Secondary 1 Polygons and Constructions Worksheet

13 of the 14 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 13 of the 14 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Classifying Polygons

   This question is built around a diagram or a table of values. Open it in the PDF.

2. Q2Congruent and Similar Figures

   A bubble-tea shop prints its loyalty card as a rectangle $9$ cm long and $5$ cm wide. For the window display, the shop orders a poster of the same card in which every length is multiplied by $4$. It also prints a small sticker measuring $5$ cm by $9$ cm.

   1. Give the length and the width of the poster.
   2. Is the poster congruent to the card, similar to it, or neither? Justify your answer by checking the angles and the sides.
   3. Is the sticker congruent to the card? Explain.
3. Q3Constructing Polygons

   Using a ruler, a compass and a protractor, construct the quadrilateral $ABCD$ below, at true size, in the space provided.

   • $AB=6$ cm;
   • $m\angle DAB={70}^{\circ }$ with $AD=4$ cm;
   • $m\angle ABC={110}^{\circ }$ with $BC=5$ cm;
   • then join $C$ to $D$.
   1. Measure $DC$, $m\angle ADC$ and $m\angle BCD$ on your drawing.
   2. Add the four angle measures. What total should you get, and does your drawing agree?
   3. Look at the sides $AD$ and $BC$. What do you notice, and what does it tell you about the kind of quadrilateral you have built?
4. Q4Constructing Regular Polygons

   A community garden is laid out as a regular hexagon inscribed in a circle. Construct a scale drawing with a compass and a ruler:

   • draw a circle of radius $3.5$ cm and mark its centre $O$;
   • without changing the compass opening, place the point anywhere on the circle and step that opening around the circle, marking each new point;
   • join the six points in order.
   1. Measure a side of your hexagon and one of its interior angles.
   2. Join $O$ to two neighbouring vertices and measure the angle formed at $O$ — the central angle.
   3. Explain why the compass opening fits exactly six times around the circle and never five or seven.
5. Q5Constructing a Rhombus and a Parallelogram

   Construct the rhombus $ABCD$ with a side of $4$ cm and $m\angle DAB={60}^{\circ }$, in the space below. Draw $AB=4$ cm, mark ${60}^{\circ }$ at $A$ and place $D$ at $4$ cm along that ray, then use the compass opened to $4$ cm from $B$ and from $D$ to locate $C$.

   1. Draw the two diagonals $AC$ and $BD$ and measure them.
   2. Verify with a protractor and a ruler that the diagonals cut each other at right angles and that each one cuts the other in half.
   3. The diagonal $BD$ separates the rhombus into two triangles. Name them and explain why each one is equilateral.
6. Q6Constructing a Square and a Rectangle

   Construct a square $ABCD$ with a side of $5$ cm using only a ruler and a compass — no protractor. Draw $AB=5$ cm on a line, then construct the perpendicular to $AB$ at $A$ like this: with centre $A$ and any opening, draw arcs cutting the line on both sides of $A$, at $X$ and $Y$; then with a larger opening draw arcs centred at $X$ and at $Y$ that meet at $Z$; the line $AZ$ is perpendicular to $AB$. Place $D$ at $5$ cm from $A$ on $AZ$, then locate $C$ with two arcs of radius $5$ cm, one centred at $B$ and one centred at $D$; those two arcs cross at $A$ and at one other point, and that other point is $C$.

   1. Measure the two diagonals of your square. What do you notice?
   2. Check where the diagonals cross and at what angle.
   3. Which of these properties would still hold if you had built a $7$ cm by $5$ cm rectangle instead?
7. Q7Constructing a Trapezoid

   Construct the trapezoid $ABCD$ below, at true size, with a ruler and a set square (or a compass).

   • $AB=8$ cm is the longer of the two parallel sides;
   • $DC=5$ cm is parallel to $AB$ and $4$ cm above it;
   • the side $AD$ is perpendicular to $AB$, so $D$ sits directly above $A$.
   1. State the measures of $m\angle DAB$ and $m\angle ADC$ without measuring them, and say how you know.
   2. Measure $m\angle ABC$ and $m\angle BCD$ on your drawing, then check that the four angles total ${360}^{\circ }$.
   3. Which pair of sides is parallel, and how can you check it on your drawing with a set square?
8. Q8Constructing a Triangle

   Construct $△ABC$ with $AB=7$ cm, $BC=5$ cm and $AC=6$ cm, using a ruler and a compass only.

