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Classifying Polygons: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 Read the ordered boundary

A simple polygon has vertices $A,B,C,D,E$ in boundary order and one reflex interior angle at $C$ . Name its side-count class and its convexity class. List the two sides incident to $E$ and every diagonal from $A$ .
1. Hint 1
  Use the number and boundary order of the vertices.
2. Hint 2
  The last vertex and first vertex are adjacent; a diagonal skips adjacent vertices.
Answer

Concave pentagon; sides $DE$ and $EA$ at $E$ ; diagonals $AC$ and $AD$ .

Full solution

Five boundary vertices give five sides, so this is a pentagon.
The reflex interior angle makes it concave.

At $E$ , the boundary arrives along $DE$ and leaves along $EA$ .
From $A$ , the adjacent vertices are $B$ and $E$ , leaving $C$ and $D$ for diagonals.

Answer

Concave pentagon; sides $DE$ and $EA$ at $E$ ; diagonals $AC$ and $AD$ .

Key idea

Boundary order determines adjacency even where the final side closes the polygon.
Problem 2 Use all the given measurements

Convex quadrilateral $PQRS$ has both opposite-side pairs parallel, all four angles right, and boundary side lengths 3 cm, 7 cm, 3 cm, 7 cm in that order. List every applicable name among trapezoid, parallelogram, rectangle, rhombus, square, and kite. Explain every exclusion.
1. Hint 1
  Check the side, angle, and parallel conditions separately.
2. Hint 2
  For a kite, the equal pairs must be adjacent and disjoint, using every side once.
Answer

Trapezoid, parallelogram, and rectangle; neither rhombus, square, nor kite.

Full solution

The two parallel pairs give a parallelogram and hence an inclusive trapezoid.
Four right angles give a rectangle.

The side lengths are not all equal, so it is neither a rhombus nor a square.
In the order 3, 7, 3, 7, every adjacent side pair has different lengths, so the kite condition fails too.

Answer

Trapezoid, parallelogram, and rectangle; neither rhombus, square, nor kite.

Key idea

An applicable broad family remains applicable after a more specific name is established.
Problem 3 Separate the conditions

Case A. Give a convex polygon that is equilateral but not regular, and explain which regularity condition fails.

Case B. Give a convex polygon that is equiangular but not regular, and explain which regularity condition fails.
1. Hint 1
  Regularity requires two different kinds of equality.
2. Hint 2
  A slanted equal-sided parallelogram and a nonsquare rectangle let you vary one condition at a time.
Answer

Case A: an oblique rhombus with unequal angles. Case B: a nonsquare rectangle with unequal adjacent side lengths. Other valid examples are accepted.

Full solution

Case A.
Choose an oblique parallelogram with four equal sides and two different angle measures, such as 60 degrees and 120 degrees.
It is equilateral but not equiangular.

Case B.
A 3-by-7 rectangle has four right angles but unequal adjacent side lengths.
It is equiangular but not equilateral.

Each example fails one required regularity condition while satisfying the other.

Examples for the two cases.

Answer

Case A: an oblique rhombus with unequal angles. Case B: a nonsquare rectangle with unequal adjacent side lengths. Other valid examples are accepted.

Key idea

To disprove that one condition guarantees two, keep the first and make the second fail.
Problem 4 Test the universal claim

Consider the claim, “Every kite is a regular polygon.” Decide whether it is true under the inclusive kite definition. Give a counterexample with two specified equal-side pairs, and identify the regularity condition it fails.
1. Hint 1
  A universal claim fails if one object meets its hypothesis and fails its conclusion.
2. Hint 2
  The two equal-side pairs of a kite are allowed to have different lengths.
Answer

False. For example, a convex kite with adjacent side pairs of 5 units and 4 units is not equilateral and therefore is not regular.

Full solution

Take convex $ABCD$ with $AB$ and $AD$ each 5 units, and $BC$ and $CD$ each 4 units.
These are two disjoint adjacent congruent pairs using all four sides, so the figure is a kite.

