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Geometric Twins

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of congruence, which means two figures have the same shape and size and can be superimposed exactly. It explains why measuring only side lengths or only angles is not always enough to recreate a figure, and establishes the conditions (SSS, SAS, ASA, AAS, RHS) that guarantee two triangles are congruent. The big idea is that congruence is a precise mathematical relationship that allows us to prove properties of shapes, such as equal angles in isosceles and equilateral triangles.
Key Concepts (14)
Congruent figures
Figures that are exact copies of each other or have the same shape and size. They can be superimposed exactly, one over the other, possibly after rotating or flipping.
e.g. Two identical ₹5 coins placed on top of each other fit exactly.
SSS (Side Side Side) condition
If two triangles have the same three side lengths, then they are congruent.
e.g. A triangle with sides 40 cm, 60 cm, 80 cm is congruent to any other triangle with those exact side lengths.
SAS (Side Angle Side) condition
If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
e.g. If AB = XY = 6 cm, AC = XZ = 5 cm, and ∠A = ∠X = 30°, then ΔABC ≅ ΔXYZ.
ASA (Angle Side Angle) condition
If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.
e.g. If BC = YZ = 5 cm, ∠B = ∠Y = 50°, and ∠C = ∠Z = 30°, then ΔABC ≅ ΔXYZ.
AAS (Angle Angle Side) condition
If two angles and a non-included side of one triangle are equal to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
e.g. If ∠A = ∠X = 35°, ∠C = ∠Z = 75°, and BC = YZ = 4 cm, then ΔABC ≅ ΔXYZ (since ∠B = ∠Y = 70° by angle sum property, making it ASA).
RHS (Right Hypotenuse Side) condition
In right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle, then the triangles are congruent.
e.g. If ∠B = ∠Y = 90°, AC = XZ = 5 cm (hypotenuse), and BC = YZ = 4 cm, then ΔABC ≅ ΔXYZ.
Hypotenuse
In a right-angled triangle, the side opposite the right angle is called the hypotenuse.
e.g. In a right triangle with legs 3 cm and 4 cm, the hypotenuse is 5 cm.
Corresponding vertices
Vertices that overlap when two congruent triangles are superimposed. They are written in the same order in the congruence statement.
e.g. If ΔABC ≅ ΔXYZ, then A corresponds to X, B to Y, and C to Z.
Corresponding sides
Sides that overlap when two congruent triangles are superimposed. They are equal in length.
e.g. In ΔABC ≅ ΔXYZ, AB corresponds to XY, BC to YZ, and AC to XZ.
Corresponding angles
Angles that overlap when two congruent triangles are superimposed. They are equal in measure.
e.g. In ΔABC ≅ ΔXYZ, ∠A corresponds to ∠X, ∠B to ∠Y, and ∠C to ∠Z.
SSA (Side Side Angle) condition
A condition where two sides and a non-included angle are equal. This does NOT guarantee congruence because two different non-congruent triangles can be formed.
e.g. If AB = 6 cm, AC = 4 cm, and ∠B = 30° (non-included), two triangles ΔPQR and ΔPQS can be drawn with these measurements that are not congruent.
Isosceles triangle
A triangle with at least two equal sides.
e.g. A triangle with AB = AC is isosceles.
Equilateral triangle
A triangle in which all three sides have equal lengths.
e.g. A triangle with sides 5 cm, 5 cm, 5 cm is equilateral.
Angles opposite to equal sides are equal
In a triangle, if two sides are equal, then the angles opposite those sides are also equal.
e.g. In ΔABC with AB = AC, we have ∠B = ∠C.
Formulas & Equations (2)
Sum of angles of a triangle = 180°
Reads as: The sum of the three interior angles of any triangle is 180 degrees.
Variables: ∠A, ∠B, ∠C are the three interior angles (measured in degrees).
Use when: To find a missing angle in a triangle when two angles are known, or to convert AAS condition to ASA.
⚠ Students forget to use this property when checking AAS congruence and directly compare the given non-included side without calculating the third angle.
