Sometimes, we are presented with geometry problems that rely on utilizing more than one basic concept, and the two concepts are unrelated, making it hard to connect the two in order to solve the … [Read more...] about Midpoints of a Quadrilateral – a Harder Geometry Problem

# 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

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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.*

## Congruent Triangles in a Circle

Using the known properties of circles, like the fact that their radii are all equal, it is easy to solve geometry problems that require proof that triangles in the circle are congruent, like this … [Read more...] about Congruent Triangles in a Circle

## Obtuse Triangle – Definition and Properties

An obtuse triangle is a triangle in which one of the angles is larger than 90°. There is only one such angle possible, since the sum of angles in a triangle is 180°.It's easy to show that in an … [Read more...] about Obtuse Triangle – Definition and Properties

## Scalene Triangle Inequality

A scalene triangle is a triangle in which all three sides have different lengths. The scalene triangle inequality theorem states that in such a triangle, the angle facing the larger side has a … [Read more...] about Scalene Triangle Inequality

## 30-60-90 Triangle

A right triangle with angles that measure 30 degrees, 60 degrees and 90 degrees has some special properties. One of them is that if we know the length of only one side, we can find the lengths of the … [Read more...] about 30-60-90 Triangle

## Converse Triangle Midsegment Theorem

The Triangle Midsegment Theorem states that the midsegment of a triangle is parallel to the third side, and its length is equal to half the length of the third side.We will now prove the converse … [Read more...] about Converse Triangle Midsegment Theorem

## Applications of Similar Triangles: Calculating the Height of Tall Objects Using Their Shadow

One application of the properties of similar triangles is to find the height of very tall objects such as buildings using the length of their shadow on the ground and comparing it to the length of the … [Read more...] about Applications of Similar Triangles: Calculating the Height of Tall Objects Using Their Shadow