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Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 6

Euclid, Congruence, Similarity, Pythagoras and the Mid-point Theorem

What to remember

  • Congruent figures are identical in shape and size; similar figures have the same shape and proportional sides. Congruence criteria are SSS, SAS, ASA, AAS and RHS; similarity criteria are AA, SSS and SAS.
  • The Basic Proportionality Theorem (Thales theorem) says a line parallel to one side of a triangle divides the other two sides in the same ratio.
  • In a right triangle, (hypotenuse)² = (base)² + (height)²; the mid-point theorem says the segment joining mid-points of two sides is parallel to the third side and half of it.

Euclid and the axiomatic method

Euclid of Alexandria wrote the *Elements*, a work of 13 books. He began with definitions (a point is that which has no part), then listed common notions (also called axioms) and postulates, and proved theorems step by step. This is the model of a logical proof still used in school geometry.

Some common notions: things equal to the same thing are equal to one another; if equals are added to equals, the wholes are equal; things which coincide with one another are equal; the whole is greater than the part.

Euclid gave five postulates. Postulate 1 says a straight line can be drawn from any point to any point. Postulate 2 says a line segment can be extended without end. Postulate 3 says a circle can be drawn with any centre and radius. Postulate 4 says all right angles are equal. Postulate 5 (the parallel postulate) says that if a line falling on two lines makes the interior angles on one side less than two right angles, the two lines meet on that side when extended. Attempts to prove the fifth postulate led later to non-Euclidean geometries.

A theorem is a statement proved using definitions, axioms and earlier theorems. A corollary follows directly from a theorem. The converse of a statement swaps its "if" and "then" parts; a converse need not be true.

Congruence of triangles

Two triangles are congruent (symbol ≅) when one can be placed exactly over the other. Their corresponding sides and angles are equal. The order of letters matters: ΔABC ≅ ΔPQR means A↔P, B↔Q, C↔R.

CriterionMeaning
SSSthree sides of one equal three sides of the other
SAStwo sides and the included angle equal
ASAtwo angles and the included side equal
AAStwo angles and any one non-included side equal
RHSright angle, hypotenuse and one side equal

SSA (two sides and a non-included angle) and AAA do not prove congruence in general. CPCT (Corresponding Parts of Congruent Triangles are equal) is used after the triangles are proved congruent.

Standard results: angles opposite equal sides of a triangle are equal, and the converse is true. In an isosceles triangle the median, altitude and angle bisector from the apex coincide. The sum of the angles of a triangle is 180°. An exterior angle equals the sum of the two interior opposite angles. In any triangle the side opposite the greater angle is longer.

Similarity of triangles

Triangles are similar (symbol ∼) if their corresponding angles are equal and their corresponding sides are proportional.

CriterionCondition
AA (AAA)two angles equal (the third is then equal)
SSSall three pairs of sides in the same ratio
SASone angle equal and the including sides proportional

Basic Proportionality Theorem: In ΔABC, if DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC. Its converse is also true: if a line divides two sides in the same ratio, it is parallel to the third side.

Area theorem: For similar triangles, area ratio = (ratio of corresponding sides)². The ratio of perimeters, medians, altitudes and angle bisectors equals the side ratio itself.

Worked example: In ΔABC, DE ∥ BC, AD = 3, DB = 6 and AE = 2. Then 3/6 = 2/EC, so EC = 4.

Worked example: Two similar triangles have areas 16 and 36. Side ratio = √(16/36) = 2/3.

Shadow problems use similar triangles: a pole of height h and shadow s gives h/s the same for every object at the same time.

In a right triangle ABC with the right angle at B, if BD ⊥ AC then triangles ABD, BCD and ABC are similar, giving BD² = AD × DC, AB² = AD × AC and BC² = DC × AC.

Pythagoras theorem

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². The converse says that if c² = a² + b² in a triangle, the angle opposite side c is 90°. If c² is greater than a² + b², the largest angle is obtuse; if less, the triangle is acute.

