1. Geometry Basics

Lesson

Suppose we draw a straight path from a single point and extend that path though another point and keep going forever. The path we've made is called a ray.

If we want to talk about the ray in the diagram above, we would refer to the ray as $\overrightarrow{AB}$›‹`A``B`or ray $AB$`A``B` (with the endpoint first) no matter what direction the ray continues.

Ray

A ray is the set of all points on a line bounded by an endpoint and extending to infinity in one direction.

If we start at the same point as before but draw a straight path in the exact opposite direction forever, we've created an opposite ray.

Can you guess what two opposite rays create?

Opposite rays

Opposite rays are rays with the same endpoint that extend in opposite directions; opposite rays form a line.

According to the diagram, which of the following rays contain the same points as $\overrightarrow{FD}$›‹`F``D`?

Select all that apply.

$\overrightarrow{FB}$›‹

`F``B`A$\overrightarrow{DG}$›‹

`D``G`B$\overrightarrow{DF}$›‹

`D``F`C$\overrightarrow{FE}$›‹

`F``E`D$\overrightarrow{FB}$›‹

`F``B`A$\overrightarrow{DG}$›‹

`D``G`B$\overrightarrow{DF}$›‹

`D``F`C$\overrightarrow{FE}$›‹

`F``E`D

If we have two rays that share a common endpoint, we can call the space between the two rays an angle.

Let's look at the angle in the diagram below:

We can refer to this angle as either $\angle AOB$∠`A``O``B` or $\angle BOA$∠`B``O``A` since the path is the same in both directions.

Notice the rays $\overrightarrow{OA}$›‹`O``A` and $\overrightarrow{OB}$›‹`O``B` intersect at their common endpoint, point $O$`O`. We can refer to the rays as the legs of the angle. Point $O$`O` is the angle vertex.

Angle

The space between two rays (called legs) that share a common endpoint (the angle vertex)

To avoid confusion, we'll only refer to the space between the two rays as an angle itself. We'll call the space outside the two rays the reflex of the angle.

The reflex of an angle

The space outside two rays that share a common endpoint.

Using the diagram below, state the vertex and name the marked angle using three points.

The vertex is point $\editable{}$

The angle is $\angle\editable{}$∠

An angle is named $\angle RST$∠`R``S``T`.

What point is the vertex of this angle?

Point $\editable{}$

Which two rays are the legs of the angle?

$\overrightarrow{ST}$›‹

`S``T`A$\overrightarrow{TR}$›‹

`T``R`B$\overrightarrow{SR}$›‹

`S``R`C$\overrightarrow{RS}$›‹

`R``S`D$\overrightarrow{ST}$›‹

`S``T`A$\overrightarrow{TR}$›‹

`T``R`B$\overrightarrow{SR}$›‹

`S``R`C$\overrightarrow{RS}$›‹

`R``S`D

Let's define the space between two opposite rays as a straight angle. Let's also say that it measures $180^\circ$180°. We can then say the following:

- Every angle smaller than a straight angle has a value between $0^\circ$0° and $180^\circ$180°.
- Every angle larger than a straight angle is the reflex of an angle between $0^\circ$0° and $180^\circ$180°.

To visualize these statements, we can use the applet below. Draw point $A$`A` around and notice the measurements of the angle and its reflex.

In other words, every angle has a measure (m) from $0^\circ$0° to $180^\circ$180° and a reflex from $180^\circ$180° to $360^\circ$360°. Since we agree on this statement moving forward, we will formalize it in the protractor postulate.

Protractor postulate

Every angle has a measure ($m$`m`) from $0^\circ$0° to $180^\circ$180°.

If $\angle AOB$∠`A``O``B` is an angle, then $0^\circ`m`∠`A``O``B`≤180°.

Let's review some descriptors of angles based on their measures.

Descriptor | Angle Measure |
---|---|

Acute | Between $0^\circ$0° and $90^\circ$90° |

Right | Exactly $90^\circ$90° |

Obtuse | Between $90^\circ$90° and $180^\circ$180° |

Straight | Exactly $180^\circ$180° |

It's important to note that a right angle can be marked in a diagram with a square.

Classify these angles as acute, right, obtuse or straight based on their measure:

Straight

AAcute

BRight

CObtuse

DStraight

AAcute

BRight

CObtuse

D

Classify $\angle ABC$∠`A``B``C` by its measure in each case.

If $m\angle ABC=$

`m`∠`A``B``C`= $143^\circ$143°, then $\angle ABC$∠`A``B``C`is:Obtuse

AAcute

BStaight

CRight

DObtuse

AAcute

BStaight

CRight

DIf $m\angle ABC=$

`m`∠`A``B``C`= $113^\circ$113°, then $\angle ABC$∠`A``B``C`is:Obtuse

AStraight

BAcute

CRight

DObtuse

AStraight

BAcute

CRight

DIf $m\angle ABC=$

`m`∠`A``B``C`= $180^\circ$180°, then $\angle ABC$∠`A``B``C`is:Straight

ARight

BAcute

CObtuse

DStraight

ARight

BAcute

CObtuse

D

We say that two angles are congruent if they have equal measures. In a diagram, angles that are congruent have the same markings.

Congruent angles

Two angles are congruent if and only if they have equal measures.

For example, if $m\angle A=m\angle B$`m`∠`A`=`m`∠`B` then $\angle A\cong\angle B$∠`A`≅∠`B` and vice versa.

Congruent angles have the same markings in a diagram.

Consider the diagram below.

Solve for $x$

`x`.

Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane.