Proofs of Basic Theorems

Grade 12 Math - Proofs of Basic Theorems

Lesson Objectives

State and explain fundamental geometric theorems clearly.
Apply logical reasoning to prove geometric theorems step by step.
Use geometric properties to establish formal proofs of triangles and angles.
Solve problems involving parallel lines, angles, and triangles using theorems.
Recognize and use previously proven theorems in more complex geometric proofs.

Lesson Introduction

Geometry is all around us—from the tiles on your floor to the pyramids of Egypt. Have you ever wondered how ancient architects were sure that angles and shapes would work together perfectly? They used geometry! In this lesson, we’ll uncover the power of geometric theorems and learn how to prove them using logic and previously known facts.

Core Lesson Content

Let’s explore and prove several fundamental geometric theorems. These include theorems about angles in a triangle, parallel lines, and properties of isosceles and equilateral triangles.

Worked Examples

Example 1: Prove that the sum of angles in a triangle is 180°.

Consider triangle ABC. Draw a line through point A parallel to BC.

Since alternate interior angles are equal: $\angle DAB = \angle ABC$ and $\angle CAE = \angle ACB$

Now the straight angle at A is: $\angle DAB + \angle BAC + \angle CAE = 180^\circ$

So: $\angle ABC + \angle BAC + \angle ACB = 180^\circ$

Example 2: Prove that vertically opposite angles are equal.

Let two straight lines intersect at point O. This forms angles AOC and BOD.

Since they are on a straight line: $\angle AOC + \angle COB = 180^\circ$ and $\angle BOD + \angle COB = 180^\circ$

Equating both: $\angle AOC = \angle BOD$

Example 3: Prove that base angles of an isosceles triangle are equal.

Let triangle ABC have AB = AC.

Draw the bisector of angle A, meeting BC at D.

Now in triangles ABD and ACD: $AB = AC,\ \angle BAD = \angle CAD,\ AD = AD$

By SAS: $\triangle ABD \cong \triangle ACD$ , hence $\angle ABD = \angle ACD$

Example 4: Prove that angles on a straight line sum to 180°.

Consider point B on a straight line AC, with angles ABD and DBC.

Then: $\angle ABD + \angle DBC = \angle ABC = 180^\circ$

Example 5: Prove that alternate angles between parallel lines are equal.

Let lines AB and CD be parallel and cut by a transversal EF.

Then: $\angle AEF = \angle FDC$ (Alternate interior angles)

Example 6: Prove that corresponding angles between parallel lines are equal.

Let lines AB and CD be parallel with transversal XY.

Then: $\angle ABX = \angle CDY$

Example 7: Prove that exterior angle of a triangle equals the sum of two interior opposite angles.

In triangle ABC, extend side BC to D.

Then: $\angle A + \angle B = \angle ACD$

Example 8: Prove that the diagonals of a parallelogram bisect each other.

Let ABCD be a parallelogram with diagonals AC and BD intersecting at E.

Triangles ABE and CDE share side AE = CE and BE = DE (opposite sides equal).

By SAS, triangles ABE and CDE are congruent, so diagonals bisect each other.

Example 9: Prove that the angles in a cyclic quadrilateral sum to 180°.

Let ABCD be a cyclic quadrilateral.

Then: $\angle A + \angle C = 180^\circ$ , $\angle B + \angle D = 180^\circ$

Example 10: Prove that the triangle exterior angle is greater than either of the opposite interior angles.

In triangle ABC, extend BC to D so that: $\angle ACD = \angle A + \angle B$

Therefore, $\angle ACD > \angle A$ and $\angle ACD > \angle B$

Exercises

[WAEC] Prove that the sum of interior angles of a triangle is $180^\circ$ . [Past Question]
[NABTEC] In a triangle, prove that the exterior angle is equal to the sum of two opposite interior angles. [Past Question]
Prove that the diagonals of a rhombus bisect each other at right angles.
[NECO] Show that opposite angles of a cyclic quadrilateral sum to $180^\circ$ . [Past Question]
Using congruent triangles, prove that base angles of an isosceles triangle are equal.
Prove that vertical angles formed by intersecting lines are equal.
In triangle XYZ, prove that $XY + YZ > XZ$ .
[JAMB] Given a parallelogram, prove that opposite angles are equal. [Past Question]
Prove that angles in a straight line add up to $180^\circ$ .
Construct and prove that corresponding angles are equal using a transversal and parallel lines.

Conclusion / Recap

In this lesson, we examined how to understand and prove fundamental geometric theorems, including angle properties, triangle relations, and parallel line behaviors. These theorems form the foundation for deeper studies in geometry, trigonometry, and coordinate geometry. In the next lesson, we will explore **Circle Geometry**, focusing on theorems involving chords, tangents, and arcs.