**Congruent figures** are figures that have the same shape and size. They are exact copies of each other. When two figures are congruent, you can superimpose them (place one exactly over the other) and they fit perfectly without any part sticking out or leaving gaps.
**Key Definition:** Two figures are congruent if and only if they can be made to coincide completely when one is placed over the other through suitable rotation or flipping.
Consider a symbol on a signboard that needs to be recreated on another board. Let us see what measurements are necessary:
**Example: The Bent Arm Symbol**
**What Additional Measurement is Needed?**
In a sports complex in Delhi, they need to install identical arrow symbols pointing to different directions. If they have:
1. **Tracing Method:** Trace the outline of one figure on tracing paper and superimpose it on the second figure. If they fit exactly, the figures are congruent.
2. **Rotation and Flipping:** Remember that figures can be rotated or flipped before checking congruence. A figure that appears different when rotated might still be congruent to another figure.
3. **Cutout Method:** Make a cardboard cutout of one figure and try to place it exactly over the second figure.
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Triangles are the simplest polygons and understanding their congruence is fundamental in geometry. In real life, carpenters, architects, and engineers use triangle congruence to create identical structural supports, frameworks, and decorative elements.
**Real-Life Example:** A contractor in Mumbai needs to create identical triangular trusses for a roof. Instead of measuring every part of the original truss, they can measure only certain key measurements and use congruence conditions to ensure all trusses are identical.
**Statement:** If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
**Abbreviation:** SSS = Side, Side, Side
**How It Works:**
When Meera and Rabia measured the triangular frame in school, they found:
With just these three measurements, they could recreate an identical triangle without measuring any angles.
**Why SSS Works:**
When you have all three side lengths, the shape of the triangle is completely determined. There is NO other way to arrange three lines of given lengths to form a different triangle shape.
**Worked Example 1: Verifying SSS Congruence**
Triangle ABC has sides: AB = 5 cm, BC = 7 cm, AC = 6 cm
Triangle PQR has sides: PQ = 5 cm, QR = 7 cm, PR = 6 cm
Are these triangles congruent?
**Solution:**
All three pairs of corresponding sides are equal.
**Therefore, ΔABC ≅ ΔPQR by SSS condition.**
**Construction Activity:**
If we draw a line segment AB of length 6 cm, then draw:
These circles intersect at two points (E and F above and below the line AB). This creates two triangles: ΔABE and ΔABF.
**Are these triangles congruent?**
Yes! ΔABE ≅ ΔABF because:
**Important:** Both triangles have the same sidelengths but are positioned differently (one above, one below). This shows that once we have the sidelengths, the shape is fixed, but the triangle can be in different positions or orientations.
When two triangles are congruent, specific parts correspond to each other:
**Definition:** **Corresponding vertices** are vertices that overlap when the triangles are superimposed.
**Corresponding sides** are sides that overlap.
**Corresponding angles** are angles that overlap.
**Example with Triangles ABC and XYZ:**
Consider ΔABC ≅ ΔXYZ
This notation tells us:
Therefore:
The **ORDER of vertices matters** when writing congruence.
**CORRECT way:** ΔABC ≅ ΔXYZ
**INCORRECT way:** ΔACB ≅ ΔXYZ
**ALTERNATIVE CORRECT way:** ΔACB ≅ ΔXZY
**Worked Example 2: Finding Corresponding Parts**
If ΔABC ≅ ΔPQR, identify the corresponding parts.
**Solution:**
**Problem:** In rectangle ABCD, diagonal BD divides it into two triangles. Are these triangles congruent?
**Given:** ABCD is a rectangle with:
**Solution:**
In rectangle ABCD:
Therefore, ΔABD and ΔCDB satisfy the **SSS condition:**
**So ΔABD ≅ ΔCDB**
**Finding Correct Correspondence:**
**Correct way to write:** ΔABD ≅ ΔCDB
---
**Statement:** If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
**Abbreviation:** SAS = Side, Angle, Side
**Important:** The angle MUST be between the two sides (included angle). This is crucial!
**Visual Understanding:**
When constructing a triangle with sides AB = 6 cm, AC = 5 cm, and ∠A = 30°:
Once you fix two sides and the angle between them, the third vertex position is completely determined. There is only ONE way to complete the triangle.
**Worked Example 3: SAS Congruence**
Triangle ABC has: AB = 6 cm, AC = 5 cm, ∠A = 30°
Triangle PQR has: PQ = 6 cm, PR = 5 cm, ∠P = 30°
Are these triangles congruent?
**Solution:**
All SAS conditions are satisfied.
