Mathematics notes, chapter 9

Basic polygons

Textbook Mathematics Form 1 (DLP), pages 200–222

What this chapter covers

State the relationship between the number of sides, vertices and diagonals of a polygon, and find the number of diagonals with n(n − 3) ÷ 2 (9.1)
Draw polygons, label their vertices in order and name them, from triangle to decagon (9.1)
Classify triangles by their sides (equilateral, isosceles, scalene) and by their angles (acute-angled, obtuse-angled, right-angled), and state their axes of symmetry (9.2)
Use the interior and exterior angles of a triangle (the interior angles add up to 180°, an interior angle and its adjacent exterior angle add up to 180°, and an exterior angle equals the sum of the two opposite interior angles) (9.2)
Solve problems involving triangles (9.2)
Describe and classify the square, rectangle, parallelogram, rhombus, trapezium and kite by their geometric properties (9.3)
Use the interior and exterior angles of a quadrilateral (sum 360°) and the equal opposite angles of a parallelogram (9.3)
Solve problems involving quadrilaterals and combinations of triangles and quadrilaterals (9.3)

Look up inside Putra Mosque in Putrajaya and you will see triangles, squares, hexagons, octagons and more, fitted together into beautiful patterns. Each of these shapes is a polygon. Architects, tile makers and designers use the properties of polygons, and the rules for their angles, to make shapes that fit exactly. In this chapter you will name polygons, count their diagonals, learn the properties of triangles and quadrilaterals, and use angle facts to find unknown angles.

Photograph of the decorated ceiling and dome of Putra Mosque, made of many polygon shapes — Polygons in the ceiling of Putra Mosque

Photograph: Mathematics Form 1 (DLP) textbook, p. 200.

English	Bahasa Melayu
polygon, side, vertex	poligon, sisi, bucu
diagonal	pepenjuru
triangle, quadrilateral	segi tiga, sisi empat
square, rectangle	segi empat sama, segi empat tepat
parallelogram, rhombus	segi empat selari, rombus
trapezium, kite	trapezium, lelayang
interior angle, exterior angle	sudut pedalaman, sudut peluaran
acute angle, obtuse angle, right angle	sudut tirus, sudut cakah, sudut tegak
conjecture	konjektur

9.1Polygons

A polygon is a closed figure on a flat surface bounded by 3 or more straight sides. A vertex (plural: vertices) is a corner where two sides meet. A diagonal is a straight line joining two vertices that are not next to each other.

A circle is not a polygon (it has no straight sides), and a shape with a gap is not a polygon (it is not closed).

Naming polygons

A polygon is named by its number of sides.

Eight regular polygons: triangle 3 sides, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10 — Names of polygons (textbook p. 204)

Sides	3	4	5	6	7	8	9	10
Name	triangle	quadrilateral	pentagon	hexagon	heptagon	octagon	nonagon	decagon

Memory hooks: a pentagon has 5 sides like the 5-sided Pentagon building; an octopus has 8 legs, so an octagon has 8 sides; decade = 10 years, so a decagon has 10 sides.

Sides, vertices and diagonals

Quadrilateral with 2 diagonals, pentagon with 5, hexagon with 9 and heptagon with 14, each with all diagonals drawn — All the diagonals of polygons with 4, 5, 6 and 7 sides (textbook pp. 202–203)

For every polygon, number of vertices = number of sides. For a polygon with n sides, number of diagonals = n(n − 3)2.

Why? From each vertex you can draw a diagonal to every other vertex except itself and its two neighbours, so n − 3 diagonals start at each vertex. That gives n(n − 3), but every diagonal has been counted twice (once from each end), so we divide by 2.

The textbook also gives a step method: let m = number of sides − 3; then number of diagonals = 2m + (m − 1) + (m − 2) + … + 1.

Example 1 (textbook p. 203)

Find the number of vertices and the number of diagonals of a polygon with 10 sides.

Number of vertices = number of sides = 10
Step method: m = 10 − 3 = 7, so diagonals = 2(7) + 6 + 5 + 4 + 3 + 2 + 1 = 35
Formula: 10(10 − 3)2 = 702 = 35 diagonals

Do not forget to divide by 2. A 9-sided polygon has 9 × 62 = 27 diagonals, not 54. And n − 3 (= 6) is only the number of diagonals from one vertex.

