Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 7

Geometry II: Polygons, Angle Sum, Quadrilaterals, Parallelograms and Constructions

What to remember

The interior angles of an n-sided polygon add up to (n − 2) × 180°, and the exterior angles of any convex polygon add up to 360°, so each exterior angle of a regular polygon is 360°/n.
A quadrilateral has an angle sum of 360°; a parallelogram has equal opposite sides and angles and diagonals that bisect each other, and the rectangle, rhombus and square are special parallelograms.
Ruler and compass constructions rest on equal arcs: a perpendicular bisector, an angle bisector, and angles of 60°, 90°, 30°, 45° and 75° come from them; a unique quadrilateral needs five measurements.

Polygons

A polygon is a closed figure made only of line segments. Its sides are segments, its vertices are the corners, and a diagonal joins two vertices that are not neighbours.

Sides	Name
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon
10	Decagon
12	Dodecagon

A convex polygon has every interior angle less than 180° and every diagonal inside. A concave polygon has at least one interior angle more than 180°.
A regular polygon has all sides equal and all angles equal (equilateral and equiangular). A square and an equilateral triangle are regular.
A polygon with n sides has n vertices, n interior angles and n(n − 3)/2 diagonals. A hexagon has 9 diagonals and an octagon has 20.

Angle sum of polygons

A polygon of n sides can be split into (n − 2) triangles from one vertex, which gives the rule.

Sum of interior angles = (n − 2) × 180°.
Each interior angle of a regular polygon = (n − 2) × 180°/n.
Sum of exterior angles (one at each vertex) = 360°, for any convex polygon, whatever the number of sides.
Each exterior angle of a regular polygon = 360°/n.
At each vertex: interior angle + exterior angle = 180°.

Polygon (regular)	Sum of interior angles	Each interior angle	Each exterior angle
Triangle	180°	60°	120°
Quadrilateral	360°	90°	90°
Pentagon	540°	108°	72°
Hexagon	720°	120°	60°
Octagon	1080°	135°	45°
Decagon	1440°	144°	36°
Dodecagon	1800°	150°	30°

Worked examples

A regular polygon has each exterior angle 36°. Number of sides = 360/36 = 10.
Each interior angle is 140°. The exterior angle is 40°, so n = 360/40 = 9.
The interior angle is 150°, so the exterior is 30° and n = 12.
Sum of interior angles is 1080°: (n − 2) = 6, so n = 8.
Sum of interior angles is 1440°: n − 2 = 8, so n = 10.

Quadrilaterals

A quadrilateral has four sides, four angles and two diagonals. The sum of its four angles is 360°, because a diagonal splits it into two triangles. If three angles are 80°, 95° and 105°, the fourth is 360 − 280 = 80°. If the angles are in ratio 1 : 2 : 3 : 4, each part is 36°, so the largest angle is 144°.

Types and properties

Quadrilateral	Sides	Angles	Diagonals
Parallelogram	Opposite sides parallel and equal	Opposite angles equal; adjacent angles add to 180°	Bisect each other
Rectangle	Opposite sides parallel and equal	All 90°	Equal and bisect each other
Rhombus	All four sides equal	Opposite angles equal	Perpendicular bisectors of each other; bisect the angles
Square	All sides equal	All 90°	Equal, perpendicular, bisect each other
Trapezium	Exactly one pair of parallel sides	None special	None special
Kite	Two pairs of adjacent sides equal	One pair of opposite angles equal	One diagonal is the perpendicular bisector of the other

Every square is a rectangle, a rhombus and a parallelogram. Every rectangle and rhombus is a parallelogram. A parallelogram need not be a rectangle.
A quadrilateral that is both a rectangle and a rhombus is a square.
In an isosceles trapezium the non-parallel sides are equal and the diagonals are equal.

Parallelograms

Properties: opposite sides equal and parallel; opposite angles equal; consecutive angles supplementary; diagonals bisect each other.

Conditions to prove a quadrilateral is a parallelogram (any one is enough):

Both pairs of opposite sides are equal.
Both pairs of opposite angles are equal.
One pair of opposite sides is equal and parallel.
The diagonals bisect each other.

