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

# 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.*

## 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

## Are All Equilateral Triangles Similar?

A regular polygon is one in which all angles are equal and all sides are equal. So by definition, all regular polygons with the same number of sides are similar to each other.An equilateral … [Read more...] about Are All Equilateral Triangles Similar?

## Ratio of Area of Triangles in a Trapezoid

The diagonals of a trapezoid form 2 similar triangles. and two other sets of triangles that share the same base and height. This property is used in many geometry problems that require you to find the … [Read more...] about Ratio of Area of Triangles in a Trapezoid

## A Harder Geometry Problem With Similar Triangles

One of my regular readers sent me the following problem, asking for help in solving it. This is a challenging problem that was fun to solve, because it uses several concepts we have discussed earlier … [Read more...] about A Harder Geometry Problem With Similar Triangles

## Another Converse Midsegment Theorem

The triangle midsegment theorem states that the midsegment is parallel to the third side, and its length is equal to half the length of the third side, and its converse states that if a line … [Read more...] about Another Converse Midsegment Theorem

## Midpoints of a Quadrilateral – a Harder Geometry Problem

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

## 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