Geometric Twins – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash II, tailored for Class 7 in Mathematics.
This one-pager compiles key formulas and equations from the Geometric Twins chapter of Ganita Prakash II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
SSS Condition: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
The lengths of sides are denoted as a, b, c for triangle A and x, y, z for triangle B. If a = x, b = y, c = z, then ΔABC ≅ ΔXYZ. This is foundational for proving triangle congruence.
SAS Condition: 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.
Let sides a and b, with angle C in triangle ABC be equal to sides x and y, with angle Z in triangle XYZ. If a = x, b = y, and ∠C = ∠Z, then ΔABC ≅ ΔXYZ.
ASA Condition: 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.
Let angles A and B, with side c in triangle ABC be equal to angles X and Y, with side z in triangle XYZ. If ∠A = ∠X, ∠B = ∠Y, and c = z, then ΔABC ≅ ΔXYZ.
AAS Condition: If two angles and a non-included side of one triangle are equal to two angles and a corresponding non-included side of another triangle, then the triangles are congruent.
If ∠A = ∠X, ∠B = ∠Y, and c = z, then ΔABC ≅ ΔXYZ, where c is the side opposite to the non-included angle.
RHS Condition (Right Angle Hypotenuse Side): In right-angled triangles, if the hypotenuse and one side are equal, then the triangles are congruent.
For right triangles ABC and XYZ, if AC = XZ (hypotenuses) and AB = XY (one side), then ΔABC ≅ ΔXYZ.
Angles Opposite Equal Sides Theorem: Angles opposite to equal sides in a triangle are equal.
If side AB = AC, then ∠B = ∠C in triangle ABC. This theorem is key in establishing angle relationships.
Congruence Notation: If triangles ABC and XYZ are congruent, write ΔABC ≅ ΔXYZ.
This notation signifies that corresponding vertices (A to X, B to Y, C to Z) and angles are equal.
Equilateral Triangle: In an equilateral triangle, each angle measures 60°.
If all sides are equal, then all angles are equal, confirming that each angle = 180° / 3 = 60°.
Isosceles Triangle Angle Theorem: The angles opposite the equal sides are equal.
If AB = AC, then ∠B = ∠C in triangle ABC. This helps in solving triangle problems involving isosceles triangles.
Sum of Angles in Triangle: ∠A + ∠B + ∠C = 180°.
This fundamental triangle property is essential for angle calculations within geometric figures.
Equations
AB = AC (Isosceles Triangle)
For triangle ABC, if AB = AC, ∠B = ∠C. Use this relation to prove angle equalities.
∠A + ∠B + ∠C = 180° (Triangle Interior Angles)
This equation holds for any triangle and is crucial for angle calculations in triangle geometry.
a² + b² = c² (Pythagorean Theorem)
In right triangle ABC, if C is the right angle, then the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).
Area = (1/2) × base × height (Triangle Area)
This formula calculates the area of triangle ABC, given its base and the corresponding height.
Perimeter = a + b + c (Triangle Perimeter)
The perimeter of triangle ABC is equal to the sum of its three sides.
∠A = 180° - (∠B + ∠C) (Angle Calculation)
This equation helps determine an unknown angle in triangle ABC when the other two angles are known.
AB = XY (Congruence Condition)
If two sides AB and XY are equal, this is part of proving congruence between two triangles.
∠B = ∠Y (Corresponding Angles in Congruent Triangles)
This equation shows that corresponding angles are equal in congruent triangles.
Side a = Side x (Congruence Condition)
For two triangles to be congruent, corresponding sides must be equal.
Angle A = Angle X (Corresponding Angles in Congruent Triangles)
This equation reinforces that corresponding angles in congruent triangles maintain equality.