Each angle in an equilateral triangle = 60°
Reads as: Each angle of an equilateral triangle measures 60 degrees.
Variables: All three angles are equal, and their sum is 180°, so each angle = 180° ÷ 3 = 60°.
Use when: Finding missing angles in equilateral triangles or proving properties of equilateral triangles.
⚠ Students assume all triangles with equal sides have angles 60° without checking if it is equilateral, or they forget that the sum is 180°.
How It Works
Checking if two figures are congruent by superimposition
  1. Step 1: Trace one figure on tracing paper or take a cutout.
  2. Step 2: Place the tracing or cutout over the other figure.
  3. Step 3: Rotate or flip the tracing if needed to make it fit exactly.
  4. Step 4: If the figures cover each other exactly, they are congruent.
💡 A figure can be rotated or flipped before superimposing; it does not have to be in the same orientation.
Constructing a triangle given three side lengths (SSS)
  1. Step 1: Draw a line segment equal to one of the given side lengths (e.g., 6 cm).
  2. Step 2: From one endpoint, draw an arc with radius equal to the second side length (e.g., 4 cm).
  3. Step 3: From the other endpoint, draw an arc with radius equal to the third side length (e.g., 8 cm).
  4. Step 4: The two arcs intersect at two points (one above, one below the segment). Join either intersection point to the endpoints to form the triangle.
💡 The two triangles formed (above and below the base) are congruent to each other, so all triangles with the same side lengths are congruent.
Constructing a triangle given two sides and a non-included angle (SSA)
  1. Step 1: Draw the base of length equal to the side adjacent to the given angle (e.g., 6 cm).
  2. Step 2: From one endpoint, draw a ray making the given angle (e.g., 30°) with the base.
  3. Step 3: From the other endpoint, draw an arc with radius equal to the other given side (e.g., 4 cm).
  4. Step 4: The arc may intersect the ray at two points, giving two different triangles that both satisfy the given measurements.
💡 SSA does NOT guarantee congruence because two non-congruent triangles can be formed.
Proving angles opposite to equal sides are equal in an isosceles triangle
  1. Step 1: Draw an isosceles triangle ABC with AB = AC.
  2. Step 2: Construct the altitude AD from vertex A to side BC (so ∠ADB = ∠ADC = 90°).
  3. Step 3: In triangles ADB and ADC: AB = AC (given), AD is common, and ∠ADB = ∠ADC = 90°.
  4. Step 4: By RHS condition, ΔADB ≅ ΔADC.
  5. Step 5: Therefore, ∠B = ∠C (corresponding parts of congruent triangles).
💡 The altitude is drawn to create two right triangles that satisfy the RHS condition.
Finding angles of an equilateral triangle
  1. Step 1: In an equilateral triangle, all sides are equal, so by the property 'angles opposite to equal sides are equal', all angles are equal.
  2. Step 2: Let each angle be x. Then x + x + x = 180° (angle sum property).
  3. Step 3: So 3x = 180°, hence x = 60°.
💡 All three angles are 60°, not just two.
Exam Traps — Don't Make These Mistakes
⚠ Confusing SSA with SAS
✗ Two sides and an angle are equal, so the triangles are congruent by SAS.
→ SAS requires the angle to be the included angle (between the two given sides). If the angle is not between the two sides, it is SSA, which does not guarantee congruence.
✓ Check if the given angle is between the two sides. If yes, use SAS. If the angle is opposite one of the sides, it is SSA and congruence is not guaranteed.
⚠ Assuming AAA guarantees congruence
✗ All three angles are equal, so the triangles are congruent.
→ Triangles with the same angles have the same shape but can be different sizes (similar, not congruent). For example, a small triangle with angles 30°, 70°, 80° and a large triangle with the same angles are not congruent.
✓ AAA only proves similarity, not congruence. To prove congruence, you need at least one side length to be equal.
