High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize High School Geometry Textbook (Common Core) > Foundations of Geometry

Angles

Length of a Line SegmentPrevious Section
PostulatesNext Section
Popular Courses
Find My Course
0:00 / 0:00

Angles

An angle is when two different rays share a common endpoint. We call the endpoint the vertex of the angle, and the rays are the sides of the angle. We can describe an angle using a variety of notations.
  • ∠B\angle BUsing the vertex
  • ∠ABC\angle ABCUsing the vertex and points on the rays. Note the vertex goes between the points.
  • ∠3\angle 3Using a number
The interior of an angle is all of the points that live between the two sides. The exterior of an angle is all of the points that live out side of the two sides.

Constructing Angles


To construct an angle we can use a marking tool and a straightedge.
  1. Mark out three points that are not in a strait line.
  2. Draw a ray through two points at a time.
  3. The common point will be the vertex.
Example 1
Construct an angle.

ANSWER


0:00 / 0:00

Measuring Angles


To measure an angle we can use a protractor.
This tool commonly looks like a half circle and has markings along the curved side much like a ruler.
The units we use for measuring an angle are usually degrees or radians.

Measure and angle using a protractor

  1. Align the vertex of the angle with the origin of the protractor.
  2. Record the numbers where the rays pass through
  3. Take the difference of these numbers
  4. Take their absolute value
When talking about he measure of an angle, we can put a small mm next to the notation.
To write that the measure of angle A is 30 degrees we write: m∠A=30∘m \angle A = 30^\circ

Special Angles

  • Acute angles have a measure less than 90 degrees
  • Right angles are exactly 90 degrees
  • Obtuse angles have a measure more than 90 degrees, but less than 180
  • Straight angles are exactly 180 degrees

Congruent Angles

We say that two angles are congruent if they have exactly the same measurement.
Angle A is congruent to angle B is written as ∠A≅∠B\angle A \cong \angle B

When drawing congruent angles we can use small arcs on the interior of the congruent angles.
0:00 / 0:00

Constructing congruent angles

Our goal in this construction is to create a new angle that is congruent to a given angle.

Using a compass and straightedge

Begin with an angle whose vertex is marked as AA.
  1. Build a ray to form the first side of your new angle. Label the endpoint as BB.
  2. Use a compass to mark out an arc on the interior of your original and new angle. Label the intersections as points C,D,C, D, and EE.
  3. Adjust the compass so that it is the same as the distance between CCand DD.
  4. Mark out an arc with this distance on the new angle. Label the intersection as FF.
  5. Use a straightedge to make the ray BF→\overrightarrow{BF}ray
From this we now have ∠DAC≅∠FBE\angle DAC \cong \angle FBE

Using Technology


Begin with an angle whose vertex is marked as AA.
  1. Build a ray to form the first side of your new angle. Label the endpoint as BB.
  2. Place a point along one side of the original angle. Label this as point CC.
  3. Use the compass tool to make a circle with a radius the length of AC‾\overline{AC} , centered at AA. Label the point where it intersects the other side as DD.
  4. Use the compass tool to make a circle with a radius the length of AC‾\overline{AC}, centered at BB. Label the point where it intersects the ray as EE.
  5. Use the compass tool to make a circle with a radius the length of CD‾\overline{CD}, centered at EE. Label the point where the two circles intersect as FF.
  6. Use the ray tool to draw a ray starting at BB, and through FF.
From this we now have ∠DAC≅∠FBE\angle DAC \cong \angle FBE
0:00 / 0:00

Astronomy and Math



A radio telescope scans the sky, looking for signals produced by things like the formation of stars or even black holes. Use the diagram and information below about a particular radio telescope to answer the questions.


Over the course of one night, this radio telescope can sweep out an angle of 84 degrees. This is marked as angle ∠ABC\angle{ABC}. The angle ∠ABM\angle{ABM} is 8 more than 3 times the angle of ∠MBC\angle{MBC}.

