a special square.

Question 12

Every trapezoid is a special parallelogram.

Question 13

Any fact that is true for every parallelogram is also true for every square.

Question 14

Any fact that is true for every rectangle is also true for every quadrilateral.

Question 15

Draw (if possible) an isosceles trapezoid with exactly one right angle. If it is not possible, explain why.

Question 16

Draw (if possible) a kite that does not have four congruent sides. If it is not possible, explain why.

Question 17

Give the best name (if the shape is possible) for the descriptions listed below. If the shape is not possible, explain why.

A) a kite that is also an isosceles trapezoid
B) a rhombus that is not equilateral
C) an equilateral (but not regular) isosceles trapezoid

Question 18

Every kite is also a parallelogram.

Question 19

There are a total of 1175 diagonals in a 50-gon.

Question 20

It is possible to make a regular pyramid using an isosceles trapezoid as a base.

Question 21

A pentagonal pyramid has a total of five vertices.

Question 22

The lateral faces of a prism are always parallelogram regions.

Question 23

A rhombus is a kite.

Question 24

The interior angles of a trapezoid add up to 360°, but only 270° if it is an isosceles trapezoid.

Question 25

Give the best name for the following shapes.

A) a regular quadrilateral
B) an isosceles trapezoid with at least one right angle

Question 26

If possible, sketch an example of each description. If it is not possible, explain why. Use tick marks and hidden edges and label equal angles to make your intent clear.

A) a trapezoid with exactly two right angles
B) a kite with equal diagonals

Question 27

For the following conjecture, draw an example that supports the claim and also one that shows the claim is false.
"The diagonals of a parallelogram are equal."

Question 28

Which shape will have ALL of the properties that every isosceles trapezoid has?

A) parallelogram
B) kite
C) rhombus
D) rectangle

Question 29

Why is it NOT safe to make a general conclusion based on a drawing?

Question 30

Why is a result from inductive reasoning NOT completely trustworthy?

Question 31

Put the following terms in the blanks below to show the relationship among them: isosceles trapezoid, parallelogram, quadrilateral, rectangle, and rhombus.

Question 32

Arrange (only) the following terms in a hierarchical diagram, with the most general at the top: kite, square, polygon, trapezoid, and rectangle.

Question 33

Complete the following statements.

A) The measure of EACH interior angle of a regular decagon is _____.
B) If an interior angle of a regular polygon is 175°, then the polygon has _____ sides.
C) The number of congruent sides on a scalene triangle is _____.
D) The number of diagonals in a 16-gon is _____.

Question 34

For each shape, sketch an example if it is possible. (Be sure to mark your picture to fit.) If it is impossible, say so and explain why or show a counterexample.

A) a parallelogram with exactly one right angle
B) an isosceles right triangle
C) a rectangle that is not a parallelogram
D) an equilateral quadrilateral that is not regular
E) a concave hexagon

Question 36

A student states that a square cannot be a rhombus. What irrelevant characteristic(s) might she be assuming to be important? How would you help her to understand her error?

Question 37

How many diagonals does each of these shapes have?

A) a pentagon
B) a 103-gon

Question 38

The sizes of three interior angles of a quadrilateral are 65°, 35°, and 60°. What is the size of the fourth angle of the quadrilateral?

A) 20°
B) 100°
C) 160°
D) 200°
E) This quadrilateral is impossible.

Question 39

An isosceles triangle has two angles that measure 100° and 40°. How large is the third angle?

A) 40°
B) 60°
C) 100°
D) 140°
E) More information is needed to answer the question.

Question 40

If a polygon is equiangular, then it must be:

A) equilateral.
B) regular.
C) a triangle.
D) equilateral and regular.
E) None of the answers is correct.

Question 41

An angle that is supplementary to an angle with size 70° has what size?

A) 20°
B) 70°
C) 90°
D) 110°
E) 180°

Question 42

Find the number of degrees in each lettered angle.

Question 43

The sum of the measures of all of the angles of a 17-gon is _____.

Question 44

Indicate whether each statement is always true, sometimes true, or never true. If the answer is sometimes true, then draw a picture of a shape that meets both requirements and a shape that meets only the first requirement.

A) A parallelogram is a rectangle.
B) A rectangle is a parallelogram.
C) A rhombus is a rectangle.
D) A prism is a pyramid.