⚠ Incorrectly writing the congruence statement
✗ ΔABC ≅ ΔXYZ without matching corresponding vertices correctly.
→ The order of vertices matters. For example, if ΔABC ≅ ΔXYZ, then A corresponds to X, B to Y, and C to Z. Writing ΔACB ≅ ΔXYZ would be incorrect because it implies A↔X, C↔Y, B↔Z, which may not match the equal sides.
✓ Write the vertices in the order that matches the correspondence. For example, if AB = XY, BC = YZ, and AC = XZ, then write ΔABC ≅ ΔXYZ.
⚠ Forgetting to check the included angle in SAS
✗ AB = XY, AC = XZ, and ∠B = ∠Y, so triangles are congruent by SAS.
→ ∠B is not the included angle between sides AB and AC. The included angle is ∠A (between AB and AC). The given angle must be between the two sides mentioned.
✓ Identify which angle is between the two given sides. If the angle is not between them, it is SSA, not SAS.
⚠ Misidentifying the hypotenuse in RHS
✗ In a right triangle, the side opposite the right angle is called the base.
→ The side opposite the right angle is specifically called the hypotenuse, not the base. The base can be any of the other two sides.
✓ In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs or arms.
Compare & Contrast
What is given
SSS condition
SAS condition
SSS requires all three side lengths to be equal. SAS requires two sides and the included angle (angle between those two sides) to be equal.
Position of the side relative to the angles
ASA condition
AAS condition
ASA requires the side to be between the two given angles (included side). AAS requires the side to be non-included (opposite one of the given angles). Both guarantee congruence, but the side is in a different position.
Position of the angle relative to the sides
SSA condition
SAS condition
In SAS, the angle is between the two given sides (included angle). In SSA, the angle is not between the two sides (non-included angle). SAS guarantees congruence; SSA does not.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 short answer (2-3 marks) on identifying congruence condition, 1 long answer (4-5 marks) on proving congruence and finding equal parts
Most asked: Identifying which congruence condition (SSS, SAS, ASA, AAS, RHS) applies to given triangles · Writing correct congruence statements with corresponding vertices · Proving angles or sides equal using congruence of triangles · Properties of isosceles and equilateral triangles
Don't skip: Given a figure with parallel lines or midpoints, prove two triangles are congruent and find equal sides/angles; Using congruence to prove angles opposite equal sides are equal in an isosceles triangle
One-Line Revision (read aloud before the exam)
  1. Congruent figures have the same shape and size and can be superimposed exactly after rotation or flipping.
  2. SSS condition: If all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.
  3. SAS condition: If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  4. ASA condition: If two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  5. AAS condition: If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
  6. RHS condition: In right triangles, if the hypotenuse and one side are equal, the triangles are congruent.
  7. SSA condition does NOT guarantee congruence because two different non-congruent triangles can be formed.
  8. The sum of the angles of a triangle is always 180°.
  9. In an isosceles triangle, angles opposite to equal sides are equal.
  10. In an equilateral triangle, all three angles are 60°.
  11. When writing a congruence statement like ΔABC ≅ ΔXYZ, the order of vertices shows the correspondence: A↔X, B↔Y, C↔Z.
  12. Corresponding sides and angles of congruent triangles are equal.
  13. The hypotenuse is the side opposite the right angle in a right-angled triangle.
  14. AAA (equal angles) does not guarantee congruence; it only guarantees similarity.

