SSC-Standard Mathematics for Forest Posts · Chapter 5

Geometry I: Lines, Angles, Triangles and Similarity

What to remember

The three angles of a triangle add up to 180°. An exterior angle equals the sum of the two interior opposite angles. The sum of any two sides is greater than the third side.
Triangles are congruent by SSS, SAS, ASA, AAS or RHS. They are similar by AA, SSS (sides in proportion) or SAS (one equal angle between proportional sides). SSA and AAA do not prove congruence.
For similar triangles, the ratio of areas equals the square of the ratio of corresponding sides. The basic proportionality theorem (BPT), Pythagoras' theorem and the four triangle centres are the most tested topics.

Lines and angles

Two angles are complementary if they add up to 90°, and supplementary if they add up to 180°. The complement of 35° is 55°. The supplement of 110° is 70°.
Angles on a straight line add up to 180°. Angles around a point add up to 360°.
Vertically opposite angles (formed when two lines cross) are equal.
A linear pair of angles is supplementary.

When a transversal cuts two lines, the following hold only if the two lines are parallel.

Pair of angles	Relation (parallel lines)
Corresponding angles	Equal
Alternate interior angles	Equal
Alternate exterior angles	Equal
Co-interior (same-side interior) angles	Supplementary (sum 180°)

Worked example. If alternate interior angles are (x + 20)° and (2x − 30)°, then x + 20 = 2x − 30 and x = 50. Each angle is 70°.

Worked example. Two angles are supplementary and one is three times the other. Then 4x = 180, so the smaller angle is 45°.

Triangles: angles and sides

Angle-sum property. A + B + C = 180°.

If the angles are in the ratio 2 : 3 : 4, the angles are 40°, 60° and 80°.
In an isosceles triangle with vertex angle 40°, each base angle is (180° − 40°)/2 = 70°.
In an equilateral triangle, each angle is 60°.

Exterior angle property. An exterior angle equals the sum of the two interior opposite angles. If the exterior angle is 120° and one interior opposite angle is 50°, the other is 70°. The three exterior angles (one at each vertex) add up to 360°.

Side properties.

The sum of any two sides is greater than the third side. The difference of any two sides is less than the third. For sides 7 and 10, the third side x satisfies 3 < x < 17.
The largest side lies opposite the largest angle, and the smallest side lies opposite the smallest angle.
If angle A = 70°, the bisectors of angles B and C meet at the incentre I, and ∠BIC = 90° + A/2 = 125°.

Types by sides: equilateral (all equal), isosceles (two equal), scalene (none equal). By angles: acute, right, obtuse.

Congruence and similarity

Test	Congruent triangles	Similar triangles
Sides	SSS	Sides in proportion (SSS similarity)
Angles	ASA, AAS	AA (two angles equal; the third follows)
Mixed	SAS (angle between the two sides), RHS (right angle, hypotenuse, one side)	SAS similarity (one equal angle between proportional sides)
Not valid	SSA (except RHS) and AAA	AAA is the same as AA and works for similarity only

All congruent triangles are similar. Similar triangles need not be congruent. Any two equilateral triangles are similar, but any two isosceles triangles need not be.

Ratios in similar triangles.

Corresponding sides are in the same ratio. Perimeters are in the same ratio as the sides. Medians, altitudes and angle bisectors are also in the same ratio.
Ratio of areas = (ratio of sides)². If sides are in the ratio 3 : 5, the areas are in the ratio 9 : 25. If the areas are 36 and 81, the sides are in the ratio 6 : 9. A side of 8 in the first triangle matches 8 × 9/6 = 12 in the second.
Two triangles with perimeters 24 and 36 have a side ratio 2 : 3.

Basic proportionality theorem and midpoint theorem

BPT (Thales). If a line is drawn parallel to one side of a triangle and meets the other two sides, it divides them in the same ratio. In triangle ABC with DE ∥ BC, AD/DB = AE/EC. The converse is also true.

Worked example. AD = 4, DB = 6, AE = 6. Then 4/6 = 6/EC, so EC = 9.

Midpoint theorem. The line joining the midpoints of two sides is parallel to the third side and equal to half of it. If the third side is 12, the joining line is 6.

Angle bisector theorem. The bisector of an angle of a triangle divides the opposite side in the ratio of the other two sides. If AB = 6, AC = 9 and BC = 10, the bisector of A meets BC at D with BD/DC = 6/9. So BD = 4 and DC = 6.

Pythagoras' theorem and triangle centres

Pythagoras. In a right triangle, (hypotenuse)² = (base)² + (perpendicular)². The converse holds: if a² + b² = c², the triangle is right-angled.

