Lines and Angles

Understanding Angles

An angle is formed by two rays that meet at the same endpoint. Angles are denoted by the symbol $\angle$. In an angle, we recognize the angle arms, the vertex, and the angle region.

In the picture, the arms of the angle are rays $OA$ and $OB$. Point $O$ is the vertex, while the shaded area is the angle region.

Notes:

• the vertex is the intersection point of the two angle arms;
• the angle region is the area bounded by the two arms.

Angle Measures

Degrees are the most common unit for measuring angles. One full turn is $360^\circ$. If a full turn is divided into 360 equal parts, each part measures $1^\circ$.

Smaller units than degrees are minutes and seconds:

Conversion Example

Find the equivalent values:

1. $3^\circ= \ldots '= \ldots ''$
2. $7200''= \ldots '= \ldots ^\circ$

Answer:

Addition and Subtraction of Angles

Find the results:

1. $12^\circ 25' + 13^\circ 11'$
2. $35^\circ 11' - 21^\circ 10'$

Answer:

Drawing and Naming Angles

Drawing Angles

To draw an angle with a known measure, we can use a ruler and a protractor.

A protractor has two scales:

• if the angle is measured clockwise, use the outer scale;
• if the angle is measured counterclockwise, use the inner scale.

Example: draw an angle of $70^\circ$.

Steps:

1. Draw a straight line $AB$.
2. Place the protractor on line $AB$ with its center exactly at point $A$.

3. Mark point $C$ at $70^\circ$.
4. Connect points $A$ and $C$ to form the $70^\circ$ angle.

Angle Notation and Names

Angles are usually named in three ways:

1. using symbols such as $\alpha$, $\beta$, or $\theta$;
2. using the name of the vertex;
3. using three capital letters, with the vertex letter in the middle.

The angle in the figure may be named $\angle \alpha$, $\angle A$, $\angle BAC$, or $\angle CAB$.

Types of Angles

Based on their measure, angles are divided into:

1. acute angles, less than $90^\circ$;
2. right angles, exactly $90^\circ$;
3. obtuse angles, more than $90^\circ$ and less than $180^\circ$;
4. straight angles, exactly $180^\circ$;
5. reflex angles, more than $180^\circ$ and less than $360^\circ$.

Relationships Between Angles

Complementary Angles

The sum of two complementary angles is $90^\circ$.

Supplementary Angles

The sum of two supplementary angles is $180^\circ$.

Vertical Angles

If two lines intersect, the opposite angles are called vertical angles. Their measures are equal.

Example Problems

1. Observe the following figure.

Find:

• the value of $x$;
• the measure of $\angle POR$.

Answer:

2. Observe the following figure.

The supplementary angle of $\angle KLN$ is:

3. In the following figure, $\angle LOM=100^\circ$. Find $\angle KOL$ and $\angle KON$.

Since $\angle KOL$ and $\angle LOM$ are supplementary:

Since $\angle KON$ and $\angle LOM$ are vertical angles:

Lines

Figure (a) is a line segment $AB$, written $\overline{AB}$. If end $B$ is extended, we get ray $AB$, written $\overrightarrow{AB}$. If both ends are extended indefinitely, we get line $AB$, written $\overleftrightarrow{AB}$.

Position of Two Lines

There are four possible positions of two lines: parallel, intersecting, coinciding, and skew.

Parallel Lines

Line $a$ is parallel to line $b$, written $a \parallel b$.

Note: two or more lines are parallel if they lie in the same plane and never intersect even when extended.

Intersecting Lines

Two lines intersect if they have exactly one common point.

Coinciding Lines

Two lines coincide if they lie on the same straight line and appear as a single line.

Skew Lines

Skew lines only appear in three-dimensional figures. They do not lie on the same plane and do not intersect even when extended.

Properties of Parallel Lines

An axiom is a statement accepted as true without proof.

One example of an axiom:

• through a point outside a line, exactly one line can be drawn parallel to that line.

Some properties of parallel lines:

1. If a line intersects one of two parallel lines, it also intersects the other.
2. If a line is parallel to two other lines, then those two lines are parallel to each other.

Relationship Between Lines and Angles

In the figure below, line $m$ intersects two parallel lines, $k$ and $l$. Line $m$ is called a transversal.

At point $P$ there are four angles: $\angle P_1$, $\angle P_2$, $\angle P_3$, and $\angle P_4$. At point $Q$ there are also four angles: $\angle Q_1$, $\angle Q_2$, $\angle Q_3$, and $\angle Q_4$.

Corresponding Angles

Each pair of corresponding angles has the same measure:

• $\angle P_1=\angle Q_1$
• $\angle P_2=\angle Q_2$
• $\angle P_3=\angle Q_3$
• $\angle P_4=\angle Q_4$

Alternate Interior and Alternate Exterior Angles

Each pair of alternate interior angles is equal:

• $\angle P_3=\angle Q_1$
• $\angle P_4=\angle Q_2$

Each pair of alternate exterior angles is also equal:

• $\angle P_1=\angle Q_3$
• $\angle P_2=\angle Q_4$

GeoGebra simulation

Same-Side Interior and Same-Side Exterior Angles

Each pair of same-side interior angles sums to $180^\circ$:

• $\angle P_3+\angle Q_2=180^\circ$
• $\angle P_4+\angle Q_1=180^\circ$

Each pair of same-side exterior angles also sums to $180^\circ$:

• $\angle P_1+\angle Q_4=180^\circ$
• $\angle P_2+\angle Q_3=180^\circ$

GeoGebra simulation

Closing

That concludes the material about lines and angles. If you notice any mistake or want to add a suggestion, please leave a comment.