Worked examples: If one angle is 70°, the adjacent angle is 110°. If adjacent angles are in the ratio 2 : 3, they are 72° and 108°. In a parallelogram with ∠A = (2x + 10)° and the opposite ∠C = (3x − 10)°, we get 2x + 10 = 3x − 10, so x = 20 and each of those angles is 50°. Joining the mid-points of the sides of any quadrilateral gives a parallelogram.

Areas and diagonals

Figure	Area	Other result
Rectangle	length × breadth	Diagonal = √(l² + b²); 12 by 5 gives 13
Square	side² = (diagonal)²/2	Diagonal = side × √2
Parallelogram	base × height	Height is perpendicular to the base
Rhombus	½ × d₁ × d₂	Side = √((d₁/2)² + (d₂/2)²)
Trapezium	½ × (sum of parallel sides) × height
Kite	½ × d₁ × d₂

Rhombus with diagonals 6 and 8: side = √(9 + 16) = 5. Rhombus with diagonals 10 and 24: area 120 and side 13.
Trapezium with parallel sides 8 and 12 and height 5: area = ½ × 20 × 5 = 50.
Parallelogram of base 15 and height 8: area 120. If area is 72 and base 9, the height is 8.
Square of diagonal 10: area = 100/2 = 50. Square with perimeter 40: side 10, area 100.
Rectangle of area 96 and length 12: breadth 8, perimeter 40.
Rhombus with perimeter 52 and one diagonal 24: side 13, half of that diagonal 12, half of the other √(169 − 144) = 5, so the other diagonal is 10.

Constructions (ruler and compass)

Tools: a ruler, a compass, a divider, a protractor and a set square. The compass draws arcs of equal radius; this is the idea behind every construction.

Basic constructions

1. Perpendicular bisector of a segment AB: draw arcs above and below from A and from B with the same radius, more than half of AB. Join the two crossing points.
2. Angle bisector: draw an arc cutting both arms, then from those two points draw equal arcs that cross. Join the vertex to the crossing.
3. 60° angle: draw an arc from the vertex, and with the same radius cut it from the point where it meets the arm. The point makes 60° (an equilateral triangle).
4. 120° angle: mark two successive 60° arcs.
5. 90° angle: build from 60° and 120° and bisect the gap between them, or use the perpendicular construction.
6. 30°, 45° and 15°: bisect 60°, 90° and 30°.
7. 75°: bisect the angle between 60° and 90°. 105°: bisect between 90° and 120°. 135°: bisect between 90° and 180°.

Constructing triangles: a triangle is fixed by SSS, SAS, ASA or RHS. Three angles alone (AAA) do not fix a size, so a unique triangle cannot be drawn from them.

Constructing quadrilaterals: a unique quadrilateral needs five independent measurements. Common sets: four sides and one diagonal; three sides and two included angles; two diagonals and three sides; two adjacent sides and three angles. A square needs one side, a rectangle two sides, a rhombus one side and one diagonal.

Classroom angle: use a geoboard and paper folding to explore polygons. Ask children to cut a quadrilateral and tear its four corners to join them: they meet around a point to make 360°. Let them measure diagonals of different parallelograms to see that they bisect each other.

Exam traps

The exterior angle sum is 360° for every polygon, not (n − 2) × 180°.
A rectangle has equal diagonals, but a rhombus has perpendicular diagonals; they are not the same property.
Diagonals of a kite do not bisect each other fully; only one is bisected.
A parallelogram is not necessarily a rectangle; the adjacent angles are supplementary, not right angles.
A trapezium has exactly one pair of parallel sides; a parallelogram has two pairs.
Each interior angle of a regular hexagon is 120°, and not 60° (that is its exterior angle).
Three angles alone cannot construct a unique triangle.
Number of diagonals is n(n − 3)/2, not n(n − 1)/2 (that counts all segments between vertices).