The two pair lengths differ, so not all four sides are congruent.
It is not equilateral and cannot be regular.
The drawing gives one realizable example; no angle calculation is needed.

One counterexample with unequal pair lengths.

Answer

False. For example, a convex kite with adjacent side pairs of 5 units and 4 units is not equilateral and therefore is not regular.

Key idea

A counterexample must satisfy the hypothesis, even when its conclusion fails.
Problem 5 Identify the matching pairs

A convex quadrilateral $JKLM$ has $JK$ congruent to $KL$ , and $LM$ congruent to $MJ$ . The first pair has length 6 cm and the second has length 10 cm. Which special-quadrilateral definition can you verify directly? Is the rhombus definition satisfied? Explain.
1. Hint 1
  Locate the shared vertex of each equal-side pair.
2. Hint 2
  Check whether all four sides can have one common length.
Answer

It is a kite. It is not a rhombus.

Full solution

Sides $JK$ and $KL$ meet at $K$ ; sides $LM$ and $MJ$ meet at $M$ .
The pairs are adjacent and disjoint and use all four sides.
With convexity given, they establish a kite.

The two pair lengths differ, so the quadrilateral does not have four congruent sides.
It therefore fails the rhombus definition regardless of any further parallel evidence.

Answer

It is a kite. It is not a rhombus.

Key idea

Adjacent congruent pairs establish a kite only when they cover the four sides without overlap.
Problem 6 Track the matching turns

An unmarked convex regular dodecagon has 12 sides. Find its number of reflection axes, number of rotations including identity once, and least positive matching angle. Decide whether a 150-degree turn about its symmetry center matches, and explain.
1. Hint 1
  The side count gives both symmetry counts for a convex regular polygon.
2. Hint 2
  Compare 150 degrees with the least positive rotation step.
Answer

12 reflection axes; 12 rotations including identity; least angle $30^\circ$ ; a $150^\circ$ turn matches.

Full solution

There is one reflection axis per side and one rotation position per vertex, giving 12 of each.

The least positive rotation angle is
$360^\circ\div12=30^\circ.$

The proposed turn is a whole number of matching steps.
$150^\circ\div30^\circ=5.$
It therefore shifts each vertex five positions and matches every joining side.

Five least rotation steps give the proposed turn.

Answer

12 reflection axes; 12 rotations including identity; least angle $30^\circ$ ; a $150^\circ$ turn matches.

Key idea

A rotation matches a convex regular polygon when it is a whole number of the least matching steps.
Problem 7 Draw and annotate a boundary

On paper, draw a simple concave pentagon $ABCDE$ in boundary order. Make $C$ its reflex corner and draw diagonal $BD$ so that some of it lies outside the polygonal region. Label all five vertices, mark the reflex interior angle, and explain why your drawing is a polygon but is not convex.
1. Hint 1
  Think of a rectangle whose right side has been pushed inward to make an extra corner.
2. Hint 2
  Put B and D at the ends of that indented side and C between them, toward the interior.
Answer

A valid labeled drawing has five straight sides, no crossings, a reflex interior angle at $C$ , and part of diagonal $BD$ outside its region.

Full solution

One sample begins with $A$ upper left, $B$ upper right, $C$ inward from the right, $D$ lower right, and $E$ lower left.
Join them in that order and close the chain.

Mark the larger interior angle at $C$ , greater than a straight angle.
Segment $BD$ crosses the missing indentation rather than remaining in the region.

The chain is closed, straight-sided, and simple, so it is a polygon.
The reflex corner and outside segment show it is concave.

One possible drawing; other valid concave pentagons are accepted.

Answer

A valid labeled drawing has five straight sides, no crossings, a reflex interior angle at $C$ , and part of diagonal $BD$ outside its region.

Key idea

A concave polygon may have outside diagonals while its boundary remains simple.
Problem 8 Compare odd and even boundaries

Case A. An unmarked convex regular heptagon has 7 sides. Describe its reflection-axis count and where each axis meets the boundary. Decide whether a half-turn matches.