Find the number of vertices and the number of diagonals of a polygon with a) 6 sides, b) 9 sides, c) 12 sides, d) 20 sides. (Textbook Self Practice 9.1a, p. 203)

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a) 6 vertices; 6(3) ÷ 2 = 9 diagonals
b) 9 vertices; 9(6) ÷ 2 = 27 diagonals
c) 12 vertices; 12(9) ÷ 2 = 54 diagonals
d) 20 vertices; 20(17) ÷ 2 = 170 diagonals

Drawing, labelling and naming a polygon

Count the number of sides needed.
Mark that many points, with no three of them on one straight line.
Join the points with straight lines to make a closed figure.
Label the vertices with capital letters in order around the shape, and name the polygon.

Example 2 (textbook p. 204)

Draw a polygon with six sides. Label and name it.

Hexagon with vertices A, B, C, D, E, F labelled in order anticlockwise

The polygon is the hexagon ABCDEF. It may also be named, for example, BCDEFA or FEDCBA, because these go round the shape in order.

Read the letters in order around the shape, clockwise or anticlockwise. "ABDCEF" is not a name for the hexagon above, because it jumps from B to D across the shape.

True or false? a) A polygon with 11 sides has 11 vertices. b) A polygon with 12 sides has 54 diagonals. c) An octagon has 8 diagonals. (Based on textbook Mastery Q 9.1, p. 205)

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a) True: vertices = sides.
b) True: 12(9) ÷ 2 = 54.
c) False: 8(5) ÷ 2 = 20 diagonals. It has 8 sides and 8 vertices.

9.2Properties of triangles and the interior and exterior angles of triangles

Types of triangles

On diagrams, sides with the same tick marks are equal in length, angles with the same arcs are equal, and a small square marks a right angle. A dashed line shows an axis of symmetry (the fold line where the two halves match exactly).

Equilateral triangle with three equal sides and 3 dashed axes of symmetry; isosceles triangle with two equal sides, equal base angles and 1 axis; scalene triangle with three different sides and no axis — Triangles classified by their sides (textbook p. 207)

By sides	Properties	Axes of symmetry
Equilateral triangle	All 3 sides are equal. Every interior angle is 60°.	3
Isosceles triangle	2 sides are equal. The 2 base angles (opposite the equal sides) are equal.	1
Scalene triangle	All sides are different. All angles are different.	none

Acute-angled triangle with angles 60, 50 and 70 degrees; obtuse-angled triangle with angles 35, 25 and 120 degrees; right-angled triangle with angles 90, 35 and 55 degrees — Triangles classified by their angles (textbook p. 207)

By angles	Property
Acute-angled triangle	Every interior angle is acute (less than 90°).
Obtuse-angled triangle	One interior angle is obtuse (between 90° and 180°).
Right-angled triangle	One interior angle is a right angle (90°).

A triangle has two names: one from its sides and one from its angles. For example, a triangle with angles 90°, 45° and 45° is a right-angled isosceles triangle, with 1 axis of symmetry. An equilateral triangle is also a special isosceles triangle.

To decide the type by angles you need all three angles. If two angles are 25° and 35°, the triangle is not acute-angled: the third angle is 180° − 25° − 35° = 120°, so it is obtuse-angled.

True or false? Explain. a) An isosceles triangle could be an obtuse-angled triangle. b) A right-angled triangle could be an isosceles triangle. c) A scalene triangle has one axis of symmetry. (Textbook p. 207)

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a) True, e.g. angles 120°, 30°, 30°.
b) True, angles 90°, 45°, 45° with the two shorter sides equal.
c) False: a scalene triangle has no axis of symmetry.

Interior and exterior angles of a triangle

An interior angle is inside the triangle at a vertex. When a side is extended, the angle between the extended line and the next side is an exterior angle.

Triangle with interior angles a, b, c; side extended at the vertex with angle c to give exterior angle d — a, b and c are interior angles; d is an exterior angle (textbook p. 209)

For every triangle:
(i) the interior angles add up to 180°: a + b + c = 180°
(ii) an interior angle and its adjacent exterior angle add up to 180° (angles on a straight line): c + d = 180°
(iii) an exterior angle equals the sum of the two opposite interior angles: d = a + b

Fact (iii) comes from (i) and (ii): both c + d and a + b + c equal 180°, so d = a + b. It saves a step in many questions.

