Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 6

Geometry I: Basic Concepts, Lines, Angles and Triangles

What to remember

The angles of a triangle add up to 180°, and an exterior angle equals the sum of the two interior opposite angles; any two sides together are longer than the third.
When a transversal cuts two parallel lines, corresponding angles and alternate interior angles are equal, and co-interior angles add up to 180°.
Triangles are congruent by SSS, SAS, ASA, AAS or RHS (never by AAA or SSA), and the four centres (centroid, incentre, circumcentre, orthocentre) come from medians, angle bisectors, perpendicular bisectors and altitudes.

Basic concepts

A point shows a position and has no size. A line extends endlessly in both directions. A ray has one starting point and goes on endlessly in one direction. A line segment has two end points and a fixed length.
Through two distinct points there passes exactly one line. Through one point there pass infinitely many lines.
Points on the same line are collinear. Lines in the same plane that never meet are parallel. Lines that meet at 90° are perpendicular.
A plane is a flat surface that extends endlessly.
The perpendicular bisector of a segment is the line that cuts it into two equal parts at 90°. Every point on it is equally far from both ends of the segment.
An angle is formed by two rays with a common starting point (the vertex). It is measured in degrees with a protractor.

Types of angles

Angle	Measure
Acute	Less than 90°
Right	Exactly 90°
Obtuse	More than 90° and less than 180°
Straight	Exactly 180°
Reflex	More than 180° and less than 360°
Complete	360°

Pairs of angles

Complementary: two angles whose sum is 90°. The complement of 65° is 25°.
Supplementary: two angles whose sum is 180°. The supplement of 110° is 70°.
Adjacent angles: share a common vertex and a common arm, and lie on opposite sides of that arm. They are not always supplementary.
Linear pair: adjacent angles whose non-common arms form a straight line. Their sum is 180°. If two angles of a linear pair are 4x and 5x, then 9x = 180° and x = 20°.
Vertically opposite angles: formed when two lines cross. They are equal. If one is 55°, the opposite one is 55°, and each of the other two is 125°.

Quick equations

An angle equal to its own supplement is 90°.
An angle that is 20° more than its supplement: x = 180 − x + 20, so x = 100°.
An angle half of its complement: x = (90 − x)/2, so x = 30°.
Two angles x and 2x that are complementary: 3x = 90°, so x = 30°.

Parallel lines and a transversal

A transversal is a line that cuts two or more lines at different points. When it cuts two parallel lines, eight angles are formed.

Pair	Position	Relation (lines parallel)
Corresponding angles	Same side of the transversal, same position at each crossing	Equal
Alternate interior angles	Inside the lines, on opposite sides of the transversal	Equal
Alternate exterior angles	Outside the lines, on opposite sides	Equal
Co-interior (consecutive interior) angles	Inside the lines, same side	Supplementary (sum 180°)

The converse is also used: if a pair of corresponding or alternate angles is equal, or co-interior angles add up to 180°, the lines are parallel.

Worked examples

Alternate interior angle 70° means the other is also 70°.
Co-interior angle 65° means the other is 115°.
If corresponding angles are (2x + 10)° and (x + 50)° and the lines are parallel, then 2x + 10 = x + 50, so x = 40.
If co-interior angles are (3x + 10)° and (2x + 20)°, then 5x + 30 = 180, so x = 30.

Triangles

A triangle has three sides, three angles and three vertices.

By sides: scalene (all sides different), isosceles (two sides equal), equilateral (all sides equal).

By angles: acute (all angles less than 90°), right (one angle 90°), obtuse (one angle more than 90°).

Properties

Angle sum property: the three angles add up to 180°. If two angles are 50° and 60°, the third is 70°.
Exterior angle property: an exterior angle equals the sum of the two interior opposite angles. If those are 40° and 55°, the exterior angle is 95°. If an exterior angle is 120° and one interior opposite angle is 50°, the other is 70°.
Triangle inequality: the sum of any two sides is greater than the third. Sides 5, 6, 10 form a triangle; 3, 4, 8 do not, because 3 + 4 is less than 8.
A triangle cannot have two right angles or two obtuse angles.
In an isosceles triangle, the angles opposite equal sides are equal. If the angle between the equal sides is 100°, each base angle is 40°. If it is 40°, each base angle is 70°.
In an equilateral triangle each angle is 60°.
The side opposite the larger angle is the longer side.
If the angles are in the ratio 1 : 2 : 3, they are 30°, 60° and 90°, so the triangle is right-angled.
Mid-point theorem: the line joining the mid-points of two sides is parallel to the third side and is half of it.

