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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.
Theorem of sum of interior angle of a triangle
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.
Theorem of 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.
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
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
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.
Theorem of three sides relation of a triangle
In a triangle, the sum of two sides is large than the third side.
AB + BC > AC
AB + AC > BC
AC + BC > AB
Corollary of three sides relation of a triangle
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}