Example 3 (textbook p. 209)

(a) Triangle with angles 92 degrees, 37 degrees and x. (b) Triangle with interior angles 79 and 28 degrees, and exterior angle x at the third vertex

Find the value of x.

(a) 92° + 37° + x = 180° (sum of interior angles)
129° + x = 180°, so x = 180° − 129° = 51°
(b) x = 79° + 28° (exterior angle = sum of the two opposite interior angles)
x = 107°

Find the value of p or x in each triangle.

(a) triangle with angles 100 and 45 degrees and p. (b) right-angled triangle with 43 degrees and p. (c) triangle with interior angles 44 and 38 degrees and exterior angle x at the third vertex

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a) p = 180° − 100° − 45° = 35°
b) p = 180° − 90° − 43° = 47°
c) x = 44° + 38° = 82°

Solving problems involving triangles

Many problems need two or three steps. Write down each angle you find with its reason, then use it in the next step.

Example 4 (textbook p. 210)

In the diagram, PQRS is a straight line and TQ = TS. Calculate the values of x and y.

Points P, Q, R, S on a straight line; T above; TQ = TS marked with ticks; angle RTS = 46 degrees, angle TRS = 84 degrees; x is angle PQT; y is angle QTR

In ΔRST: ∠RST + 84° + 46° = 180°, so ∠RST = 180° − 84° − 46° = 50°
TQ = TS, so ΔQST is isosceles: ∠TQS = ∠RST = 50° (base angles)
x + 50° = 180° (angles on a straight line), so x = 130°
In ΔQST: ∠QTS = 180° − 50° − 50° = 80°
y + 46° = 80°, so y = 34°

The base angles of an isosceles triangle are the angles opposite the equal sides, not the angle between them. Look at the tick marks before you decide which angles are equal.

9.3Properties of quadrilaterals and the interior and exterior angles of quadrilaterals

Types of quadrilaterals

Arrow marks show parallel sides: sides with the same arrows are parallel.

Square with 4 axes of symmetry; rectangle with 2; parallelogram with none; rhombus with 2 (its diagonals); trapezium with none; kite with 1 — The special quadrilaterals and their axes of symmetry (textbook p. 213)

Quadrilateral	Geometric properties	Axes of symmetry
Square	All sides equal; opposite sides parallel; all angles 90°; diagonals equal and are perpendicular bisectors of each other.	4
Rectangle	Opposite sides parallel and equal; all angles 90°; diagonals equal and bisect each other.	2
Parallelogram	Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.	none
Rhombus	All sides equal; opposite sides parallel; opposite angles equal; diagonals are perpendicular bisectors of each other.	2
Trapezium	Only one pair of opposite sides is parallel.	none
Kite	Two pairs of adjacent sides equal; one pair of opposite angles equal; one diagonal is the perpendicular bisector of the other and bisects the angles at its ends.	1

Special cases: a square is also a rectangle and a rhombus (it has all their properties). Rectangles and rhombuses are special parallelograms. A trapezium usually has no axis of symmetry, but an isosceles trapezium (with its two slanting sides equal) has 1.

"Bisect each other" (cut each other in half) is not the same as "perpendicular" (meet at 90°). The diagonals of a rectangle bisect each other but are not perpendicular unless the rectangle is a square.

a) Which quadrilaterals have all sides equal? b) Which quadrilaterals have all angles 90°? c) State two properties a rhombus has that every parallelogram does not have. (Textbook pp. 213, 218, 221)

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a) Square and rhombus
b) Square and rectangle
c) All four sides are equal; the diagonals are perpendicular; it has 2 axes of symmetry (any two)

Interior and exterior angles of a quadrilateral

Quadrilateral with interior angles a, b, c, d and exterior angle e next to c; parallelogram with angles p, q, r, s where p is opposite r and q is opposite s — Angles of a quadrilateral and of a parallelogram (textbook p. 215)

For every quadrilateral:
(a) the interior angles add up to 360°: a + b + c + d = 360°
(b) an interior angle and its adjacent exterior angle add up to 180°: c + e = 180°
In a parallelogram (or rhombus) the opposite angles are equal: p = r and q = s. Two angles next to each other add up to 180°: p + q = 180°.