What is the measure of angles ∠ABM\angle{ABM}, and ∠MBC\angle{MBC}?

What kind of angles are these?

ANSWER: ∠ABM=65∘,∠MBC=19∘\angle{ABM} = 65^\circ, \\\angle{MBC} = 19^\circ

These are both less than 90∘90^\circ, so we call them acute angles.

Starting with the total of both angles we have

m∠ABM+m∠MBC=m∠ABCm∠ABM+m∠MBC=84∘\begin{aligned} m\angle{ABM} + m\angle{MBC} &= m\angle{ABC} \\ m\angle{ABM} + m\angle{MBC} &= 84^\circ \end{aligned}

Now from the given information we know that

m∠ABM=8+3(m∠MBC)\begin{aligned} m\angle{ABM} = 8 + 3(m\angle{MBC}) \end{aligned}
Substituting this in we have

8+3(m∠MBC)+m∠MBC=84∘8+4(m∠MBC)=84∘4(m∠MBC)=76∘m∠MBC=19∘\begin{aligned} 8 + 3(m\angle{MBC}) + m\angle{MBC} &= 84^\circ \\ 8 + 4(m\angle{MBC}) &= 84^\circ \\ 4(m\angle{MBC}) &= 76^\circ \\ m\angle{MBC} &= 19^\circ \\ \end{aligned}
Again we can substitute this to find the other angle.

m∠ABM=8+3(m∠MBC)m∠ABM=8+3(19)m∠ABM=65∘\begin{aligned} m\angle{ABM} &= 8 + 3(m\angle{MBC}) \\ m\angle{ABM} &= 8 + 3(19) \\ m\angle{ABM} &= 65^\circ \end{aligned}

Identifying Angles


Use the diagram to match the angle with a type of angle.


A.
Acute angle
B.
Side of angle
C.
Obtuse angle
D.
Straight angle
E.
Congruent angles
∠AMB\angle AMB
∠AMD\angle AMD
MB→\overrightarrow {MB}
∠BMD\angle BMD
the pair ∠AMB\angle AMB and ∠CMD\angle CMD

Angles and Algebra

Use the diagram below, and the fact that ∠ABD=138∘\angle ABD = 138^\circ to answer the questions.


1. What is the value of x ?

2. What is the measure of ∠ABC\angle ABC?

3. What is the measure of ∠CBD\angle CBD?



1. What is the value of x ?
0:00 / 0:00

Pairs of Angles


Certain pairs of angles can have special properties. To better identify these pairs we can give them names.

Adjacent Angles

Adjacent angles share a common vertex and a side.
If angles are not adjacent we simply call them nonadjacent angles.

Special Pairs

The total measure of complementary angles is 90 degrees.

The total measure of supplementary angles is 180 degrees.

A linear pair of angles are adjacent angles such that the sides that is not shared forms a strait line.

Vertical angles form two pairs of opposite rays.


Watch Out!
Complementary and supplementary angles do not need to be adjacent to one another. They key is that their measure adds to 90 or 180 degrees.

Example
Draw an example of
  • a pair of angles that are adjacent, and supplementary.
  • a pair of angles that are not adjacent, and complementary.
  • vertical angles
ANSWER
0:00 / 0:00

Constructing an angle bisector

Our goal in this construction is to split a given angle into two congruent angles. In this way we bisect the original angle.

Using a compass and straightedge



Begin with an angle whose vertex is labeled AA.
  1. Using the compass, draw an arc on the inside of the angle. Label the points where the arc intersects the sides of the angle as AA and CC.
  2. Adjust the compass so it is a little bit wider than half the distance of BE‾\overline{BE}.
  3. Put the compass at BB, and make an arc.
  4. Put the compass at CC, and make an arc. Label where these two arcs cross as point DD.
  5. Draw a ray starting at AA, and through point DD.
From this we now have that ∠BAD≅∠DAC\angle BAD \cong \angle DAC