**Therefore, ΔABC ≅ ΔPQR by SAS condition.**
**Why Students Often Make Mistakes:**
Students sometimes write ∠C (not between the two given sides) instead of ∠A (between AB and AC). This is WRONG because SAS requires the angle to be BETWEEN the two sides.
**Statement:** If two sides and a NON-included angle are equal, the triangles are NOT necessarily congruent.
**Important:** SSA does NOT guarantee congruence in general.
**Why SSA Fails:**
Consider two triangles with:
**Construction:**
1. Draw base PQ = 6 cm
2. Draw a ray from P making angle 30° with PQ
3. From Q, draw an arc of radius 4 cm
This arc will intersect the ray at TWO different points (R and S), creating:
Both triangles have the same side lengths and angle measurement, but they are NOT congruent to each other!
**Conclusion:** We can construct TWO non-congruent triangles with SSA measurements.
**Exception to SSA:** In RIGHT TRIANGLES, SSA becomes valid (called RHS condition, discussed later).
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**Statement:** If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
**Abbreviation:** ASA = Angle, Side, Angle
**Important:** The side MUST be between the two angles (included side).
**How It Works:**
When you know two angles, the third angle is automatically determined because:
Sum of angles in a triangle = 180°
Once all three angles are known, and one side is fixed, the scale and shape of the triangle are completely determined.
**Worked Example 4: ASA Congruence**
Triangle ABC has: ∠B = 50°, ∠C = 30°, BC = 5 cm
Triangle PQR has: ∠Q = 50°, ∠R = 30°, QR = 5 cm
Are these triangles congruent?
**Solution:**
All ASA conditions are satisfied.
**Therefore, ΔABC ≅ ΔPQR by ASA condition.**
**How to Construct with ASA:**
1. Draw the base BC = 5 cm
2. At B, draw a line making angle 50° with BC
3. At C, draw a line making angle 30° with BC
4. These two lines intersect at point A
5. Connect to form triangle ABC
The position of point A is uniquely determined, so only ONE triangle can be formed.
**Problem:** In the figure, point O is the midpoint of both AD and BC. Show that AB = CD.
**Given:**
**Solution:**
In ΔAOB and ΔDOC:
By **SAS condition:** ΔAOB ≅ ΔDOC
Therefore, the corresponding sides are equal:
**AB = DC** (corresponding sides of congruent triangles)
**How to Find Corresponding Vertices:**
So the congruence is written as: **ΔAOB ≅ ΔDOC**
---
**Statement:** If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
**Abbreviation:** AAS = Angle, Angle, Side
**Important:** The side is NOT between the two angles (non-included side).
**Why AAS Works (Key Insight):**
When you know two angles of a triangle, the third angle is automatically determined:
∠A + ∠B + ∠C = 180°
So if ∠A and ∠C are known, then:
∠B = 180° - ∠A - ∠C
Now we effectively have:
This becomes an **ASA condition** (two angles and the included side), which guarantees congruence.
**Worked Example 5: AAS Congruence**
Triangle ABC has: ∠A = 35°, ∠C = 75°, BC = 4 cm
Triangle XYZ has: ∠X = 35°, ∠Z = 75°, YZ = 4 cm
Are these triangles congruent?
**Solution:**
Step 1: Find the third angle in each triangle.
Step 2: Now we have two angles and the included side:
This satisfies the **ASA condition**.
**Therefore, ΔABC ≅ ΔXYZ by AAS condition.**
**How AAS is Different from ASA:**
| ASA | AAS |
|-----|-----|
| Two angles and the INCLUDED side | Two angles and a NON-included side |
| Side is BETWEEN the two angles | Side is NOT between the angles |
| Direct application | Need to find third angle first |
---
**Statement:** If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two triangles are congruent.
**Abbreviation:** RHS = Right angle, Hypotenuse, Side
**Important Terms:**
**When RHS is Used:**
RHS is a special case of SSA (which normally doesn't work). It works for right triangles because the right angle constrains the geometry in a special way.
**Worked Example 6: RHS Congruence**
Right triangle ABC has: ∠B = 90°, AC = 5 cm (hypotenuse), BC = 4 cm
Right triangle PQR has: ∠Q = 90°, PR = 5 cm (hypotenuse), QR = 4 cm
Are these triangles congruent?
**Solution:**
All RHS conditions are satisfied.
**Therefore, ΔABC ≅ ΔPQR by RHS condition.**
**Why RHS Works (Geometric Insight):**
When constructing a right triangle with:
1. A perpendicular line from Q (creating 90° angle)
2. A base QR = 4 cm
3. A line from R to meet the perpendicular at distance 5 cm
The point P where these meet is uniquely determined. There is only ONE such position for P (unlike in SSA where two positions are possible).
