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8. Geometry Basics
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Math I

8.05 Review: Angles and angle measure

Lesson
Practice
Lesson

Identifying rays

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.

A ray from endpoint $A$A through point $B$B

If we want to talk about the ray in the diagram above, we would refer to the ray as $\overrightarrow{AB}$›‹ABor ray $AB$AB  (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.

$\overrightarrow{AC}$›‹ACand $\overrightarrow{AB}$›‹AB are opposite rays

 

Practice question

Question 1

 

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

Two straight lines that intersect each other at point E. The first line runs from lower left to upper right with points labeled F at the lower end, then intersecting at point E, passing through point D, and continuing through point B at the upper end. The second line is a vertical line runs from bottom to top with points labeled G at the lower end, intersecting at point E, passing through point C, then and continuing through point A at the upper end. Points F, E, D, and B are aligned along one line, while points A, C, E, and G are aligned along the other, showing the collinear nature of the points on each individual line. The two lines have arrows at both ends, indicating that they extend indefinitely.
$\text{Two straight lines that cross each other, at point E. The first line runs from lower left to upper right with points labeled F at the lower end, then intersecting at point E, passing through point D, and continuing through point B at the upper end. The second line is a vertical line runs from bottom to top with points labeled G at the lower end, intersecting at point E, passing through point C, then and continuing through point A at the upper end. Points F, E, D, and B are aligned along one line, while points A, C, E, and G are aligned along the other, showing the collinear nature of the points on each individual line. The two lines have arrows at both ends, indicating that they extend indefinitely.}$Two straight lines that cross each other, at point E. The first line runs from lower left to upper right with points labeled F at the lower end, then intersecting at point E, passing through point D, and continuing through point B at the upper end. The second line is a vertical line runs from bottom to top with points labeled G at the lower end, intersecting at point E, passing through point C, then and continuing through point A at the upper end. Points F, E, D, and B are aligned along one line, while points A, C, E, and G are aligned along the other, showing the collinear nature of the points on each individual line. The two lines have arrows at both ends, indicating that they extend indefinitely.

  1. Select all that apply.

    $\overrightarrow{FB}$›‹FB

    A

    $\overrightarrow{DG}$›‹DG

    B

    $\overrightarrow{DF}$›‹DF

    C

    $\overrightarrow{FE}$›‹FE

    D

 

Identifying angles

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:

$\angle AOB$∠AOB or $\angle BOA$∠BOA

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

Notice the rays $\overrightarrow{OA}$›‹OA and $\overrightarrow{OB}$›‹OB 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 $\angle AOB$∠AOB

The reflex of an angle

The space outside two rays that share a common endpoint.

 

Practice questions

Question 2

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

  1. The vertex is point $\editable{}$

  2. The angle is $\angle\editable{}$∠

Question 3

An angle is named $\angle RST$∠RST.

  1. What point is the vertex of this angle?

    Point $\editable{}$

  2. Which two rays are the legs of the angle?

    $\overrightarrow{ST}$›‹ST

    A

    $\overrightarrow{TR}$›‹TR

    B

    $\overrightarrow{SR}$›‹SR

    C

    $\overrightarrow{RS}$›‹RS

    D

 

Measuring angles

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.

Created with Geogebra

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 $\angle AOB$∠AOB is an angle, then $0^\circ0°<m∠AOB≤180°.

 

Angle descriptors

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.

 

Practice questions

Question 4

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

  1. An angle measuring $44^\circ$44° as labeled.

    Straight

    A

    Acute

    B

    Right

    C

    Obtuse

    D

Question 5

Classify $\angle ABC$∠ABC by its measure in each case.

  1. If $m\angle ABC=$m∠ABC= $143^\circ$143°, then $\angle ABC$∠ABC is:

    Obtuse

    A

    Acute

    B

    Staight

    C

    Right

    D

  2. If $m\angle ABC=$m∠ABC= $113^\circ$113°, then $\angle ABC$∠ABC is:

    Obtuse

    A

    Straight

    B

    Acute

    C

    Right

    D

  3. If $m\angle ABC=$m∠ABC= $180^\circ$180°, then $\angle ABC$∠ABC is:

    Straight

    A

    Right

    B

    Acute

    C

    Obtuse

    D

 

Identifying congruent angles

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.

 

Practice question

Question 6

Consider the diagram below.

Two separate angles with double-arc indicator are shown. One angle is measured $152^\circ$152°, and the other angle is labeled with "$\left(x\right)$(x)".

  1. Solve for $x$x.

 

Outcomes

I.G.CO.1

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

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