Common triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25) and their multiples.

Legs 8 and 15 give hypotenuse 17.
A 13 m ladder with its foot 5 m from a wall reaches 12 m.
A person who walks 9 m east and 12 m north is 15 m from the start.
In a right triangle with legs 6 and 8, the altitude to the hypotenuse is (6 × 8)/10 = 4.8.

The four centres of a triangle.

Centre	Formed by	Special facts
Centroid	Medians	Always inside; divides each median 2 : 1 from the vertex
Incentre	Angle bisectors	Always inside; centre of the inscribed circle
Circumcentre	Perpendicular bisectors of the sides	Inside for acute, at the hypotenuse midpoint for right, outside for obtuse
Orthocentre	Altitudes	Inside for acute, at the right-angle vertex for right, outside for obtuse

For a median of length 15, the centroid is 10 from the vertex and 5 from the side.
For a right triangle 6, 8, 10, the circumradius is half the hypotenuse, 5. The inradius is (a + b − c)/2 = (6 + 8 − 10)/2 = 2. For the 3, 4, 5 triangle, the inradius is 1.
In an equilateral triangle of side a, the centroid, incentre, circumcentre and orthocentre coincide. The altitude is (√3/2)a, so for a = 10 it is 5√3.

Areas. A median divides a triangle into two triangles of equal area. Triangles on the same base and between the same parallel lines have equal areas.

Useful relations in right triangles

If an altitude is drawn from the right-angle vertex to the hypotenuse, it divides the triangle into two triangles. Each is similar to the whole triangle and to each other. From this, (altitude)² = (part 1) × (part 2) of the hypotenuse, and (leg)² = (hypotenuse) × (adjacent part).
In a right triangle with legs 6 and 8, the hypotenuse is 10. The leg 6 satisfies 6² = 10 × x, so the adjacent part of the hypotenuse is 3.6.
The median to the hypotenuse of a right triangle is half the hypotenuse.
For a triangle with longest side c: if c² < a² + b² the triangle is acute, if c² = a² + b² it is right-angled, and if c² > a² + b² it is obtuse.

Exam traps

Believing that SSA or AAA proves congruence. Only RHS is a special SSA case.
Saying similar triangles are always congruent. The converse is the true statement.
Using the ratio of sides for the ratio of areas. Areas follow the square of the side ratio.
Mixing up the BPT ratio. It is AD/DB = AE/EC (part to part), not AD/AB.
Placing the orthocentre or circumcentre inside an obtuse triangle. Both lie outside.
Forgetting that the centroid divides the median 2 : 1 from the vertex, not from the side.
Treating co-interior angles as equal. They add up to 180°.
Applying a² + b² = c² with the wrong side as hypotenuse. The hypotenuse is always the longest side.

One-liners

1. The angle sum of a triangle is 180°; the complement of θ is 90° − θ.
2. Vertically opposite angles are equal.
3. For parallel lines, corresponding and alternate angles are equal; co-interior angles add to 180°.
4. An exterior angle equals the sum of the two interior opposite angles.
5. SSS, SAS, ASA, AAS and RHS prove congruence.
6. AA is enough to prove two triangles similar.
7. Ratio of areas of similar triangles = (ratio of corresponding sides)².
8. BPT: a parallel to one side divides the other two sides in the same ratio.
9. The line joining the midpoints of two sides is parallel to the third and half its length.
10. The centroid divides a median in the ratio 2 : 1.
11. The circumcentre of a right triangle is the midpoint of its hypotenuse.
12. The orthocentre of a right triangle is at the vertex of the right angle.