**Construction Steps:**
1. Draw base QR = 4 cm
2. At Q, draw a perpendicular line l (making 90°)
3. From R, draw an arc of radius 5 cm
4. This arc intersects line l at exactly ONE point P (on the side of the perpendicular where the triangle exists)
5. Connect P to R to form the triangle
**Real-Life Example from India:**
In a construction site in Bangalore, carpenters need to ensure right-angled corners for walls. They can verify that two corner angles are truly 90° by checking if:
If all these are equal, the corners are congruent.
| Condition | Requirement | Valid? |
|-----------|-------------|--------|
| **SSS** | All three sides equal | ✓ YES |
| **SAS** | Two sides and included angle equal | ✓ YES |
| **ASA** | Two angles and included side equal | ✓ YES |
| **AAS** | Two angles and non-included side equal | ✓ YES |
| **RHS** | Right angle, hypotenuse, and one side equal | ✓ YES |
| **SSA** | Two sides and non-included angle equal | ✗ NO (except in right triangles) |
| **AAA** | All three angles equal | ✗ NO (only similar, not congruent) |
---
**Definition:** An **isosceles triangle** is a triangle with at least two sides of equal length.
**Naming Convention:**
**Theorem:** If two sides of a triangle are equal, then the angles opposite to these equal sides are also equal.
**Statement:** In ΔABC, if AB = AC, then ∠B = ∠C
**Proof Using Congruence:**
Given: ΔABC with AB = AC, and ∠A = 80°
**Step 1:** Construct the altitude (perpendicular) from A to BC. Let the foot of the perpendicular be D on BC.
**Step 2:** Consider triangles ΔADB and ΔADC.
In these two triangles:
**Step 3:** By RHS condition: ΔADB ≅ ΔADC
**Step 4:** Since the triangles are congruent, corresponding angles are equal:
**∠B = ∠C** (corresponding angles of congruent triangles)
**This proves the theorem!**
**Worked Example 7: Finding Angles in Isosceles Triangle**
In ΔABC, AB = AC and ∠A = 80°. Find ∠B and ∠C.
**Solution:**
Step 1: Since AB = AC, the triangle is isosceles.
By the theorem above: ∠B = ∠C
Step 2: Use the angle sum property of triangles:
∠A + ∠B + ∠C = 180°
Step 3: Substitute known values:
80° + ∠B + ∠C = 180°
80° + ∠B + ∠B = 180° (since ∠B = ∠C)
80° + 2∠B = 180°
2∠B = 100°
∠B = 50°
**Therefore, ∠B = ∠C = 50°**
**Worked Example 8: Another Isosceles Triangle Problem**
In ΔPQR, if PQ = PR and ∠Q = 65°, find ∠P and ∠R.
**Solution:**
Step 1: Since PQ = PR, triangle is isosceles.
The angles opposite to equal sides are equal: ∠Q = ∠R = 65°
Step 2: Find ∠P using angle sum property:
∠P + ∠Q + ∠R = 180°
∠P + 65° + 65° = 180°
∠P + 130° = 180°
∠P = 50°
**Therefore, ∠P = 50°**
**Definition:** An **equilateral triangle** is a triangle with all three sides of equal length.
**Key Property:** All angles of an equilateral triangle are equal and measure 60° each.
**Proof:**
Given: ΔABC with AB = AC = BC
**Step 1:** Since AB = AC, by the isosceles triangle theorem:
∠B = ∠C ... (equation 1)
**Step 2:** Since AB = BC, by the isosceles triangle theorem:
∠A = ∠C ... (equation 2)
**Step 3:** From equations 1 and 2:
∠A = ∠B = ∠C
**Step 4:** By the angle sum property:
∠A + ∠B + ∠C = 180°
3∠A = 180° (since all angles are equal)
∠A = 60°
**Therefore, ∠A = ∠B = ∠C = 60°**
**This is true for ALL equilateral triangles!**
**Worked Example 9: Working with Equilateral Triangle**
In an equilateral triangle, each side measures 7 cm. Find all angles.
**Solution:**
In an equilateral triangle, all three sides are equal.
Therefore, by the property of equilateral triangles:
**All angles = 60°**
The answer is independent of side length.
**Example from India:**
The Mysore Palace in Karnataka features many triangular roof supports. Some are:
For isosceles support beams:
Congruent triangles appear in many famous structures around the world and in India:
1. **Louvre Museum, Paris:** The glass pyramid structure uses many congruent triangular sections arranged in a pattern.