Case B. Repeat for an unmarked convex regular octagon with 8 sides.
1. Hint 1
  When the side count is odd, a vertex faces a side rather than another vertex.
2. Hint 2
  A half-turn would use half of the equal vertex-to-vertex rotation steps.
Answer

Case A: 7 axes, each through a vertex and an opposite side midpoint; no half-turn. Case B: 8 axes, four through opposite vertices and four through opposite side midpoints; a half-turn matches.

Full solution

Case A.
Each of seven vertices determines one mirror to the opposite side midpoint.
A half-turn would require 3.5 vertex steps, which is not a whole number, so it does not match.

Case B.
Eight vertices form four opposite pairs, and eight sides form four opposite pairs.
These supply four mirrors of each type.
A half-turn is four vertex steps and matches.

Answer

Case A: 7 axes, each through a vertex and an opposite side midpoint; no half-turn. Case B: 8 axes, four through opposite vertices and four through opposite side midpoints; a half-turn matches.

Key idea

Parity determines the mirror types and whether a regular polygon has a half-turn.
Problem 9 Establish an inclusion

Prove from the definitions that every rhombus is a kite under the course conventions. Name a convex rhombus $ABCD$ in boundary order and identify the two side-pairs you use. Explain why each side is used exactly once.
1. Hint 1
  Start with the congruent-side part of the rhombus definition.
2. Hint 2
  Group the two sides at A and the two remaining sides at C.
Answer

Every rhombus is a kite; use adjacent pairs $AB,AD$ and $BC,CD$ .

Full solution

Let $ABCD$ be any rhombus.
It is convex and has four congruent sides.
In particular, $AB$ is congruent to $AD$ , and $BC$ is congruent to $CD$ .

The first pair meets at $A$ , and the second meets at $C$ .
The pairs share no side, and together they contain $AB,BC,CD,DA$ , each once.

These are exactly the convexity and disjoint adjacent equal-side conditions in the kite definition.
No diagonal theorem is used.

Answer

Every rhombus is a kite; use adjacent pairs $AB,AD$ and $BC,CD$ .

Key idea

A family inclusion is proved by checking every condition of the broader definition.
Problem 10 Combine the names and motions

An unmarked convex quadrilateral has both opposite-side pairs parallel, four congruent sides, and four right angles.

Part A. List every applicable name among trapezoid, parallelogram, rectangle, rhombus, square, kite, and regular polygon. Justify the inclusions.

Part B. Give its reflection-axis count and describe both axis types. List all matching counterclockwise rotation angles from zero degrees up to but excluding a full turn.
1. Hint 1
  For Part A, keep every definition the givens meet.
2. Hint 2
  For Part B, first identify the number of sides of this regular polygon.
Answer

Part A: all seven names apply. Part B: 4 axes, two through opposite vertices and two through opposite side midpoints; rotations $0^\circ$ , $90^\circ$ , $180^\circ$ , $270^\circ$ .

Full solution

Part A.
The parallel pairs establish a parallelogram and hence a trapezoid.
Four right angles establish a rectangle; four congruent sides establish a rhombus.
Together these establish a square.

Part A.
Group the equal sides into two disjoint adjacent pairs to obtain a kite.
Equal sides and equal angles establish a regular polygon.

Part B.
The regular polygon has four sides, so it has four mirrors.
Two join opposite vertices, and two join opposite side midpoints.

Part B.
Its least positive step is
$360^\circ\div4=90^\circ.$
The distinct matching angles in the requested interval are zero degrees, 90 degrees, 180 degrees, and 270 degrees.
A full turn repeats identity.

The same definitions apply in this turned position.

Answer

Part A: all seven names apply. Part B: 4 axes, two through opposite vertices and two through opposite side midpoints; rotations $0^\circ$ , $90^\circ$ , $180^\circ$ , $270^\circ$ .

Key idea

Classification supplies the hypotheses needed for the regular-polygon symmetry count.

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