Practice questions

The complement of 35° is
1. 145°
2. 35°
3. 55°
4. 65°
Answer
C. 55°
90° − 35° = 55°.
The supplement of 110° is
1. 70°
2. 250°
3. 80°
4. 20°
Answer
A. 70°
180° − 110° = 70°.
The sum of the angles of a triangle is
1. 90°
2. 270°
3. 360°
4. 180°
Answer
D. 180°
Angle sum property of a triangle.
An exterior angle of a triangle is 120° and one interior opposite angle is 50°. The other interior opposite angle is
1. 60°
2. 70°
3. 80°
4. 130°
Answer
B. 70°
The exterior angle equals the sum of the two interior opposite angles: 120 − 50 = 70.
The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle is
1. 100°
2. 90°
3. 80°
4. 60°
Answer
C. 80°
The angles are 40°, 60° and 80°.
Two parallel lines are cut by a transversal. If one co-interior angle is 65°, the other is
1. 115°
2. 25°
3. 65°
4. 105°
Answer
A. 115°
Co-interior angles add up to 180°.
Vertically opposite angles are always
1. supplementary
2. equal
3. unequal
4. complementary
Answer
B. equal
Two crossing lines form equal vertically opposite angles.
The hypotenuse of a right triangle with legs 8 and 15 is
1. 23
2. 13
3. 16
4. 17
Answer
D. 17
√(64 + 225) = √289 = 17.
Two sides of a triangle are 7 and 10. The largest whole number value of the third side is
1. 18
2. 16
3. 17
4. 15
Answer
B. 16
The third side must be less than 17, so the largest integer is 16.
Which of the following is NOT a test of congruence of triangles?
1. RHS
2. SSS
3. ASA
4. SSA
Answer
D. SSA
SSA does not fix a triangle (except in the right-angle case, RHS).
Two triangles with two angles equal are
1. similar
2. congruent
3. equilateral
4. equal in area
Answer
A. similar
AA is a similarity test.
In triangle ABC, DE ∥ BC with D on AB and E on AC. If AD = 4, DB = 6 and AE = 6, then EC is
1. 4
2. 9
3. 8
4. 10
Answer
B. 9
AD/DB = AE/EC gives 4/6 = 6/EC, so EC = 9.
The sides of two similar triangles are in the ratio 3 : 5. Their areas are in the ratio
1. 6 : 10
2. 3 : 5
3. 9 : 25
4. 27 : 125
Answer
C. 9 : 25
Areas are in the ratio of the squares of corresponding sides.
Two similar triangles have areas 36 cm² and 81 cm². A side of the smaller triangle is 8 cm. The corresponding side of the larger one is
1. 9 cm
2. 10 cm
3. 12 cm
4. 18 cm
Answer
C. 12 cm
Side ratio = 6 : 9, so 8 × 9/6 = 12 cm.
In a triangle, the line joining the midpoints of two sides is parallel to the third side. If the third side is 12 cm, the line is
1. 12 cm
2. 24 cm
3. 3 cm
4. 6 cm
Answer
D. 6 cm
The midpoint theorem gives half the third side.
A median of a triangle is 15 cm. The centroid is at what distance from the vertex?
1. 10 cm
2. 12 cm
3. 7.5 cm
4. 5 cm
Answer
A. 10 cm
The centroid divides the median 2 : 1 from the vertex.
The orthocentre of a right-angled triangle lies
1. outside the triangle
2. at the vertex of the right angle
3. at the centroid
4. at the midpoint of the hypotenuse
Answer
B. at the vertex of the right angle
The two legs are altitudes meeting at the right-angle vertex.
The circumradius of a right triangle with sides 6, 8 and 10 is
1. 4
2. 6
3. 10
4. 5
Answer
D. 5
The circumcentre is the midpoint of the hypotenuse; R = 10/2 = 5.
The inradius of a right triangle with sides 3, 4 and 5 is
1. 2
2. 1
3. 1.5
4. 1/2
Answer
B. 1
r = (a + b − c)/2 = (3 + 4 − 5)/2 = 1.
The incentre of a triangle is the meeting point of its
1. altitudes
2. perpendicular bisectors
3. angle bisectors
4. medians
Answer
C. angle bisectors
It is the centre of the inscribed circle.
The altitude of an equilateral triangle of side 10 cm is
1. 5√3 cm
2. 10√3 cm
3. 5 cm
4. 10/√3 cm
Answer
A. 5√3 cm
Altitude = (√3/2) × 10 = 5√3 cm.
The vertex angle of an isosceles triangle is 40°. Each base angle is
1. 40°
2. 80°
3. 70°
4. 140°
Answer
C. 70°
(180 − 40)/2 = 70°.
In triangle ABC, AB = 6, AC = 9 and BC = 10. The bisector of angle A meets BC at D. Then BD is
1. 5
2. 6
3. 4
4. 3.75
Answer
C. 4
BD/DC = 6/9, so BD = 4 and DC = 6.
A 13 m ladder has its foot 5 m from a wall. It reaches a height of
1. 18 m
2. 10 m
3. 8 m
4. 12 m
Answer
D. 12 m
√(169 − 25) = 12 m.
A person walks 9 m east and then 12 m north. The distance from the starting point is
1. 15 m
2. 3 m
3. 17 m
4. 21 m
Answer
A. 15 m
√(81 + 144) = 15 m.
The perimeters of two similar triangles are 24 cm and 36 cm. The ratio of their corresponding sides is
1. 1 : 12
2. 2 : 3
3. 3 : 2
4. 4 : 9
Answer
B. 2 : 3
Perimeters are in the same ratio as corresponding sides.
In a right triangle with legs 6 and 8, the altitude drawn to the hypotenuse is
1. 5
2. 4.8
3. 4
4. 6.4
Answer
B. 4.8
Altitude = (6 × 8)/10 = 4.8.
In triangle ABC, ∠A = 70°. The bisectors of ∠B and ∠C meet at I. Then ∠BIC is