Common triples
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
20, 21, 29

Any multiple of a triple is also a triple, such as 6, 8, 10. A triple can be built from m > n as (m² − n², 2mn, m² + n²).

Useful results: diagonal of a square of side a is a√2. Diagonal of a rectangle is √(l² + b²). Altitude of an equilateral triangle of side a is (√3/2)a, and its area is (√3/4)a². In a rhombus, the diagonals bisect each other at right angles, so side² = (d₁/2)² + (d₂/2)². Distance between tops of two poles on level ground is √(distance² + (height difference)²).

Mid-point theorem

The segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. The converse: a line through the mid-point of one side, parallel to a second side, bisects the third side. Joining the three mid-points divides a triangle into four congruent triangles, and the new triangle has half the perimeter of the original and one-fourth of its area. The quadrilateral formed by joining mid-points of consecutive sides of any quadrilateral is a parallelogram.

More worked examples

  • 1. A ladder 13 m long reaches a window 12 m high. The foot is 13² − 12² = 25, so 5 m from the wall.
  • 2. Sides 9, 12, 15: 9² + 12² = 225 = 15², so the triangle is right-angled with the right angle opposite 15.
  • 3. Two poles of heights 6 m and 11 m stand 12 m apart. The tops are √(12² + 5²) = 13 m apart.
  • 4. In ΔABC the mid-points of AB and AC are D and E, and BC = 14 cm. Then DE = 7 cm and DE ∥ BC.
  • 5. Triangle ABC ∼ triangle PQR with AB = 6, PQ = 9, BC = 8. The scale factor is 3/2, so QR = 12.
  • 6. A right triangle with hypotenuse 20 cm and sides in the ratio 3 : 4 : 5 has legs 12 and 16, so its area is 96 cm².

Classroom angle

Geometry is best taught from concrete to abstract. For congruence, let learners cut paper triangles and superimpose them. For the mid-point theorem, let them draw several triangles, measure, and form a conjecture before the proof. Show counter-examples for SSA and AAA so that learners see why some tests fail. Use shadows and mirrors for similarity. Common misconceptions: writing vertices in the wrong order, treating "similar" as "equal", using Pythagoras for non-right triangles, and assuming a picture drawn to scale proves a result.

Exam traps

  • SSA is not a congruence test; RHS is a special case that works only for right triangles.
  • AAA gives similarity, not congruence.
  • Area ratio is the square of the side ratio, but the perimeter ratio is the side ratio itself.
  • The converse of Pythagoras is a separate theorem and must not be confused with the theorem.
  • In BPT the ratios are part to part (AD/DB), not part to whole, although AD/AB = AE/AC also holds.
  • The mid-point segment is half the third side, not equal to it.
  • Congruent triangles are similar, but similar triangles need not be congruent.
  • A common notion (axiom) is a general truth; a postulate is specific to geometry.

One-liners

  • 1. Euclid's *Elements* has 13 books.
  • 2. Euclid gave five postulates.
  • 3. The fifth postulate is the parallel postulate.
  • 4. The congruence symbol is ≅ and the similarity symbol is ∼.
  • 5. CPCT is valid only after congruence is proved.
  • 6. Triangle angle sum is 180°.
  • 7. BPT is also called Thales theorem.
  • 8. Area ratio of similar triangles equals the square of the side ratio.
  • 9. 5, 12, 13 is a Pythagorean triple.
  • 10. Diagonal of a square of side a is a√2.
  • 11. Altitude of an equilateral triangle of side a is (√3/2)a.
  • 12. The mid-point segment is parallel to the third side and half of it.

Practice questions

  1. Euclid's famous work on geometry is called

    1. Arithmetica
    2. Lilavati
    3. Almagest
    4. Elements
    Answer

    D. Elements

    Euclid compiled his geometry in the Elements.

