Lesson 7-2: Angle Relationships and Triangles

Property

A triangle is a closed two-dimensional polygon consisting of three line segments (sides) that meet at three points (vertices), creating three interior angles.

Examples

• Given a triangle named $\triangle ABC$, the three vertices are the individual points $A$, $B$, and $C$.
• The three sides are the line segments connecting these vertices, denoted with a bar over the letters: $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$.
• The three interior angles are formed inside the triangle where the sides meet, named using the vertex letter, such as $\angle A$, $\angle B$, and $\angle C$, or with three letters like $\angle ABC$.

Explanation

A triangle is formed when three straight line segments connect three points that do not lie on the same line. These points are called vertices, and they act as the corners of the geometric figure. At each vertex, the two meeting sides form an interior angle on the inside of the triangle. Understanding how to correctly name and identify these vertices, sides, and angles is the foundation for exploring all other triangle properties.

Property

The sum of the angles of a triangle is $180^\circ$.
If you cut out the angles and put all the vertices together, so that the angles are adjacent and not overlapping, you will get a straight angle.

Examples

• A triangle has angles measuring $50^\circ$ and $70^\circ$. The third angle is $180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ$.
• A right triangle has one angle of $90^\circ$. If another angle is $35^\circ$, the third angle must be $180^\circ - 90^\circ - 35^\circ = 55^\circ$.
• An isosceles triangle has two equal angles. If the unique angle is $40^\circ$, the other two angles together are $180^\circ - 40^\circ = 140^\circ$. Each equal angle is $140^\circ / 2 = 70^\circ$.

Explanation

No matter what a triangle looks like—tall, short, wide, or skinny—if you add its three corner angles together, you will always get exactly $180^\circ$. It is a fundamental rule for all triangles!