# Triangles Formulas and Notes

Types Triangles Properties of Triangles Right Angle Triangles Exterior Angles Similarity Congruence Median of a Triangle Altitude of a Triangle

Types of Triangles

Equilateral Triangle

All sides are equal. All angles are equal to 60 °

\displaystyle a = b = c = 60 \degree

Scalene Triangle

Angles are not equal. Sides are not equal.

Isosceles Triangle

Two sides are equal. Two angles are equal.

a = b

Acute Triangle

All angles are less than 90 °

\displaystyle a < 90 \degree ,b < 90 \degree ,c < 90 \degree

Obtuse Triangle

A triangle with an angle greater than 90°

\displaystyle x > 90 \degree

Right Angle Triangle

A triangle with an angle that is equal to 90°

Properties of Triangles

Sum of Angles

Angles in triangle have a sum of 180°

\displaystyle a + b + c = 180 \degree

Area

Area \displaystyle = \frac{1}{2} base × height

Area \displaystyle = \frac{1}{2} bc \sin(A)

\displaystyle = \frac{1}{2} ab \sin(C)

\displaystyle = \frac{1}{2} ac \sin(B)

Right Angle Triangles

Theorem of Pythagoras

\displaystyle a^{2} + b^{2} = c^{2}

\displaystyle \rightarrow c = \sqrt{a^{2} + b^{2}}

\displaystyle \rightarrow a = \sqrt{c^{2} - b^{2}}

\displaystyle \rightarrow b = \sqrt{c^{2} - a^{2}}

Finding Missing Angles and Sides

Exterior Angles

Exterior Angles

\displaystyle c = a + b

Similarity

Similarity

Triangles are similar if they have the same exact shape but not necessarily the same size.

1.Corresponding angles have the same measure.

2.Corresponding sides are proportional to each other.

Angle-Angle-Angle

Triangles are similar if 3

pairs of triangles have

the same measure.

Side-Side-Side

Triangles are similar if all

3 pairs of corresponding

sides are proportional.

Side-Angle-Side

Triangles are similar if

two pairs of corresponding

sides are proportional and the

included angles have the

same measure.

Congruence

Congruence

Triangles are congruent if they have the same exact size and shape.

Side-Side-Side

Triangles are congruent if all

3 pairs of corresponding

sides have the same measure.

Side-Angle-Side

Triangles are congruent if

two pairs of corresponding

sides have the same measure and the

included angles have the

same measure.

Angle-Side-Angle

Triangles are congruent if

two pairs of corresponding

angles have the same measure and the

included sides have the

same measure.

Right Angle Hypotenuse

Triangles are congruent if

the corresponding hypotenuses

have the same measure and a

pair of other sides have the

same measure.

Median of a Triangle

Median of a Triangle

A median of a triangle is a line segment joining a vertex to the opposite side, bisecting it.

Altitude of a Triangle

Altitude of a Triangle

A median of a triangle is a line segment through a vertex and perpendicular to a line containing the opposite side.