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# 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.

# Angle bisector theorem

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.

# Inequalities regarding sides and angles in a triangle theorem

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 below, 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 below, 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