Properties of triangles

If this was named “prior learning topics” you might just skip it, so it’s “Properties of triangles”.

Contents

Angles and sides

Theorem: The angles in a triangle sum to 180°180\degree.

Corollary: There can be at most one right angle (90°90\degree) or one obtuse angle (between 90°90\degree and 180°180\degree). Each triangle must have at least two acute angles (between 0°0\degree and 90°90\degree).

Types of triangles

nameanglessides
scaleneall differentall different
isosceles2 equal, 1 different2 equal, 1 different
equilateralall 60°60\degree3 equal
acuteall less than 90°90\degree
righthas exactly 11 right angle
obtusehas exactly 11 obtuse angle

The terminology are unimportant except for maybe “equilateral”. Yet you should recognize and be able to use properties of different classes of triangles in exam. For example, if a triangle has 22 equal sides, then you should know the two angles not between these 22 sides are equal.

The sides of a triangle satisfy the triangle inequality.

Theorem: In a triangle, the sum of any two side lengths is always strictly greater than the third. It is sufficient to verify the two shorter sides sum to longer than the third, to ensure the triangle exists.

This is same as saying the straight-line distance is shorter than a crooked path.

Similarity

Similar triangles pave the foundation of trigonometry.

Triangle △ABC\triangle ABC is similar to (∼\sim) △XYZ\triangle XYZ means the following are true

AB=kXYAB = kXY
BC=kYZBC = kYZ
AC=kXZAC = kXZ
A^=X^\hat A = \hat X
B^=Y^\hat B = \hat Y
C^=Z^\hat C = \hat Z
Area(△ABC)=k2Area(△XYZ)\text{Area}(\triangle ABC) = k^2 \text{Area}(\triangle XYZ)

where kk is some constant of proportionality.

For lengths, the order of the letters does not matter. Eg

AB=BAAB = BA

but for triangle similarity the order does matter. If △ABC\triangle ABC is similar to △XYZ\triangle XYZ, then it is not similar to △BAC\triangle BAC or △ZYX\triangle ZYX. The name tells us which angles are paired up.

If two of the angles are the same as those of another triangle, then the triangles are similar, as long as they are properly named.

Definitions and conventions

The word opposite has two closely-related meanings. The opposite side to an angle is the one not next to the angle. In a right triangle specifically, the opposite side is only relative to either acute angle; the opposite side to the right angle is called hypotenuse instead.

In a right triangle, the adjacent side is the leg next to the angle in question.

By convention, we label the opposite side using the lowercase letter. So the side opposite A^\hat A is aa.

Uppercase letter for angle is opposite to corresponding lowercase letter for side
Triangle naming convention

Bisectors

Angle bisectors are not medians, even though both connect an angle with the opposite side.

An angle bisector only bisects the angle (evenly divides it by two), but not the side.

A median, which bisects the opposite side, does not bisect the angle.

Median is not angle bisector in triangle XYZ
BZ is the angle bisector to angle Z, while MZ is the median to side XY.

An angle bisector is only a median if and only if it is also an altitude (height), such as in an isosceles triangle, the angle bisector to the angle between the two equal sides is a median and an altitude.