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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.

HL Theorem

HL Theorem (Hypotenuse-Leg Postulate)

medians to in right triangles

Medians to Legs of a Right Triangle

isosceles triangle with bisector

Angle bisector is perpendicular to the base in an isosceles triangle

Reverse Triangle Inequality

Reverse Triangle Inequality Theorem

triangle with given sides

Heron's Formula

constructing ΔDEF as a right triangle

Converse of the Pythagorean Theorem

Hinge open to 2 different angles

Hinge Theorem

Scalene Triangle

Converse of the Scalene Triangle Inequality

Base angles theorem in Geometry

Converse Base Angle Theorem

Angle bisector in triangle

Angle Bisector in a Triangle

Properties of Isosceles Triangles

Converse Angle Bisector Theorem for Isosceles Triangles

perpendicular bisector

Perpendicular Bisector Theorem

angle bisector

Angle Bisector Theorem

Pythagorean Theorem

Squares On the Sides of a Triangle

Equilateral triangles

Are All Equilateral Triangles Similar?

trapezoid with diagonals

Ratio of Area of Triangles in a Trapezoid

harder geometry problem with similar triangles

A Difficult Geometry Problem With Similar Triangles

a triangle with midsegment

Another Converse Midsegment Theorem

quadrilateral with connected midpoints and triangle

Midpoints of a Quadrilateral - a Difficult Geometry Problem

Congruent triangles in a circle

Congruent Triangles in a Circle

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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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