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More Triangle Properties

Right triangle

An right triangle is a triangle that has a right angle in its interior.

What is a right triangle? How to determine that a triangle is a right triangle?

Isosceles right triangle

An isosceles right triangle is a triangle in which two legs are equal in measure.

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

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.

What is the property of an angle bisector in a triangle? How to determine the distance from a point lies on the angle bisector to a side?

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.

What is the property of an isosceles triangle?

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.

What is the property of a vertex bisector of an isosceles triangle?

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

What degree are the interior angles in an equilateral triangle?

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.

How to find that a triangle is an isosceles triangle?

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.

How to find that a triangle is an equilateral triangle?

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.

How to find that a triangle is an equilateral triangle?

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.

What is the relationship between an angle and the side that the angle opposite?

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.

What is the relationship between two sides and the angles that the two sides opposite?

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.

What is 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.

What is the property of the perpendicular bisector of a segment?

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.

How to find that a 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

How to find that a triangle is an anisosceles triangle? What is the sum of the interior angles in a triangle?

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.

How to find the lengths of two sides in a triangle when given the degree measure of the two angles that the two sides opposite?
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

How to find the angles relation in a triangle when given the three sides that the three angles opposite?
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