In another post, we saw how to calculate the area of a triangle whose sides were all given, using the fact that those 3 given sides made up a Pythagorean Triple, and thus the triangle is a right … [Read more...] about Heron’s Formula

# 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

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”:

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

--

*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.*

## Converse of the Pythagorean Theorem

One of the most useful theorems in Euclidean geometry, which we have used often in other proofs is the Pythagorean Theorem.The Pythagorean Theorem states that in a right triangle, the following … [Read more...] about Converse of the Pythagorean Theorem

## Hinge Theorem

Let's say you have a pair of triangles with two congruent sides but a different angle between those sides. Think of it as a hinge, with fixed sides, that can be opened to different angles:The … [Read more...] about Hinge Theorem

## Converse of the Scalene Triangle Inequality

Having proven the Scalene Triangle Inequality- that if in a scalene triangle ΔABC, AB>AC then m∠ACB> m∠ABC - proving the converse is very simple.ProblemIn scalene triangle ΔABC, … [Read more...] about Converse of the Scalene Triangle Inequality

## Converse Base Angle Theorem

From the definition of an isosceles triangle as one in which two sides are equal, we proved the Base Angles Theorem - the angles between the equal sides and the base are congruent.Now we'll prove … [Read more...] about Converse Base Angle Theorem

## Angle Bisector in a Triangle

This geometry problem presents an application of the angle bisector theorem in a triangle. It also incorporates several other concepts we've explored, like the properties of isosceles triangles and … [Read more...] about Angle Bisector in a Triangle

## Converse Angle Bisector Theorem for Isosceles Triangles

One of the properties of an isosceles triangle is that the height to the base bisects the apex angle.The converse of this is also true - If the bisector of an angle in a triangle is perpendicular … [Read more...] about Converse Angle Bisector Theorem for Isosceles Triangles

## Perpendicular Bisector Theorem

A line that splits another line segment (or an angle) into two equal parts is called a "bisector." If the intersection between the two line segment is at a right angle, then the two lines are … [Read more...] about Perpendicular Bisector Theorem

## Angle Bisector Theorem

The angle bisector is a line that divides an angle into two equal halves, each with the same angle measure.The angle bisector theorem state that in a triangle, the angle bisector partitions the … [Read more...] about Angle Bisector Theorem

## Squares On the Sides of a Triangle

The Pythagorean Theorem shows us an important property of the squares constructed on the sides of a right triangle, namely that c2 = a2+b2:But there are interesting properties of squares … [Read more...] about Squares On the Sides of a Triangle