  2. How many postulates did Euclid list in the Elements?

    1. 5
    2. 3
    3. 7
    4. 10
    Answer

    A. 5

    Euclid gave five postulates and several common notions (axioms).

  3. 'If equals are added to equals, the wholes are equal' is an example of

    1. a construction
    2. a theorem
    3. a definition
    4. a common notion (axiom)
    Answer

    D. a common notion (axiom)

    This is one of Euclid's common notions.

  4. Which of the following is NOT a valid criterion for congruence of triangles?

    1. ASA
    2. SAS
    3. RHS
    4. SSA (two sides and a non-included angle)
    Answer

    D. SSA (two sides and a non-included angle)

    SSA does not fix a triangle; only the RHS special case works for right triangles.

  5. The RHS criterion for congruence is used only for

    1. obtuse-angled triangles
    2. right-angled triangles
    3. isosceles triangles
    4. equilateral triangles
    Answer

    B. right-angled triangles

    R = right angle, H = hypotenuse, S = one side.

  6. In the abbreviation CPCT, the letter 'T' at the end stands for

    1. transversals
    2. theorems
    3. triangles
    4. tangents
    Answer

    C. triangles

    CPCT = Corresponding Parts of Congruent Triangles are equal.

  7. In a triangle, the angles opposite to equal sides are

    1. complementary
    2. supplementary
    3. equal
    4. always right angles
    Answer

    C. equal

    This is the isosceles triangle theorem.

  8. Two triangles are similar if two angles of one are equal to two angles of the other. This is the

    1. SAS congruence
    2. AA criterion
    3. RHS criterion
    4. ASA congruence
    Answer

    B. AA criterion

    AA (same as AAA) similarity needs only two angles because the third follows.

  9. The Basic Proportionality Theorem is also known as

    1. Ptolemy theorem
    2. Apollonius theorem
    3. Brahmagupta theorem
    4. Thales theorem
    Answer

    D. Thales theorem

    BPT is attributed to Thales.

  10. Which of the following is a Pythagorean triple?

    1. 5, 10, 13
    2. 9, 12, 16
    3. 6, 8, 11
    4. 8, 15, 17
    Answer

    D. 8, 15, 17

    8²+15²=64+225=289=17².

  11. The line segment joining the mid-points of two sides of a triangle is

    1. perpendicular to the third side and half of it
    2. parallel to the third side and half of it
    3. parallel to the third side and equal to it
    4. perpendicular to the third side and equal to it
    Answer

    B. parallel to the third side and half of it

    This is the mid-point theorem.

  12. The ratio of the areas of two similar triangles is equal to

    1. the ratio of their perimeters
    2. the ratio of the cubes of corresponding sides
    3. the ratio of the squares of their corresponding sides
    4. the ratio of their corresponding sides
    Answer

    C. the ratio of the squares of their corresponding sides

    Area scales as the square of the linear ratio.

  13. The hypotenuse of a right triangle with legs 6 cm and 8 cm is

    1. 10 cm
    2. 9 cm
    3. 12 cm
    4. 14 cm
    Answer

    A. 10 cm

    6²+8²=36+64=100, so the hypotenuse is 10.

  14. The diagonal of a square of side 5 cm is

    1. 10 cm
    2. 5 cm
    3. 5√3 cm
    4. 5√2 cm
    Answer

    D. 5√2 cm

    d² = 5²+5² = 50, so d = 5√2.

  15. The altitude of an equilateral triangle of side 10 cm is

    1. 5√3 cm
    2. 5 cm
    3. 10√3 cm
    4. 5√2 cm
    Answer

    A. 5√3 cm

    h² = 10² − 5² = 75, so h = 5√3.

  16. Which statement about congruent and similar figures is correct?

    1. Congruent triangles can have different angles
    2. Similar triangles always have equal areas
    3. Congruent triangles are always similar
    4. Similar triangles are always congruent
    Answer

    C. Congruent triangles are always similar

    Congruent means identical in shape and size; similar means same shape only.

