What are the measurements of a triangle

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.

More Math Homework Help:

Updated April 24, 2017

By Athena Hessong

Calculating the sides of a triangle helps you to determine the perimeter of a triangle even if you only have the measure of two of the angles and one of the sides. To find the sides of the triangle, you need to use the Law of Sines. A scientific calculator with trigonometric functions will help you to find the sine of each of the angles. According to the Law of Sines, the ratio of the sines of each angle divided by the length of the opposite side are all equal. This helps you to find the sides of the triangle.

Add the two angles together and subtract the sum from 180 degrees to find the third angle. For instance, if angle A equals 30 degrees and angle B equals 45 degrees: 30 + 45 = 75; 180 – 75 = 105 degrees = angle C.

Press the measure of angle B followed by the sine button on your scientific calculator. For the example: sine 45 = 0.71.

Multiply the sine of angle B by the length of the side opposite angle A (side A). For the example, if side A measured 10 inches: 0.71 x 10 = 7.1.

Divide this number by the sine of angle A to find the length of side B. For the example, angle A measured 30: sine 30 = 0.5: 7.1/0.5 = 14.2 inches for the length of side B.

Repeat the procedure using angle C instead of angle B to find the measure of the side opposite angle C (side C). For the example: Multiply the sine of angle C (105) by the length of side A and divide the answer by the sine angle A (30): sine 105 = 0.97 x 10 = 9.7/0.5 = 19.4 inches for side C.

Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users.

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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