   1. Write down, in order, the compass steps you used.
   2. Measure the three angles of your triangle and check that they total ${180}^{\circ }$.
   3. Anh builds a triangle from the same three lengths but starts from the $6$ cm side. Will her triangle be congruent to yours? Explain.
9. Q9Plane figures

   Sort the six figures below into three groups: (i) plane figures whose boundary is made only of line segments; (ii) plane figures whose boundary includes at least one curved part; (iii) drawings that are not closed figures at all. Justify each choice in one line.

   1. A circle of radius $3$ cm.
   2. A pentagon.
   3. A path of four segments joined end to end whose last endpoint does not return to the start.
   4. A rectangle with a half-disc drawn on top of one side, forming a single closed outline.
   5. A single segment of length $8$ cm.
   6. A right triangle.
10. Q10Polygons

    A polygon is a closed plane figure formed by line segments joined end to end, where the segments meet only at their endpoints. Four figures are described below. Say which ones are polygons, and for each one that is not, name the condition of the definition that it fails.

    1. Five segments joined end to end, the last returning to the starting point, none of them crossing.
    2. A closed figure made of three segments and one arc.
    3. Six segments joined end to end and closed, but two of them cross each other partway along.
    4. Four segments joined end to end, the fourth ending $2$ cm away from the starting point.
11. Q11Regular Polygons

    A stop-motion studio builds a set piece in the shape of a regular octagon with a side of $4.5$ cm.

    1. State the two conditions that make this polygon regular, and give the total length of its outline.
    2. Draw a regular octagon roughly and join one chosen vertex to every other vertex it is not already joined to. Into how many triangles is the octagon cut?
    3. Use your answer to b) to find the sum of the eight interior angles, then the measure of one interior angle.
12. Q12Similarity, Congruence, and Equivalence

    Two figures are congruent when they have the same shape and the same size, similar when they have the same shape, and equivalent when they have the same area. For each pair below, say which of the three words apply — more than one may apply — and justify briefly.

    1. Two squares, each with a side of $6$ cm.
    2. A rectangle $4$ cm by $9$ cm and a square with a side of $6$ cm.
    3. A rectangle $3$ cm by $5$ cm and a rectangle $6$ cm by $10$ cm.
    4. Two triangles cut from the same sheet of cardboard, one being the mirror image of the other.
13. Q13The Construction of a Height

    A height of a triangle drawn from a vertex is the perpendicular segment from that vertex to the line containing the opposite side.

    Construct $△DEF$ with $DE=8$ cm, $m\angle EDF={30}^{\circ }$ and $m\angle DEF={20}^{\circ }$. Then construct the height from $D$ to the side $EF$.

    1. Give $m\angle DFE$ and say what kind of triangle this is.
    2. Describe what you had to do to the side $EF$ before the foot of the height from $D$ could be marked, and explain why.
    3. Of the three heights of this triangle, which one has its foot inside the triangle? Justify.
14. Q14Synthesis — drawing on several sheets in this topic

    Draw a circle with centre $O$ and radius $4$ cm, and construct the regular hexagon $ABCDEF$ inscribed in it by stepping the radius around the circle. Then join $A$ to $C$, $C$ to $E$ and $E$ to $A$.

    1. Explain why $△ACE$ has three congruent sides, using the triangles $△ABC$, $△CDE$ and $△EFA$ that you have just cut off.
    2. Is $△ACE$ a regular polygon? Is it similar to the hexagon? Justify each answer.
    3. Give the central angle of the hexagon and the central angle of $△ACE$, and check that each one divides ${360}^{\circ }$ exactly.

The 10 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Three conditions, and polygons that fail exactly one (Q1, Q10, Q11)

A polygon is a closed plane figure made of line segments joined end to end, meeting only at their endpoints. Q10 is that definition used as a checklist, and each of its figures fails a different clause: a figure with an arc in it is not made only of segments; six segments that cross partway along break the “only at their endpoints” clause; four segments that stop 2 cm short of the start are not closed at all.

