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### Triangle Properties More

Right triangle
An right triangle is a triangle that has a right angle in its interior. Isosceles right triangle
An isosceles right triangle is a triangle in which two legs are equal in measure. In triangle ABC, if AC = BC and angle C = 90o, then triangle ABC is an isosceles right triangle.
Throrem of angle bisector
The property of an angle bisector is that the distance from a point in the angle bisector to both sides of the triangle are equal." In the figure above, AD is an angle bisector which divide the angle A in half
P is a point which lies on AD
Then PE = PF
That is, if P is a point that lies on the angle bisector AD, then PE = PF
Property of an isosceles triangle
An isosceles triangle has two equal base angles. In triangle ABC, if AB = AC, then triangle ABC is an isosceles triangle.
In triangle ABC, AB opposite the angle C and AC opposite the angle B.
If AB = AC, then angle C = angle B.
That is, if AB = AC, then angle B = angle C.
Property of a vertex bisector of an isosceles triangle
The vertex angle bisector of an isosceles triangle bisects its base and is perpendicular to its base. In triangle ABC, if AB = AC, then triangle ABC is an isosceles triangle.
If AD is angle bisector, then angle 1 = angle 2
If AB = AC and angle 1 = angle 2, then BD = DC and AD is perpendicular to BC.
Property of the interior angles of an equilateral triangle
If a triangle is equilateral, then each of the interior angles are equal to 60 degrees. In triangle ABC, If AB = BC = AC, then angle A = angle B = angle C = 60o
Determining an isosceles triangle
If two angles of a triangle are equal, then the sides opposite these angles are also equal. In triangle ABC, if angle B = angle C, then AB = AC. Then triangle ABC is an isosceles triangle.
In a triangle, equal angles opposite equal sides.
Determine an equilateral triangle
If three angles of a triangle are equal, then the triangle is an equilateral. In triangle ABC, if angle A = angle B = angle C, then AB = BC = AC. Then triangle ABC is an equilateral triangle.
Determine an equilateral triangle
If an angle of an isosceles triangle is 60 degrees, then the triangle is an equilateral. If triangle ABC is an isosceles triangle and angle A or angle B or angle C = 60o, then triangle ABC is an equilateral triangle.
Theorem of inequalities regarding sides and angles in a triangle
If two sides of a triangle are unequal, then the angles opposite these sides are also unequal, and the large angle opposite the longer side. In triangle ABC, if AC > AB, then angle B > angle C.
In triangle ABC, AC opposite the angle B and AB opposite the angle C.
In a triangle, larger side opposite to larger angle.
Inequalities regarding sides and angles in a triangle
"If two angles of a triangle are unequal, then the sides opposite these angles are also unequal, and the longer sides opposite the large angle. In triangle ABC, if angle B > angle C, then AC > AB
note: angle B opposite AC and angle C opposite AB
In a triangle, large angle opposite to large side.
Perpendicular bisector of a segment
If a line is perpendicular to a segment and intersects the segment at its midpoint, then the line is called the perpendicular bisector of the segment. In the figure above, line l is perpendicular to line segment AB
AO = BO
line l is the perpendicular bisector of AB.
Property of the perpendicular bisector of a segment
The distance from a point of perpendicular bisector of a segment to the two endpoints of the segment are equal. In the figure above, if line l is the perpendicular bisector of AB and P is a point which lies on the line l,
then PA = PB
Converse theorem of erpendicular bisector of a segment
If the distances from a point to the two endpoint of a segment are equal, then the point lies on the perpendicular bisector of a segment. In the figure above, if PA = PB
then the point P lies on the perpendicular bisector of AB.
Example 1: In the figure above, if AB = AC and the degree measure of the angle A is 98o, what is the degree measure of the angle 1?
Solution:
Since AB = AC (Given), so triangle ABC is an isosceles triangle.
angle B = angle ACB = (180o - angle A)/2 = (180o - 98o)/2 = 41o
Since the angle 1 is an exterior angle of the triangle ABC, so
angle 1 = angle B + angle A = 41o + 98o = 139o
Example 2: In the figure above, if the degree measure of the angle A is 70o and the degree measure of the angle C is 50o, find the relation between AB and AC.
Solution:
In triangle ABC, angle B = 180o - (angle A + angle C) = 180o - (70 + 50o) = 180o - 120o = 60o
Since angle B = 60o and angle A = 50o
so AC > AB
note: angle B opposite AC and angle C opposite AB
Example 3: In the figure above, a triangle with sides of different measure. List the angles of this triangle in order from least to greatest.
Solution:
Given: AC = 14, AB = 17 and BC = 23
Since 14 < 17 < 23
so AC < AB < BC
so angle B < angle C < angle A
note: AC opposite angle B, AB opposite angle C and BC opposite angle A