Pythagoras theorem: in a right triangle, (hypotenuse)² = (base)² + (height)². Triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and their multiples (6, 8, 10). A ladder of 10 m that reaches 8 m up a wall has its foot 6 m from the wall. The converse is true: if a² + b² = c², the triangle is right-angled.

Area of a triangle: ½ × base × height. For an equilateral triangle of side a, area = (√3/4) a². Perimeter is the sum of the sides.

Congruence and similarity

Two triangles are congruent if all three pairs of sides and angles match.

Criterion	Meaning
SSS	Three sides equal
SAS	Two sides and the included angle equal
ASA	Two angles and the included side equal
AAS	Two angles and a non-included side equal
RHS	Right angle, hypotenuse and one side equal (right triangles only)

AAA proves only similarity (same shape, different size); SSA does not prove congruence. Similar triangles have equal angles and sides in the same ratio.

Special points of a triangle

Point	Formed by	Notes
Centroid	Medians (vertex to mid-point of opposite side)	Divides each median 2 : 1
Incentre	Angle bisectors	Centre of the inscribed circle
Circumcentre	Perpendicular bisectors of the sides	Centre of the circle through all vertices
Orthocentre	Altitudes (perpendicular from a vertex to the opposite side)	Right triangle: at the right-angle vertex

The centroid always lies inside the triangle. A triangle has three medians and three altitudes.

Classroom angle: use paper folding to show an angle bisector, a perpendicular bisector and a right angle (fold a paper twice to get 90°). Let children cut the three corners of a paper triangle and place them side by side to see a straight line, which shows the angle sum of 180°.

Exam traps

Adjacent angles need not be supplementary; only a linear pair is.
Co-interior angles add to 180°; they are not equal.
AAA is not a congruence criterion, and SSA is not either.
The orthocentre of a right-angled triangle lies at the vertex of the right angle, not inside.
The centroid divides a median 2 : 1 from the vertex, not 1 : 2 and not 1 : 1.
Perpendicular bisectors give the circumcentre; angle bisectors give the incentre.
Sides 3, 4, 8 do not form a triangle; the sum of the two smaller must exceed the largest.
An obtuse angle is between 90° and 180°; a reflex angle is above 180°.

One-liners

The complement of 65° is 25°.
The supplement of 110° is 70°.
Angle sum of a triangle: 180°.
An exterior angle equals the sum of the two interior opposite angles.
Hypotenuse of a right triangle with legs 6 and 8 is 10.
Sides 8, 15, 17 form a right triangle.
RHS means right angle, hypotenuse, side.
The centroid divides each median 2 : 1.
Circumcentre: perpendicular bisectors meet.
Incentre: angle bisectors meet.
Isosceles right triangle: acute angles are 45° each.
Vertically opposite angles are equal.