Question 45

When asked to find the sum of the interior angles in a hexagon, a student writes the statement alongside the sketch below. Comment on whether the student's mathematical reasoning is correct or incorrect. If it is correct, explain how you know. If it is incorrect, explain what was incorrect about the student's thinking and what he/she would have to do to correct the error.

Question 46

Sketch an example of each description if it is possible. If any sketches are not possible, explain why.

A) an equilateral quadrilateral that is not equiangular (What is this shape usually called?)
B) an equilateral obtuse triangle
C) a pentagon with exactly three right angles

Question 47

Create a Venn diagram or a hierarchy diagram for only the following terms: quadrilateral, square, rectangle, polygon, and rhombi.

Question 48

A) How many different arrangements of three identical rhombi are possible in which each rhombus matches up edge to edge with at least one other rhombus? Two arrangements are considered the same if one of the arrangements can be flipped and/or rotated to obtain the other arrangement. Sketch all possible arrangements.
B) Explain a counting strategy you used to justify that you have found all the different arrangements.

Question 49

Why is 10 called a triangular number?

Question 50

Sketch, if possible, an obtuse isosceles triangle that has an angle with 30°. If such a triangle exists, give the measures of the other angles. If such a triangle is impossible, explain why.

Question 51

State a fact that is true for all rhombi but not true for all kites.

Question 52

State a fact that is true for all rectangles but not true for all parallelograms.

Question 53

State two facts that are true for the diagonals of every rhombus.

Question 54

State two facts that are true for the diagonals of every rectangle.

Question 55

State two facts that are true for every parallelogram.

Question 56

The diagonals of a parallelogram are equal.

A) always true
B) sometimes true
C) never true

Question 57

A square is a rhombus.

A) always true
B) sometimes true
C) never true

Question 58

The diagonals of every parallelogram:

A) bisect the angles of the parallelogram.
B) are parallel to each other.
C) are perpendicular to each other.
D) are equal in length.
E) None of the answers is correct.

Question 59

Which statement is TRUE for every rhombus?
I. The diagonals of a rhombus must be equal.
II. The sides of a rhombus must be equal.

A) I only
B) II only
C) I and III
D) neither I nor II

Question 60

If IJKLMN is a regular polygon, then:

A) all the diagonals are equal in length.
B) IJKLMN has seven sides.
C) angle JKL has twice as many degrees as angle LMN.
D) drawing all the diagonals from K gives six triangles.
E) None of the answers is correct.

Question 61

How are the diagonals of every rectangle related?
I. The diagonals are the same length.
II. The diagonals are perpendicular.
III. The diagonals bisect the angles of the rectangle.

A) I only
B) II only
C) III only
D) I and III only
E) None of the answers is correct.

Question 62

Every square is a special quadrilateral.

Question 63

Every rectangle is a parallelogram.

Question 64

Every rhombus is a parallelogram.

Question 65

Every rectangle is a special square.

Question 66

Every trapezoid is a special parallelogram.

Question 67

Any fact that is true for every parallelogram is also true for every square.

Question 68

Any fact that is true for every rectangle is also true for every quadrilateral.

Question 69

What is the BEST name for every four-sided polygon with equal sides?

A) square
B) parallelogram
C) rectangle
D) rhombus
E) kite

Question 70

Fill in each blank with the BEST choice from the list of shapes A-E. You may re-use a choice.
_______has parallel lateral edges
_______a regular quadrilateral
_______has an equal number of faces and vertices
_______a polygon that is never a kite
_______possibility of all its faces being regular
is always a kite
_______has lateral faces that are triangular regions
has two bases

A) pyramid
B) non-square rectangle
C) parallelogram
D) square
E) oblique prism

Question 71

Is it possible for the sum of the angles of a polygon to be 180,000°? Explain.

Question 72

What is the measure of one interior angle of a regular 18-gon?

Question 73

Find the measure of angle A of the triangle.

A) 156°
B) 24°
C) 57°
D) 20°
E) More information is needed to answer the question.

Question 74

Which, if either, of the following is IMPOSSIBLE for x, y, and z in the sketch below? (The drawing is not to scale.)
I. x = 100, y = 110, z = 150
II. x = 80, y = 130, z = 150

A) Only I is impossible.
B) Only II is impossible.
C) Both I and II are impossible.
D) Both I and II are possible.