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Lines and Angles

Introduction to Lines and Angles

An Overview

Angle Sum Property of Triangles

If we join any three non-collinear points in a plane, then we get a triangle. There are three angles in a triangle.

 

The sum of the three [[mn:glossary]]interior angles[[/mn:glossary]] of a triangle is 180° and this property of a triangle is known as the angle sum property. This property holds true for all types of triangles, i.e., [[mn:glossary]]acute-angled triangles[[/mn:glossary]], [[mn:glossary]]obtuse-angled triangles[[/mn:glossary]] and [[mn:glossary]]right-angled triangles[[/mn:glossary]]. The angle sum property was identified by the Pythagorean school of Greek mathematicians (or the Pythagoreans) and proved by Euclid.

We will study the proof of the angle sum property of triangles and then solve some examples based on this property.

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Proving the Angle Sum Property of Triangles

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Pythagoras

Pythagoras (570 BC−495 BC) was a great Greek mathematician and philosopher, often described as the first pure mathematician. He was born on the island of Samos and is best known for the Pythagoras theorem about right-angled triangles. He also made influential contributions to philosophy and religious teaching. He led a society that was part religious and part scientific. This society followed a code of secrecy, which is the reason why a sense of mystery surrounds the figure of Pythagoras.

Euclid

Euclid of Alexandria(325 BC−265 BC) was a great Greek mathematician. He is referred to as ‘the father of geometry’. Euclid taught at Alexandria during the reign of Ptolemy I, who ruled Egypt from 323 BC to 285 BC. Euclid wrote a series of books which are collectively known as the Elements.It is considered one of the most influential works in the history of mathematics. The Elements served as the main textbook for teaching mathematics (especially geometry) from the time of its publication up until the early 20th century. In the Elements, Euclid defined most of the basic geometrical figures and deduced the principles of geometry through different sets of axioms.

 

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[[mn:know]]

In 1942, a Dutch mathematics teacher Albert E. Bosman invented a plane fractal constructed from a square. He named it the Pythagoras tree because of the presence of right-angled triangles in the figure.

 

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Facts about the Angle Sum Property

An important fact deduced through the angle sum property of triangles is that there can be no triangle with two right angles or two obtuse angles. This fact can be proved as is shown.

Consider a ΔABC such that ∠A = 90° and ∠B = 90°.

According to the angle sum property, we have:

A + ∠B + ∠C = 180°

90° + 90° + ∠C = 180°

⇒ ∠C = 180° − 180°

⇒ ∠C = 0°

However, the above is not possible. So, ΔABC (or any other triangle) cannot have two right angles.

Similarly, we can prove that a triangle cannot have two obtuse angles.

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Relationship between the side lengths and the angle measurements of a triangle

The largest interior angle is opposite the largest side.

The smallest interior angle is opposite the smallest side.

The middle-sized interior angle is opposite the middle-sized side.

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Facts about the Angle Sum Property

By the angle sum property, we can deduce the fact that there can be no triangle with all angles less than or greater than 60°. This fact can be proved as is shown.

Consider a ΔABC with all angles equal to 59°.

According to the angle sum property, we should have ∠A + ∠B + ∠C = 180°.

By adding the given angles, we obtain:

59° + 59° + 59° = 177° ≠ 180°

Since ΔABC does not satisfy the angle sum property, it cannot exist.

Now, consider a ΔABC with all angles equal to 61°.

According to the angle sum property, we should have ∠A + ∠B + ∠C = 180°.

By adding the given angles, we obtain:

61° + 61° + 61° = 183° ≠ 180°

Since ΔABC does not satisfy the angle sum property, it cannot exist.

Thus, we have proved that a triangle can...

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