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Triangles

A triangle is one of the most basic shapes in geometry- and object with three straight sides (“edges”) and three angles, formed where each of the two sides meet. These meeting points are called “vertices”.

Triangle Notation

A triangle is often noted by using the points at its vertices, for example: ΔABC

Triangle Notation

And the angles in the triangle are often defined by the points, as well, so ∠1 can be written as ∠ABC and angle ∠2 can be written as ∠ACB. The vertices where the angle is the middle letter in this type of notation.

Basic properties of triangles

One of the basic properties of triangles is that the sum of the measure of angles, in every triangle, is 180°, as we will now prove, using what we know about parallel lines and the angles formed by a transversal line.

Proof: The sum of the angles in a triangle is 180°.

If we extend the triangle’s sides beyond the triangle, we form angles between the line’s extension and the angle inside the triangle, like angle1 below. These angles are called ”exterior angles”:

Triangle Exterior Angles

Having just proven that the sum of the angles in a triangle is 180°, it is now simple to prove a corollary theorem, that the measure of an exterior angle at a vertex of a triangle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle (called the remote interior angles). Proof: The exterior angle is equal to the sum of the two remote interior angles

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Now that we've explained the basic concept of triangles in geometry, let's scroll down to work on specific geometry problems relating to this topic.

obtuse triangle

Obtuse Triangle - Definition and Properties

Scalene triangle

The Scalene Inequality Theorem

30-60-90 triangle

30-60-90 Triangle

Converse Triangle Midsegment Theorem

Geometry drawing: objects and their shadows

Calculating the Height of Tall Objects

Geometry shapes: Median to the hypotenuse

Right Triangles: Median to the Hypotenuse is Equal to Half the Hypotenuse

a triangle with midsegment

Triangle Midsegment Theorem

Right Triangle

Prove the Pythagorean Theorem Using Triangle Similarity

Similar triangles in geometry

How to Prove that Triangles are Similar

Triangle inequality theorem - A to B is shortest

Triangle Inequality Theorem

area of equilateral triangle

Finding the Area of an Equilateral Triangle from its Perimeter

Geometry: two right triangles

45 45 90 Triangle

isosceles triangle with median

Isosceles Triangles: the Median to the Base is Perpendicular to the Base

HL Theorem

Congruent Right Triangles

Properties of Isosceles Triangles

Isosceles Triangles: the Height to the Base Bisects the Apex Angle

Properties of Isosceles Triangles

The Height to the Base of an Isosceles Triangle Bisects the Base

Base angles theorem in Geometry

Base Angles Theorem

Right Triangle

Pythagorean Theorem: Lengths of Edges in a Right Triangle

Triangle Exterior Angles

Sum of Angles in a Triangle

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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