Some of the most important geometry proofs are demonstrated here. I will try to provide you with solid and thorough examples.

Observations, measurements, and experimentation are not proofs. If a formula works for 1 million specific examples, this is still not a proof.

My approach is to explain everything at the same time I am writing the proof. If you are interested in proofs, you will find the proofs below a pleasure to read.

Proof of the area of a circle

Here, we prove that the area of a circle is pi × r^{2} by inscribing circles into polygons.

Proof of the area of a triangle

A crystal clear proof of the area of a triangle. We will know why it makes sense to multiply the base by the height and divide the result by two.

Proof of the area of a parallelogram

Here, we prove that the area of a parallelogram is base times height by turning a parallelogram into a rectangle.

Proof of the area of a trapezoid

Here, we prove that the area of a trapezoid is the average of the two parallel bases times the height by turning a trapezoid into a rectangle.

Here, we prove that the area of a circle is pi × r

Proof of the area of a triangle

A crystal clear proof of the area of a triangle. We will know why it makes sense to multiply the base by the height and divide the result by two.

Proof of the area of a parallelogram

Here, we prove that the area of a parallelogram is base times height by turning a parallelogram into a rectangle.

Proof of the area of a trapezoid

Here, we prove that the area of a trapezoid is the average of the two parallel bases times the height by turning a trapezoid into a rectangle.

Proof of the triangle midsegment theorem

A crystal clear proof of the triangle midsegment theorem.

Proof of the angle sum theorem

A crystal clear proof of the angle sum theorem.

Triangle inequality theorem proof

A crystal clear proof of the triangle inequality theorem.

Base angles theorem proof

A crystal clear proof of the base angles theorem.

Right isosceles triangle

Prove that the median of a right isosceles triangle is half the hypotenuse.

A crystal clear proof of the triangle midsegment theorem.

Proof of the angle sum theorem

A crystal clear proof of the angle sum theorem.

Triangle inequality theorem proof

A crystal clear proof of the triangle inequality theorem.

Base angles theorem proof

A crystal clear proof of the base angles theorem.

Right isosceles triangle

Prove that the median of a right isosceles triangle is half the hypotenuse.

Prove that opposite sides of a parallelogram are congruent

See how to prove that opposite sides of a parallelogram are equal.

Prove that the diagonals of a rectangle are congruent

See how to prove that the diagonals of a rectangle are equal.

Prove that the diagonals of a trapezoid are congruent

See how to prove that the diagonals of a trapezoid are equal.

Prove that the diagonals of a kite are perpendicular

See how to prove that the diagonals of a kite are perpendicular.

See how to prove that opposite sides of a parallelogram are equal.

Prove that the diagonals of a rectangle are congruent

See how to prove that the diagonals of a rectangle are equal.

Prove that the diagonals of a trapezoid are congruent

See how to prove that the diagonals of a trapezoid are equal.

Prove that the diagonals of a kite are perpendicular

See how to prove that the diagonals of a kite are perpendicular.