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Definition and Properties of a Triangles

Triangle

A triangle is a three-side figure. It has three angles in its interior and each angle corresponding to a vertex labeled by a letter.

Sum of interior angle of a triangle theorem

The sum of the interior angles of any triangle is 180 degrees.

Exterior angle of a triangle

An exterior angle of a triangle is formed when extended one side of the triangle. The angle outside of the triangle but adjacent to an interior angle is an exterior angle of the triangle

In the figure above, the angle 2 is an exterior angle of the triangle ABC. The angle 2 is adjacent to angle 1 which is an interior angle of the triangle ABC. One side of the angle 2 is CD which is comming from extending the side BC.

Exterior angle of a triangle theorem

The measure of an exterior angle of a triangle is equal to the sum of the measure of the two nonadjacent interior angles.

Angle 2 is an exterior angle of the triangle ABC. Angle A and angle B are two interior angles of the triangle ABC. Angle 2 is not adjacent to angle A and angle B

Equilateral triangle

An equilateral triangle is a triangle with all three sides are equal in measure.

If triangle ABC is an equilateral trangle, then AB = AC = BC

Isosceles triangle

An isosceles triangle is a triangle with two sides are equal in measure.

If triangle ABC is an isosceles triangle, then AB = AC.

Scalene triangle

A scalene triangle is a triangle with all three sides of different in measure.

Obtuse triangle

An obtuse triangle is a triangle having an obtuse angle which is large than 90 degrees and less than 180 degrees.

In the triangle ABC, since 90^o < angle A < 180^o, then triangle ABC is an obtuse triangle.

Acute triangle

An acute triangle is a triangle having all acute angles which is less than 90 degrees.

In the figure above, since angle A < 90^o, angle B < 90^o and angle C < 90^o, so triangle ABC is an acute triangle.

Angle bisector in a triangle

An angle bisector in a triangle is a line segment drawn fron a vertex and which cut in half that vertex angle.

In the figure above, since angle 1 = angle 2, so the line segment AD is an angle bisector which divide the angle A in half.

Median in a triangle

A median in a triangle is a line segment drawn from a vertex to the midpoint of its opposite side.

In the figure above, since BD = DC, so line segment AD is a median of the triangle ABC. The median divide the side BC in half.

Altitudes and base

Altitude in a triangle is a perpendicular segment drawn from a vertex to its opposite side or to the txtension of the opposite side.

Three sides of a triangle relation theorem

In a triangle, the sum of two sides is large than the third side.

AB + BC > AC

AB + AC > BC

AC + BC > AB

Three sides of a triangle relation corollary

In a triangle, the difference of any two sides is less than the third side.

|AB - BC| < AC

|AB - AC| < BC

|AC - BC| < AB

Example 1

In triangle ABC, if angle A : angle B : angle C = 3 : 4 : 5, then what is the degree measure of the angle A?

Solution

Let angle A = 3x, angle B = 4x, angle C = 5x

In triangle ABC, angle A + angle B + angle C = 180^o

[Theorem: The sum of the interior angles of any triangle is 180 degrees.]

3x + 4x + 5x = 180^o

12x = 180^o

x = 15^o

angle A = 3x = 3 × 15^o = 45^o

Therefore, the degree measure of the angle A is 45^o.

Example 2

In the figure below, find the value of x.

Solution

Since the angle ACD is adjacent to the angle 1 which is an interior angle of the triangle ABC.

and since one side of the angle ACD is CD which is the extended line of BC

So the angle ACD is an exterior angle of the triangle ABC.

[Theorem of the exterior angle of a triangle: The measure of an exterior angle of a triangle is equal to the sum of the measure of the two nonadjacent interior angles.]

So angle ACD = angle A + angle B = 77^o + 45^o = 122^o

In the figure above, angle ACD = x = 122^o