Practice questions

What is the complement of 65°?
1. 115°
2. 35°
3. 90°
4. 25°
Answer
D. 25°
Complementary angles add up to 90°, so 90° − 65° = 25°.
What is the supplement of 110°?
1. 20°
2. 250°
3. 70°
4. 80°
Answer
C. 70°
Supplementary angles add up to 180°, so 180° − 110° = 70°.
Two angles of a triangle are 50° and 60°. What is the third angle?
1. 130°
2. 70°
3. 60°
4. 80°
Answer
B. 70°
The angles of a triangle add to 180°; 180 − 110 = 70°.
Two angles of a linear pair are 3x and 2x. What is the larger angle?
1. 72°
2. 120°
3. 90°
4. 108°
Answer
D. 108°
5x = 180°, so x = 36° and the larger angle is 3 × 36 = 108°.
Two straight lines cross each other, and one angle formed is 55°. What is the angle vertically opposite to it?
1. 125°
2. 55°
3. 35°
4. 45°
Answer
B. 55°
Vertically opposite angles are equal.
A transversal cuts two parallel lines and one co-interior angle is 65°. What is the other co-interior angle?
1. 65°
2. 25°
3. 115°
4. 105°
Answer
C. 115°
Co-interior angles are supplementary: 180 − 65 = 115°.
The angles of a triangle are in the ratio 1 : 2 : 3. The triangle is
1. right-angled
2. obtuse-angled
3. acute-angled
4. equilateral
Answer
A. right-angled
The parts are 30°, 60° and 90°, so there is a right angle.
The two interior opposite angles of a triangle are 40° and 55°. What is the exterior angle?
1. 105°
2. 95°
3. 85°
4. 135°
Answer
B. 95°
Exterior angle = sum of the interior opposite angles = 95°.
Which set of lengths can form the sides of a triangle?
1. 5 cm, 6 cm, 10 cm
2. 3 cm, 4 cm, 8 cm
3. 2 cm, 3 cm, 5 cm
4. 4 cm, 4 cm, 9 cm
Answer
A. 5 cm, 6 cm, 10 cm
The sum of two sides must exceed the third; 5 + 6 > 10, while the others fail.
In an isosceles triangle, the angle between the two equal sides is 40°. What is each base angle?
1. 60°
2. 80°
3. 40°
4. 70°
Answer
D. 70°
(180 − 40)/2 = 70°.
The two legs of a right triangle are 6 cm and 8 cm. What is the hypotenuse?
1. 9 cm
2. 12 cm
3. 10 cm
4. 14 cm
Answer
C. 10 cm
√(36 + 64) = √100 = 10.
The hypotenuse of a right triangle is 13 cm and one leg is 5 cm. What is the other leg?
1. 11 cm
2. 12 cm
3. 8 cm
4. 10 cm
Answer
B. 12 cm
√(169 − 25) = √144 = 12.
Which of the following is NOT a criterion for congruence of triangles?
1. AAA
2. SAS
3. SSS
4. RHS
Answer
A. AAA
AAA gives only similar triangles; the others prove congruence.
The abbreviation RHS in congruence of triangles stands for
1. Right angle, Height, Sum
2. Radius, Hypotenuse, Side
3. Right angle, Hypotenuse, Side
4. Ratio, Height, Side
Answer
C. Right angle, Hypotenuse, Side
RHS applies only to right-angled triangles.
The medians of a triangle meet at a point that divides each median in the ratio
1. 1 : 2 from the vertex
2. 1 : 1
3. 3 : 1 from the vertex
4. 2 : 1 from the vertex
Answer
D. 2 : 1 from the vertex
The centroid is two-thirds of the way from each vertex.
Where does the orthocentre of a right-angled triangle lie?
1. Outside the triangle
2. At the vertex of the right angle
3. Inside, at the centroid
4. At the mid-point of the hypotenuse
Answer
B. At the vertex of the right angle
The two legs are altitudes and meet at the right-angle vertex.
The circumcentre of a triangle is the point where which lines meet?
1. Perpendicular bisectors of the sides
2. Angle bisectors
3. Altitudes
4. Medians
Answer
A. Perpendicular bisectors of the sides
The perpendicular bisectors meet at the point equidistant from all three vertices.
The incentre of a triangle is the point of intersection of its
1. perpendicular bisectors
2. altitudes
3. angle bisectors
4. medians
Answer
C. angle bisectors
It is the centre of the inscribed circle.
What is each angle of an equilateral triangle?
1. 30°
2. 90°
3. 45°
4. 60°
Answer
D. 60°
Three equal angles adding to 180° give 60° each.
How many straight lines pass through two distinct points?
1. Infinitely many
2. Exactly one
3. Two
4. None
Answer
B. Exactly one
Two points fix exactly one straight line.
An angle of 200° is called a
1. obtuse angle
2. straight angle
3. complete angle
4. reflex angle
Answer
D. reflex angle
A reflex angle lies between 180° and 360°.
An angle that measures between 90° and 180° is called
1. obtuse
2. reflex
3. right
4. acute
Answer
A. obtuse
Obtuse means more than 90° and less than 180°.
Two angles x and 2x are complementary. What is x?
1. 90°
2. 45°
3. 30°
4. 60°
Answer
C. 30°
3x = 90°, so x = 30°.
An angle is 20° more than its supplement. What is the angle?
1. 110°
2. 120°
3. 80°
4. 100°
Answer
D. 100°
x = (180 − x) + 20 gives 2x = 200, so x = 100°.
An angle is half of its complement. What is the angle?
1. 30°
2. 45°
3. 22.5°
4. 60°
Answer
A. 30°
x = (90 − x)/2 gives 3x = 90, so x = 30°.