Why 360°? A diagonal cuts any quadrilateral into two triangles, and 2 × 180° = 360°.

Example 5 (textbook p. 215)

In each diagram, PST is a straight line. Calculate the values of x and y.

(a) Quadrilateral PQRS with exterior angle TSR = 100 degrees, angle PQR = 145 degrees, angle SPQ = 52 degrees, x = angle RSP, y = angle QRS. (b) Parallelogram QRSP with angle Q = 42 degrees, x = angle RSP, y = exterior angle RST

(a) x + 100° = 180° (interior + adjacent exterior angle), so x = 80°
y + 80° + 52° + 145° = 360° (sum of interior angles of a quadrilateral)
y + 277° = 360°, so y = 83°
(b) x = 42° (opposite angles of a parallelogram are equal)
y + 42° = 180°, so y = 138°

In the diagram, PQT is a straight line. Calculate the value of x. (Textbook Self Practice 9.3b, p. 216)

Quadrilateral PQRS with angle P = 68 degrees, angle S = 115 degrees, angle R = 118 degrees; PQ extended to T; x is exterior angle RQT

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∠PQR = 360° − 68° − 115° − 118° = 59°
x = 180° − 59° = 121°

Solving problems involving quadrilaterals

Example 6 (textbook p. 216)

PQRU is a square and QRSV is a parallelogram. PVR is a straight line. Find the values of x and y.

Square PQRU with diagonal PR; parallelogram QRSV with V on PR and S to the right of the square; angle VSR = 72 degrees; x = angle RVS; y = angle PVQ

∠PQR = 90° and PQ = QR, so the diagonal makes ∠QRV = 90° ÷ 2 = 45°.
VS is parallel to QR, so x = ∠QRV = 45° (alternate angles)
∠QVS + 72° = 180° (adjacent angles of a parallelogram), so ∠QVS = 108°
∠QVR = 108° − 45° = 63°
PVR is a straight line, so y = 180° − 63° = 117°

Combinations of triangles and quadrilaterals

Example 7 (textbook p. 217)

QRTU is a parallelogram and PQRS is a straight line. Find the values of x and y.

Parallelogram QRTU on the line PQRS with angle PQU = 112 degrees; diagonal QT with angle QTR = 36 degrees and x = angle TQR; triangle RST with equal angles at R and S, y = angle RTS

∠QRT = 112° (corresponding angles, QU ∥ RT)
In ΔQRT: x + 112° + 36° = 180°, so x = 32°
∠TRS + 112° = 180° (angles on a straight line), so ∠TRS = 68°
In ΔRST, ∠TSR = ∠TRS = 68°: y + 68° + 68° = 180°, so y = 44°

Give a reason for every angle (straight line, sum of angles, opposite angles, isosceles, parallel lines). If you cannot give a reason, you have probably assumed something that is not given, such as two lines being parallel.

The four interior angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the largest angle. (Textbook p. 221)

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Total parts = 3 + 4 + 5 + 6 = 18
One part = 360° ÷ 18 = 20°
Largest angle = 6 × 20° = 120°

Summary

Idea	Key fact	Example
Polygon	Closed figure with 3 or more straight sides; named by its number of sides	5 sides: pentagon; 8 sides: octagon
Vertices, diagonals	Vertices = sides; diagonals = n(n − 3) ÷ 2	10 sides: 10 vertices, 35 diagonals
Naming	Letters in order round the shape	hexagon ABCDEF
Triangles by sides	Equilateral (3 axes), isosceles (1 axis), scalene (none)	Isosceles: base angles equal
Triangles by angles	Acute-angled, obtuse-angled, right-angled	25°, 35°, 120°: obtuse-angled
Triangle angles	Interior sum 180°; interior + exterior = 180°; exterior = sum of opposite interior angles	79° + 28° = 107°
Quadrilaterals	Square (4 axes), rectangle (2), rhombus (2), kite (1), parallelogram and trapezium (none)	A square is also a rhombus
Quadrilateral angles	Interior sum 360°; interior + exterior = 180°; parallelogram: opposite angles equal	360° − 80° − 52° − 145° = 83°