MCQs — 10 Questions with Answers

Q1. Two figures are said to be congruent if they have:

  • A. The same shape and size ✓
  • B. The same shape only
  • C. The same size only
  • D. The same colour

Answer: A — Congruent figures must have both the same shape and the same size so they can be superimposed exactly over each other.

Q2. Which of the following is the correct notation if triangle PQR is congruent to triangle MNO with P corresponding to M, Q to N, and R to O?

  • A. ∆PQR ≅ ∆MNO ✓
  • B. ∆PQR ≅ ∆NMO
  • C. ∆PQR ≅ ∆ONM
  • D. ∆PQR ≅ ∆MON

Answer: A — In congruence notation, the order of vertices must match the correspondence, so first vertex P matches first vertex M, and so on.

Q3. If two triangles have sides of length 3 cm, 4 cm, and 5 cm, what can you say about them?

  • A. They are always congruent ✓
  • B. They are always similar but not congruent
  • C. They must have the same angles
  • D. They are congruent only if the angles are also measured

Answer: A — By the SSS condition, if two triangles have all three sides equal, they must be congruent regardless of angle measurements.

Q4. Two sides of a triangle are 6 cm and 8 cm. Which additional measurement would guarantee that another triangle with these two measurements is congruent to the first?

  • A. The third side
  • B. The angle between the two sides
  • C. Either the third side or the included angle ✓
  • D. The angle opposite to one of the sides

Answer: C — Both SSS (measuring the third side) and SAS (measuring the included angle) guarantee congruence, so either works.

Q5. A signboard symbol has arms AB = 4 cm and BC = 8 cm. Why is measuring only these two lengths not enough to recreate an exact copy?

  • A. Because we also need to know the colour
  • B. Because different angles at B create different shapes ✓
  • C. Because we need to measure AC as well
  • D. Because the location of the symbol matters

Answer: B — With only two side lengths, many different triangular shapes can be formed depending on the angle between them, so the angle ∠ABC must also be measured.

Q6. In rectangle ABCD, the diagonal AC divides it into two triangles. Which congruence condition applies to these triangles?

  • A. SSS only
  • B. SAS only
  • C. Both SSS and SAS ✓
  • D. ASA only

Answer: C — Triangle ABC and triangle ACD have all three sides equal (SSS) and also have two equal sides with an equal included angle (SAS), so both conditions apply.

Q7. Can two triangles with angles of 50°, 60°, and 70° always be congruent?

  • A. Yes, because all angles are equal
  • B. No, they may have the same shape but different sizes ✓
  • C. Yes, if we also measure one side
  • D. No, because angles cannot determine congruence

Answer: B — Triangles with the same angles have the same shape but can have different sizes, so angle measurements alone do not guarantee congruence.

Q8. Meera has a triangular frame with sides 40 cm, 60 cm, and 80 cm. If she creates a smaller frame with sides 4 cm, 6 cm, and 8 cm, are these triangles congruent?

  • A. Yes, because the ratios are the same
  • B. No, because the sizes are different ✓
  • C. Yes, if the angles are also the same
  • D. No, because they do not have the same shape

Answer: B — Congruent figures must have the same size; these triangles are similar (same shape) but not congruent (different sizes).

Q9. In triangle PQR, if PQ = 5 cm, PR = 7 cm, and ∠P = 40°, and in triangle ABC, if AB = 5 cm, AC = 7 cm, and ∠A = 40°, which condition proves these triangles are congruent?

  • A. SSS condition
  • B. SAS condition ✓
  • C. ASA condition
  • D. SSA condition

Answer: B — We have two sides (PQ = AB and PR = AC) and the included angle (∠P = ∠A), which satisfies the SAS condition for congruence.

Q10. Two triangles have two sides measuring 6 cm and 4 cm, and an angle of 30°. If the angle is not between these two sides in both triangles, can the triangles be congruent?

  • A. Yes, always
  • B. No, never
  • C. Yes, but not always—they might or might not be congruent
  • D. No, because SSA does not guarantee congruence ✓

Answer: D — When two sides and a non-included angle are given (SSA condition), an arc can intersect a line at two different points, creating two non-congruent triangles, so congruence is not guaranteed.

Flashcards

What does congruent mean?

Two figures are congruent if they have exactly the same shape and size and can be superimposed perfectly over each other.

What is the SSS condition for triangle congruence?

If all three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.

What is the SAS condition for triangle congruence?

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

What is the ASA condition for triangle congruence?

If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Why does SSA not guarantee congruence?

With two sides and a non-included angle, an arc can intersect a line at two different points, creating two different non-congruent triangles.

What are corresponding vertices in congruent triangles?

Corresponding vertices are the vertices that overlap and fit exactly when two congruent triangles are superimposed on each other.

If triangle ABC is congruent to triangle XYZ, what does the notation ∆ABC ≅ ∆XYZ mean?

It means vertex A corresponds to X, vertex B corresponds to Y, vertex C corresponds to Z, and their corresponding sides and angles are equal.

Can two triangles with the same three angles always be congruent?

No, two triangles with the same three angles have the same shape but can have different sizes, so they are not necessarily congruent.

What measurement would you take to check if two circles are congruent?

Compare their radii; if the radii are equal, the circles are congruent.

In the rectangle ABCD, if we draw diagonal BD, which triangles formed are congruent?

Triangle ABD and triangle CDB are congruent because AB = CD, AD = CB, and BD is common to both.

Important Board Questions

Define congruent figures with an example. [1 mark]

State that congruent figures have the same shape and size and can be superimposed perfectly. Give any real example like two identical signboard symbols.

