What relationship exist between the measures of the angles of a triangle and the lengths of the sides of the triangle?

Updated April 24, 2017

By Sreela Datta

Euclidean geometry, the basic geometry taught in school, requires certain relationships between the lengths of the sides of a triangle. One cannot simply take three random line segments and form a triangle. The line segments have to satisfy the triangle inequality theorems. Other theorems that define relationships between the sides of a triangle are the Pythagorean theorem and the law of cosines.

According to the first triangle inequality theorem, the lengths of any two sides of a triangle must add up to more than the length of the third side. This means that you cannot draw a triangle that has side lengths 2, 7 and 12, for instance, since 2 + 7 is less than 12. To get an intuitive feel for this, imagine first drawing a line segment 12 cm long. Now think of two other line segments 2 cm and 7 cm long attached to the two ends of the 12 cm segment. It is clear that it would not be possible to make the two end segments meet. They would have to add up at least to 12 cm.

The longest side in a triangle is across from the largest angle. This is another triangle inequality theorem and it makes intuitive sense. You can draw various conclusions from it. For example, in an obtuse triangle, the longest side has to be the one across from the obtuse angle. The converse of this is true as well. The largest angle in a triangle is the one that is across from the longest side.

The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse (the side across from the right angle) is equal to the sum of the squares of the other two sides. So if the length of the hypotenuse is c and the lengths of the other two sides are a and b, then c^2 = a^2 + b^2. This is an ancient theorem that has been known for thousands of years and has been used by builders and mathematicians through the ages.

The law of cosines is a generalized version of the Pythagorean theorem that applies to all triangles, not just the ones with right angles. According to this law, if a triangle had sides of length a, b and c, and the angle across from the side of length c is C, then c^2 = a^2 + b^2 - 2abcosC. You can see that when C is 90 degrees, cosC = 0 and the law of cosines is reduced to the Pythagorean theorem.

Triangle rules and theorems allow us to understand the properties of this shape. As one of the most central elements of trigonometry, triangles have many geometric rules. Among other things, these help us to distinguish right triangles from equilateral triangles and isosceles triangles.

Let's review some of the most notable trigonometric triangle rules.

Interior Angles Rule

The interior angles rule states that the three angles of a triangle must equal 180°. As you can see below, the three angle measurements of obtuse triangle ABC add to 180°.

Sides of a Triangle

The sides of a triangle rule asserts that the sum of the lengths of any two sides of a triangle has to be greater than the length of the third side. See the side lengths of the acute triangle below. The sum of the lengths of the two shortest sides, 6 and 7, is 13. That length is greater than the length of the longest side, 8.

Triangle Congruence Rules

Congruent triangles are triangles whose corresponding sides and angles are equal. In trigonometric fashion, equal sides and equal angles are proven congruent through the four triangle rules of congruence. We’ll go through these one at a time.

#1: SSS Rule

The side-side-side (SSS) rule says that when the three side measurements of a triangle match the three side measurements of another triangle, these two shapes are congruent.

See the right-angled triangles below. The sides of the triangle DEF are the same exact lengths as triangle GHI, making them congruent.

#2: ASA Rule

The angle-side-angle (ASA) rule states that when two angles and one side of a triangle are equal to that of another triangle, they are congruent triangles.

See triangles JKL and MNO. Angles J and M, K and N (the opposite angles to the length of the hypotenuse), and the hypotenuse-legs of both triangles are all equal. Therefore, triangles JKL and MNO are congruent.

#3: AAS Rule

The angle-angle-side (AAS) rule asserts that when two triangles have the following matching properties, they must be congruent:

• Two angles
• One opposite side length with no vertices

#4: SAS Rule

The side-angle-side (SAS) rule states that if the included angle and the two included side lengths of a triangle are equal to that of another triangle, then the two are congruent. See below triangles CDE and FGH. Right angle C and angle F, the length of d and g, and the hypotenuse length of c and f are equal. Therefore, triangle CDE=FGH.

The Importance of Triangle Rules

Expanding your knowledge of triangle rules will make it easier to learn other trigonometric ideas like Pythagoras theorem and cosine, tangent, and sine rules. This knowledge will also help you master the area of a triangle and polygon.

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Triangles are one of the most fundamental geometric shapes and have a variety of often studied properties including:

Rule 1: Interior Angles sum up to $$ 180^0 $$

Rule 2: Sides of Triangle -- Triangle Inequality Theorem : This theorem states that the sum of the lengths of any 2 sides of a triangle must be greater than the third side. )

Rule 3: Relationship between measurement of the sides and angles in a triangle: The largest interior angle and side are opposite each other. The same rule applies to the smallest sized angle and side, and the middle sized angle and side.

Rule 4 Remote Extior Angles-- This Theorem states that the measure of a an exterior angle $$ \angle A$$ equals the sum of the remote interior angles' measurements. more)

This question is answered by the picture below. You create an exterior angle by extending any side of the triangle.

This may be one the most well known mathematical rules-The sum of all 3 interior angles in a triangle is $$180^{\circ} $$. As you can see from the picture below, if you add up all of the angles in a triangle the sum must equal $$180^{\circ} $$.

To explore the truth of this rule, try Math Warehouse's interactive triangle, which allows you to drag around the different sides of a triangle and explore the relationship between the angles and sides. No matter how you position the three sides of the triangle, the total degrees of all interior angles (the three angles inside the triangle) is always 180°.

This property of a triangle's interior angles is simply a specific example of the general rule for any polygon's interior angles.

Interior Angles of Triangle Worksheet

What is m$$\angle$$LNM in the triangle below?

$$ \angle $$ LMN = 34°
$$ \angle $$ MLN = 29°

Use the rule for interior angles of a triangle:

m$$ \angle $$ LNM +m$$ \angle $$ LMN +m$$ \angle $$ MLN =180° m$$ \angle $$ LNM +34° + 29° =180° m$$ \angle $$ LNM +63° =180°

m$$ \angle $$ LNM = 180° - 63° = 117°

A triangle's interior angles are $$ \angle $$ HOP, $$ \angle $$ HPO and $$ \angle $$ PHO. $$ \angle $$ HOP is 64° and m$$ \angle $$ HPO is 26°.
What is m$$ \angle $$ PHO?

In any 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

To explore the truth of the statements you can use Math Warehouse's interactive triangle, which allows you to drag around the different sides of a triangle and explore the relationships betwen the measures of angles and sides. No matter how you position the three sides of the triangle, you will find that the statements in the paragraph above hold true.

(All right, the isosceles and equilateral triangle are exceptions due to the fact that they don't have a single smallest side or, in the case of the equilateral triangle, even a largest side. Nonetheless, the principle stated above still holds true. !)

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