The angles of a triangle are x, x + 10° and x + 20°. What is the smallest angle?
1. 60°
2. 70°
3. 50°
4. 40°
Answer
C. 50°
3x + 30 = 180 gives x = 50°.
What is the area of a triangle with base 12 cm and height 5 cm?
1. 60 cm²
2. 30 cm²
3. 17 cm²
4. 36 cm²
Answer
B. 30 cm²
Area = ½ × 12 × 5 = 30.
A line segment that joins a vertex of a triangle to the mid-point of the opposite side is called a
1. angle bisector
2. altitude
3. median
4. perpendicular bisector
Answer
C. median
This is the definition of a median.
Consider these statements. 1. Vertically opposite angles are equal. 2. Adjacent angles are always supplementary. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Only a linear pair is always supplementary, so 2 is wrong.
Consider these statements. 1. The sum of any two sides of a triangle is greater than the third side. 2. An exterior angle of a triangle equals the sum of the two interior opposite angles. 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.
Consider these statements. 1. A triangle can have two right angles. 2. A triangle can have two obtuse angles. 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
Two such angles would add to 180° or more, leaving nothing for the third angle.
Consider these statements. 1. Corresponding angles are equal for any two lines cut by a transversal. 2. Co-interior angles add up to 180° when the lines are parallel. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Corresponding angles are equal only when the lines are parallel.
Match: (1) Centroid (2) Orthocentre (3) Circumcentre (4) Incentre with (a) altitudes meet (b) medians meet (c) perpendicular bisectors meet (d) angle bisectors meet. Choose the correct matching.
1. 1-b, 2-a, 3-c, 4-d
2. 1-d, 2-a, 3-b, 4-c
3. 1-a, 2-b, 3-c, 4-d
4. 1-b, 2-c, 3-a, 4-d
Answer
A. 1-b, 2-a, 3-c, 4-d
Medians give the centroid; altitudes the orthocentre; perpendicular bisectors the circumcentre; angle bisectors the incentre.
Which set of numbers gives the sides of a right-angled triangle?
1. 7, 24, 26
2. 8, 15, 17
3. 9, 12, 16
4. 6, 8, 11
Answer
B. 8, 15, 17
8² + 15² = 64 + 225 = 289 = 17².
A 10 m ladder reaches a point 8 m above the ground on a vertical wall. How far is its foot from the wall?
1. 5 m
2. 4 m
3. 7 m
4. 6 m
Answer
D. 6 m
√(100 − 64) = √36 = 6.
Each acute angle of an isosceles right-angled triangle is
1. 45°
2. 30°
3. 60°
4. 90°
Answer
A. 45°
The two equal angles share 90°, so each is 45°.
In triangle ABC, AB = AC and angle A = 100°. What is angle B?
1. 50°
2. 40°
3. 30°
4. 80°
Answer
B. 40°
Base angles are equal: (180 − 100)/2 = 40°.
An exterior angle of a triangle is 120° and one interior opposite angle is 50°. What is the other interior opposite angle?
1. 80°
2. 130°
3. 60°
4. 70°
Answer
D. 70°
The exterior angle is the sum of the two: 120 − 50 = 70°.
A triangle has sides 5 cm, 5 cm and 8 cm. It is
1. scalene
2. right-angled
3. isosceles
4. equilateral
Answer
C. isosceles
Two sides are equal. 25 + 25 is not 64, so it is not right-angled.
A transversal cuts two parallel lines. Corresponding angles are (2x + 10)° and (x + 50)°. What is x?
1. 50
2. 40
3. 30
4. 60
Answer
B. 40
Corresponding angles are equal: 2x + 10 = x + 50 gives x = 40.
Co-interior angles between two parallel lines are (3x + 10)° and (2x + 20)°. What is x?
1. 35
2. 40
3. 30
4. 20
Answer
C. 30
5x + 30 = 180 gives x = 30.
A ray OC stands on a straight line AB at O. If angle AOC = 4x and angle COB = 5x, what is x?
1. 36°
2. 30°
3. 18°
4. 20°
Answer
D. 20°
They form a linear pair: 9x = 180°, so x = 20°.
A child folds a sheet of paper twice so that the second fold lies along the first. The corner formed is a
1. right angle
2. straight angle
3. acute angle
4. reflex angle
Answer
A. right angle
Folding the straight crease onto itself divides 180° into two equal angles of 90°.
Every point on the perpendicular bisector of a line segment is
1. on the segment itself
2. equidistant from the two ends of the segment
3. at a fixed distance from its mid-point
4. closer to the left end
Answer
B. equidistant from the two ends of the segment
This is the defining property of the perpendicular bisector.
The legs of a right triangle are 9 cm and 12 cm. What is the length of the altitude drawn to the hypotenuse?
1. 7.2 cm
2. 6 cm
3. 7.5 cm
4. 9 cm
Answer
A. 7.2 cm
Hypotenuse = 15; area = ½ × 9 × 12 = 54 = ½ × 15 × h, so h = 108/15 = 7.2.

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Chapter quiz · 10 questions

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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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