That pair of conditions is worth dwelling on, because dropping either one lets in a figure that is plainly not regular. A rhombus has four congruent sides and is not regular; a rectangle has four congruent angles and is not regular. Only requiring both at once picks out the square.

Regular polygons: the central angle does the work (Q4, Q11)

The central angle of a regular polygon is the angle at the centre between the radii drawn to two neighbouring vertices. Because those angles share the full turn equally, one line settles almost every regular-polygon question on the sheet:

Read forwards it gives the angle; read backwards it gives the number of sides. And it carries a consequence students rarely expect: the number of sides has to be a whole number, so a central angle is only possible when it divides 360° exactly. That is the whole content of the hexagon question in Q4 — the compass opening equal to the radius steps round the circle exactly six times, never five and never seven, because the triangle formed by two radii and one side has three equal sides and therefore a 60° angle at the centre, and 60° goes into 360° exactly six times.

Congruent, similar, equivalent — three different words (Q2, Q12)

They are used interchangeably in conversation and they mean three different things. Congruent is same shape and same size: every corresponding side and every corresponding angle equal. Similar is same shape, with every length multiplied by the same number. Equivalent is same area, and says nothing whatever about shape.

Q12 asks which of the three words apply to each pair, and more than one may apply, which is the part that catches people. The implications only run one way: congruent figures are always similar and always equivalent, and similar figures whose factor is 1 are congruent — but a 4 cm by 9 cm rectangle and a 6 cm square are equivalent while being neither similar nor congruent.

What a construction proves (Q3, Q5, Q6, Q7, Q8)

Five questions ask you to build a figure at true size and then interrogate it. The drawing itself takes a couple of minutes; the parts that follow are the reason the question exists. Work them in the order they are asked, because each one checks the one before.

Q13 is the construction that goes wrong most often, and it is worth previewing. A height is the perpendicular segment from a vertex to the line containing the opposite side — and in a triangle with a very wide angle that line has to be extended beyond the triangle before the foot of the height can be marked at all. The height then lands outside the figure. That is not a mistake in your drawing; it is what the definition says, and the question asks you to explain it.

Which lengths actually determine a figure (Q3, Q8)

Q8 builds a triangle from three given sides with a compass, and then asks whether starting from a different side would have produced a different triangle. It would not: three lengths fix a triangle completely, up to sliding and turning it. Two arcs drawn from the ends of the first side cross at one point, and there is nothing left to choose.

Q3 quietly makes the opposite point about quadrilaterals, where four lengths alone leave the figure free to flex — which is why that question hands you two angles as well as the sides. Whenever a construction feels under-determined, count what you were given: a figure you can push out of shape without changing any of its stated measurements has not been pinned down.

Q14: the synthesis question

The last free question builds a regular hexagon inside a circle, joins alternate vertices, and asks what the triangle in the middle is. Every claim in it is settled by the same observation — the six radii are all equal, so the six triangles cut off are all congruent, so the three chords you drew are all equal. Then the vocabulary from earlier in the sheet has to be applied precisely: the triangle is regular, and it is not similar to the hexagon, because similar figures must have the same shape, and a three-sided figure and a six-sided one do not. Finish by checking that both central angles divide 360° exactly, which is the relation from Q4 used as a proof-read.

Getting the most out of it

Sharpen the pencil and open the compass carefully

On this set the drawing is the data: you measure your own figure to answer the later parts, so a compass that slips or a 4 cm side drawn at 4.3 cm produces answers that are wrong for reasons no amount of checking will reveal. Draw large, keep the compass opening fixed while you need it fixed, and leave your construction arcs on the page — they are evidence of method, not mess.

Verify with a total, not with an impression

Three angles of a triangle to 180°, four angles of a quadrilateral to 360°, the central angles of a regular polygon to 360°. Every construction on this sheet can be audited against one of those, and doing it is how you find out that the figure drifted before you build a conclusion on top of it.

Answer the vocabulary questions in full sentences

Convex, regular, congruent, similar, equivalent: each is a definition with conditions, and the answer that earns the mark names the condition that is met or the condition that fails. Write “not regular, because its angles are not all congruent”, never just “not regular”.

Common questions