Geometric Proofs for Polygons

Geometric Proofs for Polygons are 2D shapes with straight edges, and they have the same number of sides as they do corners (vertices). Because there are so many different kinds of polygons, there aren’t many features that apply to all of them. However, when we need to prove something about polygons, there are a couple of reliable properties.

The sum of the interior angles of a polygon can always be calculated using the formula: (n − 2) × 180°, where n is the number of sides of the polygon. For convex polygons, the sum of the exterior angles is always 360°.

In this article, we will study and learn about these Geometric Proofs for Polygons.

What is a Geometric Proof?

A geometric proof is a chain of elements that can be verified with logical statements to show the reality or falsehood of some mathematical statement about any aspect along the similar lines. It uses definitions, postulates, theorems and results proven already to create a logical chain of reasoning that will reach the result they want. Geometric proofs: Geometric proof is crucial for theories of shapes, sizes and relationships among various geometric objects.

History of Geometric Proofs

The concept of geometric proofs dates back to ancient Greece, with mathematicians like Euclid being pivotal in laying the foundations of geometry. Euclid's work, "The Elements," is one of the most influential mathematical texts ever written. It systematically presents geometric proofs, starting from basic definitions and postulates and building up to more complex theorems. These early proofs were crucial in establishing the rigorous logical framework that geometry relies on today.

Some of the Notable Historical Contributions are:

• Euclid's Elements: A comprehensive compilation of geometric knowledge, presenting proofs in a systematic manner.
• Pythagoras: Known for the Pythagorean Theorem, which provides a fundamental relationship between the sides of a right triangle.
• Archimedes: Contributed to the understanding of areas and volumes, developing proofs that are still studied today.

Types of Geometric Proofs

Geometric proofs can be categorized into several types, each with its unique approach to demonstrating the truth of a statement.

• Direct Proofs
• Indirect Proofs (Proof by Contradiction)
• Proof by Construction

Let's discuss each in detail.

Direct Proofs

A direct proof starts with known facts or assumptions and uses logical steps to reach a conclusion. Each step in the proof follows logically from the previous one, leading directly to the statement that needs to be proven.

Example: Proving the Midpoint Theorem

Statement: Line segment joining mid-points of two sides of a triangle is parallel to the third side of the triangle and is half of it.

Proof:

In ∆AED and ∆CEF 

DE = EF (construction)
∠1 = ∠2 (vertically opposite angles)
AE = CE (E is the mid-point)

△AED ≅ △CEF by SAS criteria

Therefore, 

∠3 =∠4   (c.p.c.t)

But these are alternate interior angles.

So, AB ∥ CF

AD = CF  (c.p.c.t)

But AD = DB (D is the mid-point)

Therefore, BD = CF

In BCFD

BD∥ CF (as AB ∥ CF)

BD = CF

BCFD is a parallelogram as one pair of opposite sides is parallel and equal.

Therefore, 

DF∥ BC (opposite sides of parallelogram)

DF = BC (opposite sides of parallelogram)

As DF∥ BC, DE∥ BC and DF = BC

But DE = EF

So, DF = 2(DE)

2(DE) = BC

DE = 1/2(BC)

Hence, proved that the line joining the mid-points of two sides of the triangle is parallel to the third side and is half of it.

Indirect Proofs (Proof by Contradiction)

An indirect proof involves assuming that the statement to be proven is false. By using logical reasoning, you show that this assumption leads to a contradiction or an impossible situation. This contradiction implies that the original assumption was incorrect, and therefore, the statement must be true.

Example: Proving the Diagonals of a Parallelogram Bisect Each Other

Statement: The diagonals of a parallelogram bisect each other.

Proof:

1. Assume that the diagonals of parallelogram ABCD do not bisect each other.
2. Let E be the point where the diagonals intersect, but assume E is not the midpoint of either diagonal.
3. Use properties of parallel lines and congruent triangles to show that this assumption leads to a contradiction.

Therefore, the diagonals of parallelogram ABCD must bisect each other.

Proof by Construction

Proof by construction involves creating a geometric figure or constructing additional elements within a figure to prove a statement. This often requires drawing new lines, circles, or angles within an existing figure and using these constructions to demonstrate the truth of the statement.

Example: Proving the Diagonals of a Rectangle Are Equal

Statement: The diagonals of a rectangle are equal in length.

Proof:

1. Start with rectangle ABCD.
2. Draw diagonals AC and BD.
3. By the properties of a rectangle, opposite sides are equal, and all angles are right angles.
4. Use the Pythagorean Theorem to show that AC and BD are equal.

Therefore, the diagonals of a rectangle are equal in length.

Common Geometric Proofs of Theorems

Pythagorean Theorem

The Pythagorean Theorem is a fundamental theorem in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Statement: In a right-angled triangle ABC, with the right angle at C, AB^2 = AC^2 + BC^2.