One-liners

The interior angle sum of an n-gon is (n − 2) × 180°.
Exterior angle sum of a convex polygon: 360°.
Each interior angle of a regular pentagon: 108°.
Each exterior angle of a regular octagon: 45°.
A hexagon has 9 diagonals.
A quadrilateral's angles add to 360°.
A rhombus's diagonals bisect each other at right angles.
Rectangle diagonals are equal.
Area of a rhombus is half the product of its diagonals.
Area of a trapezium is half of (sum of parallel sides) times height.
A 75° angle comes from bisecting the angle between 60° and 90°.
A unique quadrilateral needs five measurements.

Practice questions

What is the sum of the interior angles of a hexagon?
1. 900°
2. 540°
3. 720°
4. 1080°
Answer
C. 720°
(6 − 2) × 180° = 720°.
What is each interior angle of a regular pentagon?
1. 120°
2. 135°
3. 72°
4. 108°
Answer
D. 108°
(5 − 2) × 180/5 = 540/5 = 108°.
What is each exterior angle of a regular octagon?
1. 60°
2. 135°
3. 45°
4. 30°
Answer
C. 45°
360°/8 = 45°.
What is the sum of the exterior angles of any convex polygon, taking one at each vertex?
1. 360°
2. 720°
3. 180°
4. (n − 2) × 180°
Answer
A. 360°
The exterior angles of any convex polygon always add to 360°.
Each exterior angle of a regular polygon is 36°. How many sides does it have?
1. 8
2. 9
3. 12
4. 10
Answer
D. 10
n = 360/36 = 10.
Each interior angle of a regular polygon is 140°. How many sides does it have?
1. 8
2. 9
3. 12
4. 10
Answer
B. 9
Exterior angle = 40°, so n = 360/40 = 9.
How many diagonals does an octagon have?
1. 24
2. 28
3. 16
4. 20
Answer
D. 20
n(n − 3)/2 = 8 × 5/2 = 20.
The sum of the interior angles of a polygon is 1080°. How many sides does it have?
1. 8
2. 9
3. 6
4. 7
Answer
A. 8
(n − 2) × 180 = 1080 gives n − 2 = 6, so n = 8.
Each interior angle of a regular polygon is 150°. How many sides does it have?
1. 10
2. 18
3. 12
4. 15
Answer
C. 12
Exterior angle = 30°, so n = 360/30 = 12.
What is each interior angle of a regular hexagon?
1. 60°
2. 120°
3. 135°
4. 108°
Answer
B. 120°
(6 − 2) × 180/6 = 120°.
What is the sum of the four angles of any quadrilateral?
1. 180°
2. 360°
3. 270°
4. 540°
Answer
B. 360°
A diagonal divides it into two triangles, 2 × 180° = 360°.
Three angles of a quadrilateral are 80°, 95° and 105°. What is the fourth angle?
1. 80°
2. 90°
3. 100°
4. 70°
Answer
A. 80°
360 − (80 + 95 + 105) = 80°.
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. What is the largest angle?
1. 160°
2. 100°
3. 144°
4. 120°
Answer
C. 144°
One part = 360/10 = 36°; the largest is 4 × 36 = 144°.
One angle of a parallelogram is 70°. What is the adjacent angle?
1. 90°
2. 70°
3. 20°
4. 110°
Answer
D. 110°
Adjacent angles of a parallelogram are supplementary: 180 − 70 = 110°.
Which property holds for every parallelogram?
1. Its diagonals are equal
2. Its diagonals are perpendicular
3. Its diagonals bisect each other
4. All its angles are right angles
Answer
C. Its diagonals bisect each other
Equal diagonals belong to rectangles and perpendicular diagonals to rhombuses, but bisecting diagonals belong to all parallelograms.
Which quadrilateral always has two equal diagonals that bisect each other?
1. Trapezium
2. Kite
3. Rhombus
4. Rectangle
Answer
D. Rectangle
A rectangle has equal diagonals that bisect each other.
The diagonals of a rhombus are 6 cm and 8 cm. What is the length of its side?
1. 5 cm
2. 7 cm
3. 10 cm
4. 4.8 cm
Answer
A. 5 cm
Half diagonals 3 and 4 form a right triangle with hypotenuse 5.
What is the area of a rhombus with diagonals 10 cm and 24 cm?
1. 130 cm²
2. 120 cm²
3. 34 cm²
4. 240 cm²
Answer
B. 120 cm²
Area = ½ × 10 × 24 = 120.
What is the area of a trapezium with parallel sides 8 cm and 12 cm and height 5 cm?
1. 100 cm²
2. 50 cm²
3. 60 cm²
4. 40 cm²
Answer
B. 50 cm²
Area = ½ × (8 + 12) × 5 = 50.
What is the area of a parallelogram with base 15 cm and height 8 cm?
1. 120 cm²
2. 60 cm²
3. 240 cm²
4. 23 cm²
Answer
A. 120 cm²
Area = base × height = 15 × 8 = 120.
The diagonal of a square is 10 cm. What is its area?
1. 25 cm²
2. 100 cm²
3. 50 cm²
4. 75 cm²
Answer
C. 50 cm²
Area = (diagonal)²/2 = 100/2 = 50.
The sides of a rectangle are 12 cm and 5 cm. What is the length of its diagonal?
1. 11 cm
2. 17 cm
3. 12 cm
4. 13 cm
Answer
D. 13 cm
√(144 + 25) = √169 = 13.
A quadrilateral with exactly one pair of parallel sides is a
1. trapezium
2. kite
3. parallelogram
4. rhombus
Answer
A. trapezium
A trapezium has exactly one pair of parallel sides.
A quadrilateral with all four sides equal but whose angles need not be right angles is a
1. trapezium
2. rectangle
3. kite
4. rhombus
Answer
D. rhombus
A rhombus has equal sides; if all angles are 90° it is a square.
