This chapter explains how to check if two shapes or triangles are exactly the same in size and shape (congruent). It covers definitions, methods, important properties, and practical examples.
Introduction to Geometric Twins
- Sometimes, we need to recreate a symbol or figure on a signboard.
- Tracing the outline is hard for big shapes.
- Measurements like corner points and side lengths help make exact copies.
- Are just the arm lengths AB and BC enough to recreate the symbol? No, more info is needed.
- If AB = 4 cm, BC = 8 cm and angle ABC = 80°, these three together can help make an exact replica.
- Shapes that are exact copies with the same size and shape are called congruent.
- Congruent figures can be placed over each other and will fit exactly, even if you rotate or flip them.
Examples of Congruence
- Two given shapes can be checked for congruence using tracing or measurements.
- For the symbols: if both have the same arm lengths and same angle, the figures are congruent.
- Otherwise, there can be many figures with the same arm lengths but different angles.
Questions & Answers
- How do you check if two circles are congruent?
- Answer: If radii are equal, circles are congruent.
- How do you check if two rectangles are congruent?
- Answer: If the length and breadth both are equal.
CONGRUENCE OF TRIANGLES
- To make an identical triangle (like a cardboard cutout): Measure the three side lengths.
- Measuring angles is not required if side lengths are known and fixed.
- All triangles with same side lengths are congruent; called SSS condition (Side-Side-Side).
Congruence Condition:
If triangle ABC has sides AB, BC, CA equal to triangle XYZ's sides XY, YZ, ZX respectively:
If AB = XY, BC = YZ, CA = ZX
Then triangle ABC ≅ triangle XYZ
Repeating the Sides Does Not Make Triangles Congruent
- If only angles are measured (not sides), triangles may have same shape, but different size.
- So, equal angles do not ensure congruence.
SSS, SAS, SSA, ASA, AAS, RHS Conditions
SSS (Side-Side-Side)
- Three sides of a triangle must match another triangle’s three sides for congruence.
SAS (Side-Angle-Side)
- If two sides and the angle between them are equal, triangles are congruent.
SAS Congruence Condition:
If AB=XY, AC=XZ and angle between them (∠A=∠X)
Triangle ABC ≅ Triangle XYZ
SSA (Side-Side-Angle)
- Two sides and a non-included angle. This does not always guarantee congruence.
- Example: You can make two different triangles with same SSA values.
ASA (Angle-Side-Angle)
- If two angles and the included side are equal, triangles are congruent.
ASA Congruence Condition:
If ∠B = ∠Y, ∠C = ∠Z, and BC = YZ
Triangle ABC ≅ Triangle XYZ
AAS (Angle-Angle-Side)
- If two angles and any one corresponding side are equal, triangles are congruent.
RHS (Right-Hypotenuse-Side)
- For right-angled triangles: hypotenuse and one side are equal to corresponding parts in another triangle, the two triangles are congruent.
RHS Congruence Condition:
If triangle ABC and XYZ are right triangles and
Hypotenuse AC = XZ, Side BC = YZ
Triangle ABC ≅ Triangle XYZ
ISOSCELES AND EQUILATERAL TRIANGLES
Isosceles Triangle Properties
- Isosceles triangles have two equal sides.
- Angles opposite to equal sides are also equal.
If AB = AC in Triangle ABC
Then ∠B = ∠C
Equilateral Triangle Properties
- All sides equal, all angles equal.
- Each angle is 60∘60^\circ60∘.
If AB = BC = CA in Triangle ABC
Each angle = \( \frac{180^\circ}{3} = 60^\circ \)
QUESTIONS & FIGURE IT OUT TASKS
- How to check congruence for figures like triangle, rectangle, circle?
- For triangles: Use SSS, SAS, ASA, AAS, RHS as required.
- For rectangle/circle: Check sides/radii as needed.
- Express congruence between two triangles with given sides:
- Example:
- If triangles have AB = DE, BC = EF, CA = DF,
- Then triangle ABC ≅ triangle DEF (SSS condition applies).
- Example:
- Given OB = OC and OA = OD. Show AB is parallel to CD.
- Since OA=OD and OB=OC, alternate angles are equal and so AB || CD.
REAL LIFE EXAMPLES
- Congruent triangles can be seen in buildings (Louvre Museum), bridges (Howrah Bridge), rangoli designs, pyramids, and more.
ANGLES IN CIRCULAR AND SQUARE FIGURES
- Example task: Find missing angles using congruence rules.
- Line segments marked with single/double line are equal.
TABLE: Congruence Conditions for Triangles
| Condition | Required equal parts | Is Congruence Guaranteed? |
|---|---|---|
| SSS | 3 sides | Yes |
| SAS | 2 sides + included angle | Yes |
| ASA | 2 angles + included side | Yes |
| AAS | 2 angles + any one side | Yes |
| SSA | 2 sides + non-included angle | No |
| RHS | Right angle, hypotenuse, side | Yes |
IMPORTANT EQUATIONS :
Sum of Angles in Triangle:
∠A + ∠B + ∠C = 180°
Equilateral Triangle Angle:
Each angle = \( \frac{180^\circ}{3} = 60^\circ \)
SSS Congruence:
If AB = XY, BC = YZ, CA = ZX
Then triangle ABC ≅ triangle XYZ
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