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Triangles Side and Angles

Triangle Calculator (This free online tool lets calculates all sides and angle measurements based on your input and draws a free downloadable image of your triangle!)

Interior & Exterior Angles and sides

Properties of Triangles


Triangles are one of the most fundamental geometric shapes and have a variety of often studied properties relating to their

What's the difference between interior and exterior angles of a triangle?

This question is answered by the picture below. You create an exterior angle by extending any side of the triangle. Web page on the relationship between exterior and interior angles of a triangle
Picture of interior and exterior angles of a triangle
Interior Angles of a Triangle Rule


This may be one the most well known mathematical rule--The sum of all 3 interior angles in a triangle is 180°. As you can see from the picture below, if you add up all of the angles in a triangle the sum must equal 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 Angle Sum of triangle is 180
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°.

Interior Angles of Triangle Worksheet


Interactive Demonstration of Interior Angle Sum


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Practice Problems on Interior Angles of a Triangle Rule




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

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


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?
Answer
Use the interior angles of a triangle rule :
m$$ \angle $$ PHO = 180° - 26° -64° = 90°


Relationship between Side Lengths and Angle Measurements


Worksheet on the relationship between the side lengths and angle measurements of a triangle
    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
Largest angle vs largest 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. !)