A quadrilateral with two pairs of adjacent sides equal is a
1. rectangle
2. trapezium
3. kite
4. parallelogram
Answer
C. kite
That is the definition of a kite.
A quadrilateral that is both a rectangle and a rhombus is a
1. kite
2. square
3. trapezium
4. parallelogram that is not a rectangle
Answer
B. square
It has all sides equal and all angles 90°.
Consider these statements. 1. Every rectangle is a parallelogram. 2. Every parallelogram is a rectangle. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A parallelogram has no right angle requirement, so 2 is wrong.
Consider these statements. 1. The diagonals of a rhombus are equal. 2. The diagonals of a rectangle are equal. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Rhombus diagonals are perpendicular but generally unequal; 1 is wrong.
Consider these statements. 1. Opposite angles of a parallelogram are equal. 2. The diagonals of a parallelogram bisect each other. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard properties of a parallelogram.
Consider these statements. 1. A trapezium has two pairs of parallel sides. 2. The diagonals of every trapezium bisect each other. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
A trapezium has exactly one pair of parallel sides, and its diagonals do not generally bisect each other.
Match the property with the figure: (1) Equal diagonals (2) Perpendicular diagonals (3) Exactly one pair of parallel sides (4) Two pairs of adjacent equal sides with (a) Trapezium (b) Kite (c) Rectangle (d) Rhombus. Choose the correct matching.
1. 1-c, 2-d, 3-a, 4-b
2. 1-a, 2-d, 3-c, 4-b
3. 1-c, 2-d, 3-b, 4-a
4. 1-d, 2-c, 3-a, 4-b
Answer
A. 1-c, 2-d, 3-a, 4-b
Rectangle: equal diagonals; rhombus: perpendicular diagonals; trapezium: one parallel pair; kite: adjacent equal sides.
Which angle can be constructed by bisecting a 60° angle?
1. 90°
2. 45°
3. 15°
4. 30°
Answer
D. 30°
Half of 60° is 30°.
A 75° angle is constructed by bisecting the angle between which two angles?
1. 45° and 90°
2. 60° and 120°
3. 60° and 90°
4. 30° and 90°
Answer
C. 60° and 90°
The angle midway between 60° and 90° is 75°.
How many independent measurements are needed to construct a unique quadrilateral?
1. 3
2. 4
3. 6
4. 5
Answer
D. 5
A quadrilateral needs five measurements, for example four sides and a diagonal.
Which of the following is NOT enough to construct a unique triangle?
1. Three sides
2. Three angles only
3. Two sides and the included angle
4. Two angles and the included side
Answer
B. Three angles only
Three angles fix only the shape, not the size.
In constructing the perpendicular bisector of a segment, the compass radius for the arcs must be
1. exactly half the length
2. less than half the length
3. more than half the length of the segment
4. equal to the whole length only
Answer
C. more than half the length of the segment
The arcs from both ends must cross, which needs a radius greater than half the length.
In a parallelogram the adjacent angles are in the ratio 2 : 3. What is the smaller angle?
1. 72°
2. 80°
3. 54°
4. 60°
Answer
A. 72°
Adjacent angles add up to 180°; 180/5 = 36, so 2 × 36 = 72°.
In parallelogram ABCD, angle A = (2x + 10)° and angle C = (3x − 10)°. What is x?
1. 30
2. 20
3. 40
4. 10
Answer
B. 20
Opposite angles are equal: 2x + 10 = 3x − 10 gives x = 20.
The sum of the interior angles of a polygon is 1440°. What is the number of its sides?
1. 9
2. 10
3. 8
4. 12
Answer
B. 10
n − 2 = 1440/180 = 8, so n = 10.
What is the sum of the interior angles of a polygon with 12 sides?
1. 1800°
2. 1620°
3. 2160°
4. 1980°
Answer
A. 1800°
(12 − 2) × 180° = 1800°.
A square has a perimeter of 40 cm. What is its area?
1. 400 cm²
2. 160 cm²
3. 100 cm²
4. 80 cm²
Answer
C. 100 cm²
Side = 10 cm, so area = 100 cm².
A rhombus has a perimeter of 52 cm and one diagonal is 24 cm. What is the length of the other diagonal?
1. 12 cm
2. 20 cm
3. 5 cm
4. 10 cm
Answer
D. 10 cm
Side = 13; half of one diagonal = 12; half of the other = √(169 − 144) = 5, so the other diagonal is 10 cm.
A rectangle has an area of 96 cm² and a length of 12 cm. What is its perimeter?
1. 48 cm
2. 36 cm
3. 20 cm
4. 40 cm
Answer
D. 40 cm
Breadth = 96/12 = 8; perimeter = 2 × (12 + 8) = 40 cm.
A parallelogram has an area of 72 cm² and a base of 9 cm. What is its height?
1. 7 cm
2. 8 cm
3. 81 cm
4. 9 cm
Answer
B. 8 cm
Height = area/base = 72/9 = 8 cm.
A polygon of n sides is divided into triangles by drawing all diagonals from one vertex. How many triangles are formed in an octagon?
1. 7
2. 8
3. 6
4. 5
Answer
C. 6
The number of triangles is n − 2 = 6, which is why the angle sum is 6 × 180°.

