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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.

What is a triangle? How to determine a triangle?

Sum of interior angle of a triangle theorem

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

What is the degree measure of the sum of the interior angles of any triangle?

What is the value of the degree measure of the sum of the interior angles of any triangle?

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

What is the degree measure of 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.

How to find an exterior angle? What angle is an exterior angle?

How to find the value of an exterior angle? What is the value of an exterior angle?

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.

What is an equilateral triangle? How to determine that a triange is an equilateral triangle?

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.

What is an isosceles triangle? How to determine that a triangle is an isosceles triangle?

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.

What is a scalene triangle? How to determine that a triangle is a 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.

What is an obtuse triangle? How to determine an obtuse triangle?

In the triangle ABC, since 90o < angle A < 180o, 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.

What is an acute triangle? How to determine an acute triangle?

In the figure above, since angle A < 90o, angle B < 90o and angle C < 90o, 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.

What is the angle bisector in a triangle? How to determine the angle bisector in a triangle?

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.

How to determine that a line is the median in a triangle?

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.

How to find the altitude in an acute triangle? What is the altitude in a triangle?

How to find the altitude in an obtuse triangle? What is the altitude in a triangle?

Three sides of a triangle relation theorem

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

What is the relationship of the three sides of a triangle?
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.

What is the relationship of the three sides of a triangle?
|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
What is the interior angle of the triangle?
Let angle A = 3x, angle B = 4x, angle C = 5x
In triangle ABC, angle A + angle B + angle C = 180o
[Theorem: The sum of the interior angles of any triangle is 180 degrees.]
3x + 4x + 5x = 180o
12x = 180o
x = 15o
angle A = 3x = 3 × 15o = 45o
Therefore, the degree measure of the angle A is 45o.

Example 2

In the figure below, find the value of x.

How to find the measure of an exterior angle of a triangle?
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 = 77o + 45o = 122o
In the figure above, angle ACD = x = 122o