  17. The symbol '∼' between two triangles means they are

    1. congruent
    2. equal in area only
    3. similar
    4. parallel
    Answer

    C. similar

    ≅ is for congruence; ∼ is for similarity.

  18. In a right triangle, the longest side is always

    1. the median
    2. the hypotenuse
    3. the side adjacent to the smallest angle
    4. the altitude
    Answer

    B. the hypotenuse

    The hypotenuse lies opposite the right angle, the largest angle.

  19. In triangle ABC, DE ∥ BC with D on AB and E on AC. If AD = 3 cm, DB = 6 cm and AE = 2 cm, then EC is

    1. 1 cm
    2. 3 cm
    3. 6 cm
    4. 4 cm
    Answer

    D. 4 cm

    By BPT, AD/DB = AE/EC → 3/6 = 2/EC → EC = 4.

  20. Two similar triangles have areas in the ratio 4 : 9. If the smaller has area 16 cm², the larger has area

    1. 36 cm²
    2. 72 cm²
    3. 24 cm²
    4. 64 cm²
    Answer

    A. 36 cm²

    16 × 9/4 = 36.

  21. A 13 m ladder leans against a wall with its foot 5 m from the wall. The height reached on the wall is

    1. 8 m
    2. 12 m
    3. 11 m
    4. 10 m
    Answer

    B. 12 m

    13² − 5² = 144, so height = 12.

  22. A triangle has sides 9 cm, 12 cm and 15 cm. It is

    1. equilateral
    2. right-angled
    3. acute-angled
    4. obtuse-angled
    Answer

    B. right-angled

    9²+12²=81+144=225=15², so by the converse of Pythagoras it is right-angled.

  23. D and E are the mid-points of AB and AC of triangle ABC. If BC = 14 cm, then DE is

    1. 7 cm
    2. 14 cm
    3. 28 cm
    4. 3.5 cm
    Answer

    A. 7 cm

    By the mid-point theorem DE = BC/2.

  24. The perimeter of a triangle is 30 cm. The triangle formed by joining the mid-points of its sides has perimeter

    1. 10 cm
    2. 60 cm
    3. 15 cm
    4. 30 cm
    Answer

    C. 15 cm

    Each new side is half the corresponding original side.

  25. An isosceles triangle has equal sides 10 cm and base 12 cm. Its altitude to the base is

    1. 6 cm
    2. 7 cm
    3. 9 cm
    4. 8 cm
    Answer

    D. 8 cm

    The altitude bisects the base: h² = 10² − 6² = 64.

  26. The diagonals of a rhombus are 10 cm and 24 cm. Its side is

    1. 14 cm
    2. 13 cm
    3. 17 cm
    4. 12 cm
    Answer

    B. 13 cm

    Diagonals bisect at right angles: side² = 5²+12² = 169.

  27. Triangle ABC ∼ triangle PQR with AB = 6, PQ = 9 and BC = 8. Then QR is

    1. 12
    2. 10
    3. 16
    4. 13.5
    Answer

    A. 12

    Scale factor 9/6 = 3/2, so QR = 8 × 3/2 = 12.

  28. Two poles of heights 6 m and 11 m stand on level ground 12 m apart. The distance between their tops is

    1. 15 m
    2. 17 m
    3. 13 m
    4. 12 m
    Answer

    C. 13 m

    Height difference 5 m: √(12²+5²) = 13.

  29. A 6 m pole casts a shadow 4 m long. At the same time a tower casts a shadow 20 m long. The tower's height is

    1. 24 m
    2. 26 m
    3. 36 m
    4. 30 m
    Answer

    D. 30 m

    Triangles are similar: 6/4 = h/20 → h = 30.