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1.The area of quadrilateral is 336 m 2 and the perpendiculars drawn to one diagonal from the opposite vertices are 16 m and 12 m long. Find the length of this diagonal
2.The base of a parallelogram is twice as long as its corrosponding height. If the area of the parallelogram is 144 cm², find the mentioned height
3.The given pie chart shows the percentage wise distribution of books in 5 different states A, B, C, D and E. The total number of books is 2500.
4.The Earth makes a complete rotation about its axis in 24 hours. Through what angle will it turn in 3 hours 30 minutes?
5.The ratio of measures of the angles of a triangle is given as 3: 4: 8. What is the measure of the smallest of the three angles?
6.The radii of two circles are 4 cm and 7 cm and the distance between their centres is 12 cm. Find the length of the direct common tangent to the circles.
7.The ratio in which a transverse common tangent drawn to two circles with radii 4 cm and 6 cm, respectively, divides the line joining their centres is:
8.The length of the tangent drawn to a circle of radius 12 cm from a point 18 cm away from the centre of the circle is:
9.PQ and RS are two parallel chords of a circle of length 10 cm and 24 cm, respectively, and lie on the same side of the centre O. If the distance between the chords is 7 cm, what is the radius (in cm) of the circle?
10.RA and RB are two tangents from a common point 'R' outside the circle with centre 'O', having a radius of 6 cm. If ARB = 70°, find the measure of the angle AOB.

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