1. 140°
2. 110°
3. 55°
4. 125°
Answer
D. 125°
∠BIC = 90° + A/2 = 125°.
The sum of the three exterior angles of a triangle (one at each vertex) is
1. 360°
2. 270°
3. 180°
4. 540°
Answer
A. 360°
The exterior angles of any convex polygon add up to 360°.
Two parallel lines are cut by a transversal. Alternate interior angles are (x + 20)° and (2x − 30)°. Each angle is
1. 70°
2. 110°
3. 50°
4. 60°
Answer
A. 70°
x + 20 = 2x − 30 gives x = 50, so each angle is 70°.
Two supplementary angles are such that one is three times the other. The smaller angle is
1. 135°
2. 60°
3. 30°
4. 45°
Answer
D. 45°
4x = 180 gives x = 45°.
The centroid of a triangle is the meeting point of its
1. angle bisectors
2. altitudes
3. medians
4. perpendicular bisectors of sides
Answer
C. medians
The three medians meet at the centroid.
Which of the statements is/are correct? 1. Congruent triangles are similar. 2. Similar triangles are congruent.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Similar triangles have the same shape but may differ in size.
Which of the statements is/are correct? 1. The circumcentre of an obtuse triangle lies inside it. 2. The orthocentre of an obtuse triangle lies outside it.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
For an obtuse triangle both the circumcentre and orthocentre lie outside.
Which of the statements is/are correct? 1. The sum of any two sides of a triangle is greater than the third side. 2. The difference of any two sides is less than the third side.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are forms of the triangle inequality.
Which of the statements is/are correct? 1. SSA is a valid test of congruence in every case. 2. AAA is a valid test of congruence.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
AAA only proves similarity, and SSA does not fix a triangle in general.
Which of the statements is/are correct? 1. Any two equilateral triangles are similar. 2. Any two isosceles triangles are similar.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Isosceles triangles can have different angles, so they need not be similar.
Which of the statements is/are correct? 1. The centroid always lies inside the triangle. 2. The centroid divides each median in the ratio 1 : 2 from the vertex.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
It divides the median 2 : 1 from the vertex, so statement 2 is false.
Which of the statements is/are correct? 1. For parallel lines, alternate interior angles are equal. 2. For parallel lines, co-interior angles are equal.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Co-interior angles are supplementary, not equal.
Which of the statements is/are correct? 1. A median divides a triangle into two triangles of equal area. 2. Triangles on the same base and between the same parallels have equal areas.
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 area theorems.
Match the centre with its construction: (a) centroid, (b) incentre, (c) circumcentre, (d) orthocentre with (i) angle bisectors, (ii) medians, (iii) altitudes, (iv) perpendicular bisectors of sides. The correct matching is
1. a-ii, b-iv, c-i, d-iii
2. a-i, b-ii, c-iii, d-iv
3. a-iii, b-i, c-ii, d-iv
4. a-ii, b-i, c-iv, d-iii
Answer
D. a-ii, b-i, c-iv, d-iii
Centroid: medians; incentre: angle bisectors; circumcentre: perpendicular bisectors; orthocentre: altitudes.
If a line drawn parallel to one side of a triangle meets the other two sides, it divides them in the same ratio. This is known as
1. the angle bisector theorem
2. the basic proportionality theorem
3. the midpoint theorem
4. Heron's theorem
Answer
B. the basic proportionality theorem
This statement is Thales' theorem or BPT.
In a right triangle, the square of the hypotenuse equals
1. the sum of the other two sides
2. the difference of the squares of the other two sides
3. the sum of the squares of the other two sides
4. twice the product of the other two sides
Answer
C. the sum of the squares of the other two sides
This is Pythagoras' theorem.
Which of the following can be the sides of a triangle?
1. 3, 4, 8
2. 5, 7, 10
3. 5, 7, 13
4. 2, 3, 6
Answer
B. 5, 7, 10
Only 5 + 7 > 10 satisfies the triangle inequality.
A triangle has sides 9 cm, 12 cm and 15 cm. It is
1. obtuse-angled
2. equilateral
3. isosceles
4. right-angled
Answer
D. right-angled
81 + 144 = 225, so the converse of Pythagoras shows a right angle.

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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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