If triangle ABC has sides AB = 3 cm, BC = 4 cm, and CA = 5 cm, and triangle PQR has sides PQ = 3 cm, QR = 4 cm, and RP = 5 cm, are these triangles congruent? Give reasons. [2 marks]

Use the SSS condition—all three sides are equal, so the triangles are congruent. Write the notation correctly showing corresponding vertices.

In rectangle ABCD, draw the diagonal AC. Name the two triangles formed and explain using a congruence condition why they are congruent. Also write the congruence notation. [3 marks]

The triangles are ABC and ACD. Use SSS (AB = CD, AD = BC, AC is common) or SAS (AB = CD, ∠BAC = ∠DCA, AC is common). Write ∆ABC ≅ ∆CDA with proper vertex correspondence.

Meera and Rabia need to recreate a triangular frame from school. They measure the three sides as 40 cm, 60 cm, and 80 cm. Can they create an exact replica using only these measurements? Explain your answer. If yes, state which congruence condition applies. If they had measured only two sides and an angle (non-included), could they still be sure of an exact replica? Give reasons. [5 marks]

For three sides, use SSS condition—yes, they can create an exact replica. For two sides and non-included angle, use SSA condition—no, because two different triangles can be formed when an arc intersects a line at two points. Draw a rough diagram showing the two possible triangle positions.

Assertion-Reason Questions for Geometric Twins

Q1. Assertion (A):

Two figures are congruent if one can be superimposed exactly on the other after appropriate rotation or flipping.

Reason (R):

Congruent figures have the same shape and size, and their corresponding parts overlap perfectly.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why superimposition works—because the figures have identical shape and size. This is the defining characteristic of congruence.

Q2. Assertion (A):

If two triangles satisfy the SSS (Side-Side-Side) condition, they must be congruent.

Reason (R):

The SSS condition ensures that two triangles have the same area and perimeter.

Answer: C — A is true but R is false — The assertion is true—SSS guarantees congruence. However, the reason is false. Equal areas and perimeters do not guarantee congruence (two different triangles can have equal area and perimeter but different shapes). The true reason is that identical three side lengths uniquely determine a triangle's shape and size.

Q3. Assertion (A):

To check if two circles are congruent, comparing their radii is sufficient.

Reason (R):

Two circles are congruent if and only if they have the same radius because the radius uniquely determines a circle's size.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains that the radius is the defining measurement of a circle's size, so equal radii guarantee congruence. This is the correct explanation for the assertion.

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Case-Based Questions (CBQ) for Geometric Twins

Case Study:

A school received two identical triangular tiles for the entrance floor. The tiles are supposed to be congruent to avoid any visible mismatch. The school supervisor measures the first tile and records its sides as 15 cm, 20 cm, and 25 cm. For the second tile, the measurements are taken but recorded in a different order: 25 cm, 15 cm, and 20 cm. The supplier claims both tiles are identical and congruent. However, a student worker questions whether the different measurement order affects congruence.

Q1. Based on the SSS congruence condition, are these two tiles congruent?

  • A. No, because the measurements are recorded in different order
  • B. Yes, because they have the same three side lengths regardless of order ✓
  • C. No, because at least one side length is different
  • D. Cannot be determined without knowing the angles

Answer: B

Q2. What principle of congruence applies to this situation?

  • A. Position and order of measurements determine congruence
  • B. The actual three side lengths matter, not the order in which they are listed ✓
  • C. Angles must be measured separately to confirm congruence
  • D. The tiles cannot be congruent due to recording error

Answer: B

Q3. If the supervisor had measured the first tile's sides as 15 cm, 20 cm, 24 cm (instead of 25 cm), what would be true about the two tiles?

  • A. They would still be congruent because three measurements are given
  • B. They would be similar but not congruent
  • C. They would not be congruent because not all three sides match ✓
  • D. Congruence cannot be determined without angles

Answer: C

Q4. What is the minimum information needed to verify that two triangular tiles in a shipment are congruent?

  • A. Any two side lengths and the material type
  • B. The area and perimeter of each tile
  • C. All three side lengths (SSS condition) ✓
  • D. The color and position of the tiles

Answer: C

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