Introduction to Euclid's Geometry

They have equal areas

Step 1: Let the two triangles be T₁ and T₂, and the rectangle be R. Step 2: Given: Area of T₁ = Area of T₂ and Area of T₁ = Area of R. Step 3: We need to find the relationship between Area of T₂ and Area of R. Step 4: Using Euclid's first axiom: 'Things which are equal to the same thing are equal to one another.' Step 5: Since both T₂ and R have areas equal to T₁, they must have equal areas to each other. Therefore, Area of T₂ = Area of R.

Zero dimensions

Step 1: Euclid defined a point as 'that which has no part,' meaning it has no measurable extent. Step 2: In the progression from solids to points (solids → surfaces → lines → points), we lose one dimension at each step. Step 3: A solid has 3 dimensions (length, breadth, height), a surface has 2 dimensions (length, breadth), a line has 1 dimension (length). Step 4: Following this pattern, a point has zero dimensions - it has no length, breadth, or height. Step 5: A point represents position only, without any measurable size in any direction.

Angle A = Angle B

Step 1: We are given that angle A = 90° and angle B = 90°. Step 2: Both angles are right angles since they measure 90°. Step 3: Euclid's fourth postulate states: 'All right angles are equal to one another.' Step 4: Since both angle A and angle B are right angles, by the fourth postulate, they must be equal. Step 5: Therefore, angle A = angle B, confirming that all right angles have the same measure regardless of their position or orientation.

Extend it indefinitely on both sides

Step 1: Euclid's second postulate states: 'A terminated line can be produced indefinitely.' Step 2: In modern terms, a 'terminated line' is what we call a line segment. Step 3: The line segment PQ of 6 cm is a terminated line with two endpoints P and Q. Step 4: According to the second postulate, this line segment can be extended beyond both endpoints P and Q to form a complete line. Step 5: This means we can extend PQ indefinitely on both sides, making it an infinite line passing through points P and Q.