In another problem, we saw that in an isosceles triangle, the height to the base from the apex is also the angle bisector. Here, we will show the opposite: that the angle bisector is perpendicular to … [Read more...] about Angle bisector is perpendicular to the base in an isosceles triangle
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.
Reverse Triangle Inequality Theorem
In this problem we will prove the Reverse Triangle Inequality Theorem, using what we have already proven In a previous problem- the Triangle Inequality. The Triangle Inequality theorem states that … [Read more...] about Reverse Triangle Inequality Theorem
Heron’s Formula
Today we will prove Heron's formula for finding the area of a triangle when all three of its sides are known. The proof is a bit on the long side, but it's very useful. In another post, we saw … [Read more...] about Heron’s Formula
Converse of the Pythagorean Theorem
In today's lesson, we will focus on the 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 … [Read more...] about Converse of the Pythagorean Theorem
Hinge Theorem
What is the 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 … [Read more...] about Hinge Theorem
Converse of the Scalene Triangle Inequality
In today's lesson, we will prove the converse of the scalene triangle inequality. Using proof by contradiction, we will show that the side facing the larger angle is longer. Having proven the … [Read more...] about Converse of the Scalene Triangle Inequality
Converse Base Angle Theorem
In today's lesson, we will prove the converse to the Base Angle theorem - if two angles of a triangle are congruent, the triangle is isosceles. We will use congruent triangles for the proof. From … [Read more...] about Converse Base Angle Theorem
Angle Bisector in a Triangle
Today's 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
In today's lesson, we will prove the Converse Angle Bisector Theorem for Isosceles Triangles. We'll show that if a triangle's angle bisector is perpendicular to the opposite side, the triangle is an … [Read more...] about Converse Angle Bisector Theorem for Isosceles Triangles
Perpendicular Bisector Theorem
In today's geometry lesson, we will show a fairly easy way to prove the perpendicular bisector theorem. A line that splits another line segment (or an angle) into two equal parts is called a … [Read more...] about Perpendicular Bisector Theorem