Proof:

Consider a right-angled triangle having sides A, B, and C. Here, AC is the longest side (hypotenuse), and AB and BC are the legs of the triangle. Draw a perpendicular line BD at AC as shown in the figure below,

In △ABD and △ACB,

∠A = ∠A (Common angle)

∠ADB = ∠ABC (90°)

Therefore, we can say △ABD ∼ △ ACB (By AA Similarity)

Similarly, △BDC ∼ △ACB

Hence, AD/AB = AB/AC

AB² = AD × AC ⇢ (1)

And, CD/BC = BC/AC

BC² = CD × AC ⇢ (2)

Adding equations (1) and (2),

AB² + BC² = AC × AD + AC × CD

AB² + BC² = AC (AD + CD)

AB² + BC² = AC × AC

AB² + BC² = AC²

Also, AC² = AB² + BC²

Hence proved.

Also Read: Pythagoras Theorem | Formula, Proof and Examples

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the angles in any triangle is always 180 degrees.

Statement: In any triangle ABC, \angle A + \angle B + \angle C = 180^\circ.

Proof:

The sum of all the angles of a triangle is equal to 180°. This theorem can be proved by the below-shown figure.

Follow the steps given below to prove the angle sum property in the triangle. 

Step 1: Draw a line parallel to any given side of a triangle let’s make a line AB parallel to side RQ of the triangle.

Step 2: We know that sum of all the angles in a straight line is 180°. So, ∠APR + ∠RPQ + ∠BPQ = 180°

Step 3: In the given figure as we can see that side AB is parallel to RQ and RP, and QP act as a transversal. So we can see that angle ∠APR = ∠PRQ and ∠BPQ = ∠PQR by the property of alternate interior angles we have studied above. 

From step 2 and step 3,

∠PRQ + ∠RPQ + ∠PQR = 180° [Hence Prooved]

Also Read: Angle Sum Property of a Triangle

Applications of Geometric Proofs

Geometric proofs are applied in various fields, including:

• Architecture: Ensuring that buildings are structurally sound by verifying geometric properties of shapes.
• Engineering: Designing machinery and structures that rely on precise geometric calculations.
• Computer Graphics: Creating realistic images and animations by ensuring that geometric shapes behave correctly.

Solved Examples of Geometric Proofs

Example 1: Use proof by contradiction to prove that the diagonals of a parallelogram bisect each other.

Solution:

Let ABCD be the parallelogram. Therefore, AB || DC and AD||BC.

Consider triangle AOD and COB.

AD = BC (opposite sides of a parallelogram)

∠DAO = ∠BCO (Alternate angles)

∠ADO = ∠CBO (Alternate angles)

Therefore, by ASA congruency, the triangle are congruent.

Now AO = OC and BO = OD because they are corresponding sides of two congruent triangle. Thus, the diagonals of a parallelogram bisect each other.

Example 2: Proving the Angle Bisector Theorem

Statement: In a triangle, the angle bisector divides the opposite side into segments that are proportional to the adjacent sides.

Proof:

Draw line RT parallel to PS and extend T so that it meets P as shown in the figure.

PR is the traversal of RT || PS. So, by the property alternate interior angles are equal.

∠ SPR = ∠ PRT [interior angles] ——-(1)

∠QPS = ∠PTR [corresponding angles] ——-(2)

In the figure, PS is the angle bisector of P.

∠QPS = ∠ SPR ——(3)

From (2) and (3)

∠PTR = ∠ SPR ——(4)

From (1) and (4)

∠PRT = ∠PTR ——(5)

Equation (5) implies that triangle PQR is isosceles triangle and PS = PR.

Now, by proportionality theorem:

QS / SR = PQ / PS

Since, PS = PR

QS / SR = PQ / PR

Therefore, the Angle Bisector Theorem is proven.

Example 3: Proving the Converse of Pythagoras theorem

Statement: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Proof:

In △EGF, by Pythagoras Theorem:

⇒ EF² = EG² + FG2² = b² + a² ⇢ (1)

In △ABC, by Pythagoras Theorem:

⇒ AB² = AC² + BC² = b² + a² ⇢ (2)

From equation (1) and (2), we have;

⇒ EF² = AB²

⇒ EF = AB

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle. Hence, we can say that the converse of the Pythagorean theorem also holds.

Hence, the Converse of the Pythagorean Theorem is proven.

Practice Problems on Geometric Proofs

Problem 1: Prove that the opposite angles of a parallelogram are equal using a direct proof.

Problem 2: Use proof by contradiction to show that the sum of the angles in a quadrilateral is 360 degrees.

Problem 3: Prove that the diagonals of a rectangle are equal in length using proof by construction.

Problem 4: Prove that the base angles of an isosceles triangle are equal using a direct proof.

Problem 5: Prove that the diagonals of a rhombus bisect each other at right angles using a direct proof.

Problem 6: Prove that the exterior angle of a triangle is equal to the sum of the two opposite interior angles using a direct proof.

Problem 7: Use proof by construction to prove that the perpendicular from the center of a circle to a chord bisects the chord.

Problem 8: Prove that the diagonals of a kite are perpendicular to each other using a direct proof.

Conclusion

This list of geometry proofs and concepts in this article is fundamental for understanding more complex theorems and problems. Learning these helps students get a good grasp of geometry, making it easier to solve problems and understand how different shapes and angles relate to each other. Once they master these basics, geometry will become much easier and more intuitive for them.

Also Read,

• Mid point theorem
• Pythagoras theorem
• Angle Bisector Theorem