2. **Egyptian Pyramid of Giza:** The four triangular faces are congruent isosceles triangles.
3. **Howrah Bridge (Rabindra Setu), Kolkata:** The triangular truss work consists of many congruent triangles for structural strength.
4. **Rangoli Designs (India):** Traditional Indian floor art often features congruent triangles arranged in symmetric patterns.
5. **Dome Designs:** Many religious structures in India (temples, mosques, churches) use triangular support systems made of congruent triangles.
6. **Market Stalls in India:** Tent structures often use triangular frames made from congruent triangles for stability.
---
**Congruent Figures:** Two figures are congruent if they have the same shape and size, and one can be superimposed exactly over the other through rotation or flipping.
**Notation:** The symbol ≅ is used to represent congruence.
1. **SSS (Side Side Side):** All three sides of one triangle equal the three sides of another triangle.
2. **SAS (Side Angle Side):** Two sides and the included angle of one triangle equal two sides and the included angle of another triangle.
3. **ASA (Angle Side Angle):** Two angles and the included side of one triangle equal two angles and the included side of another triangle.
4. **AAS (Angle Angle Side):** Two angles and a non-included side of one triangle equal two angles and the corresponding non-included side of another triangle.
5. **RHS (Right Hypotenuse Side):** The hypotenuse and one side of a right-angled triangle equal the hypotenuse and corresponding side of another right-angled triangle.
**Angle Sum Property:** The sum of all three angles in any triangle is 180°.
∠A + ∠B + ∠C = 180°
**Isosceles Triangle Theorem:** If two sides of a triangle are equal, then the angles opposite to these sides are also equal.
If AB = AC, then ∠B = ∠C
**Equilateral Triangle:** In an equilateral triangle (all sides equal), all angles are 60°.
If AB = BC = CA, then ∠A = ∠B = ∠C = 60°
When ΔABC ≅ ΔPQR:
All corresponding parts are equal.
---
1. **Confusing congruence with similarity:** Congruent figures have the same size AND shape. Similar figures have the same shape but different sizes.
2. **Using SSA incorrectly:** Don't assume two sides and a non-included angle guarantee congruence unless it's in a right triangle (RHS).
3. **Forgetting to check included angles:** In SAS, the angle MUST be between the two sides. If it's not, the condition doesn't work.
4. **Writing congruence statements carelessly:** Always check that the order of vertices matches the actual correspondence of parts.
5. **Assuming angles alone determine congruence:** Two triangles with the same three angles are SIMILAR but may not be CONGRUENT (they could have different sizes).
6. **Ignoring rotation and flipping:** Remember that figures can be rotated or flipped and still be congruent. The shape should match, even if orientation differs.
7. **Confusing corresponding angles with any equal angles:** Corresponding angles are specific angles that match when triangles are superimposed, not just any two equal angles.
8. **Not using the angle sum property in AAS:** Remember that if two angles are known, the third can always be found using the 180° sum.
---
1. List all three sides of each triangle
2. Compare each pair of sides
3. If all three pairs are equal, apply SSS
4. Write the congruence statement with correct vertex order
1. Identify two sides and check if they're equal
2. Identify the angle between these two sides
3. Check if this angle is equal in both triangles
4. Verify that the angle is BETWEEN the two sides
5. Apply SAS if conditions are met
1. Identify two angles and check if they're equal
2. For ASA: verify the side between them is also equal
3. For AAS: verify any other side (not between the angles) is equal
4. Apply appropriate condition
1. Check if both triangles have right angles
2. Identify and compare the hypotenuses
3. Identify and compare one other side
4. Apply RHS if all conditions match
1. Use angle sum property: ∠A + ∠B + ∠C = 180°
2. Use isosceles triangle property when applicable
3. Use congruence to find corresponding angles
4. Use equilateral triangle property when applicable
---
1. In a right-angled triangle, the hypotenuse is the longest side.
2. An isosceles triangle has at least two equal sides and two equal angles.
3. An equilateral triangle has all sides equal and all angles equal to 60°.
4. Vertically opposite angles are always equal.
5. When a line is perpendicular to another line, it makes a 90° angle.
6. The sum of angles in any triangle is always 180°.
7. Congruent figures can be superimposed to fit exactly one over the other.
8. Two triangles are congruent if and only if one of the five conditions (SSS, SAS, ASA, AAS, RHS) is satisfied.
9. Corresponding sides and angles of congruent triangles are equal.
10. A diagonal of a rectangle divides it into two congruent triangles.
Q1. Two figures are said to be congruent if they have:
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?
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?
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?
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?
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?
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?
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?
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?
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?
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.
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.
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.
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