  30. In right triangle ABC (right angle at B), BD is perpendicular to AC. If AD = 4 cm and DC = 9 cm, then BD is

    1. 6.5 cm
    2. 6 cm
    3. 13 cm
    4. 5 cm
    Answer

    B. 6 cm

    BD² = AD × DC = 36, so BD = 6.

  31. A right triangle has legs 9 cm and 12 cm. The area of the square drawn on its hypotenuse is

    1. 225 cm²
    2. 21 cm²
    3. 180 cm²
    4. 144 cm²
    Answer

    A. 225 cm²

    Hypotenuse = 15, so square area = 225 (= 81 + 144).

  32. A right triangle has hypotenuse 20 cm and its sides are in the ratio 3 : 4 : 5. Its area is

    1. 120 cm²
    2. 150 cm²
    3. 96 cm²
    4. 60 cm²
    Answer

    C. 96 cm²

    Legs are 12 and 16; area = ½ × 12 × 16 = 96.

  33. Two similar triangles have corresponding sides in the ratio 2 : 3. The ratio of their corresponding medians is

    1. 8 : 27
    2. 4 : 9
    3. 3 : 2
    4. 2 : 3
    Answer

    D. 2 : 3

    All corresponding linear elements are in the same ratio as the sides.

  34. Consider the statements: 1. SAS congruence needs two sides and the angle included between them. 2. AAA is a criterion for congruence of triangles. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    AAA only gives similarity, not congruence.

  35. Consider the statements: 1. All circles are similar. 2. All congruent figures are similar. 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

    Same shape holds for all circles and for any congruent pair.

  36. Consider the statements: 1. If the square of one side of a triangle equals the sum of the squares of the other two, the angle opposite the first side is a right angle. 2. The Pythagoras theorem is true for every triangle. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Statement 1 is the converse; Pythagoras holds only for right triangles.

  37. Consider the statements: 1. The segment joining mid-points of two sides is parallel to the third side. 2. The segment joining mid-points of two sides is double the third side. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    It is half, not double, the third side.

  38. Consider the statements about similar triangles: 1. Corresponding angles are equal. 2. Corresponding sides are in the same ratio. 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

    These are the two defining conditions of similarity.

  39. Consider the statements: 1. SSA is a valid test for congruence of any two triangles. 2. ASA is a valid test for congruence. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    SSA can give two different triangles; ASA is valid.

  40. Consider the statements: 1. Euclid's fifth postulate is connected with parallel lines. 2. Euclid's Elements is divided into 13 books. 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 facts about Euclid's work.

  41. Consider the statements: 1. The ratio of areas of similar triangles equals the ratio of their corresponding sides. 2. Two triangles with equal angles are always congruent. 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

    Area ratio is the square of the side ratio; equal angles give similarity only.

  42. Which pair is correctly matched?

    1. SAS similarity : two angles and a side equal
    2. Pythagoras theorem : holds for all triangles
    3. Mid-point theorem : segment is equal to the third side
    4. BPT : a line parallel to one side divides the other two sides in the same ratio
    Answer

    D. BPT : a line parallel to one side divides the other two sides in the same ratio

    Only the BPT description is correct.

  43. A Class IX teacher wants learners to discover the mid-point theorem. The most effective first activity is to

    1. give a long algebraic proof first
    2. let learners draw triangles, mark mid-points and measure the joining segment and the base
    3. state the theorem and ask them to memorise it
    4. ask them to copy the proof from the board
    Answer

    B. let learners draw triangles, mark mid-points and measure the joining segment and the base

    Hands-on measurement builds a conjecture before formal proof.

  44. A learner claims two triangles are congruent because two sides and a non-included angle match. The best teacher response is to

    1. skip the topic as too hard
    2. show a counter-example by drawing two different triangles with such data
    3. say it is wrong without explanation
    4. accept it because it is nearly right
    Answer

    B. show a counter-example by drawing two different triangles with such data

    A counter-example makes the limits of SSA visible.

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