Using Technology


Begin with an angle whose vertex is labeled AA.
  1. Using the circle tool, draw a circle centered at AA . Label where this circle intersects the sides of the angle as BB and CC.
  2. Using the circle too, draw a circle centered at BB, whose radius is a bit more than half the length of BC‾\overline{BC}. Label the point where it intersects the first circle inside the angle as point EE.
  3. Use the compass tool to make a circle with a radius the length of BE‾\overline{BE}, centered at CC. Label where these two circles cross as point DD.
  4. Draw a ray starting at AA, and through point DD.
From this we now have that ∠BAD≅∠DAC\angle BAD \cong \angle DAC
0:00 / 0:00

Bridge Design



The beams on a bridge are build in a way to help distribute the load placed upon it, and to help the bridge keep it shape.

Use the diagram about one of these bridges, and the information given to answer the questions.


Given that
  • BE→\overrightarrow{BE} bisects ∠ABC\angle{ABC}
  • BD→\overrightarrow{BD} bisects ∠ABE\angle{ABE}
  • The angles ∠CBE\angle{CBE}and ∠BCE\angle{BCE}are congruent
  • The sum of the angles ∠CBE,∠BEC,\angle{CBE}, \angle{BEC}, and ∠ECB\angle{ECB} is 180∘180^\circ
  • m∠BEC=32∘m\angle{BEC} = 32^\circ
Find the measure of angle ∠ABD\angle{ABD}.

ANSWER: ∠ABD=37∘\angle{ABD} = 37^\circ

Since the sum of the angles ∠CBE,∠BEC,\angle{CBE}, \angle{BEC}, and ∠ECB\angle{ECB} is 180∘180^\circ
We can start with

m∠CBE+m∠BEC+m∠ECB=180∘m\angle{CBE} + m\angle{BEC} + m\angle{ECB} = 180^\circ

Substituting in m∠BEC=32∘m\angle{BEC} = 32^\circ, and the fact that ∠CBE≅∠BCE\angle{CBE} \cong \angle{BCE}

m∠CBE+m∠BEC+m∠ECB=180∘m∠CBE+m∠CBE+32∘=180∘2(m∠CBE)+32∘=180∘2(m∠CBE)=148∘m∠CBE=74∘\begin{aligned} m\angle{CBE} + m\angle{BEC} + m\angle{ECB} &= 180^\circ \\ m\angle{CBE} + m\angle{CBE} + 32^\circ &= 180^\circ \\ 2(m\angle{CBE}) + 32^\circ &= 180^\circ \\ 2(m\angle{CBE}) &= 148^\circ \\ m\angle{CBE} &= 74^\circ \end{aligned}

Since BE→\overrightarrow{BE} bisects ∠ABC\angle{ABC}, we have ∠CBE≅∠ABE\angle{CBE} \cong \angle{ABE}, so m∠ABE=74∘m\angle{ABE} = 74^\circ.

Since BD→\overrightarrow{BD} bisects ∠ABE\angle{ABE}, we have that ∠ABD=12(74∘)=37∘\angle{ABD} = \displaystyle\frac{1}{2}(74^\circ) = 37^\circ.

Game of pool


In the game of pool, players hit colorful balls around the table hoping to eventually strike them into a hole along the side.

When a pool ball hits the side of the table, it bounces off at an angle that is congruent to the angle it struck the side as seen in the diagram.

Use this information and the diagram to match the angles with their best description.

A.
Supplementary angles
B.
Complimentary angles
C.
Adjacent angles
D.
Linear Pair
∠AFB\angle AFB and ∠BFE\angle BFE
∠BFC\angle BFC and ∠CFD\angle CFD
∠DFE\angle DFE and ∠CFD\angle CFD
∠AFB\angle AFB and ∠AFD\angle AFD

Airplane Wings


The sweep of an airplane wing can be modeling using various angles.

In the diagram we have

  • ∠ABC=60∘\angle ABC = 60^\circ
  • ∠ABC≅∠DCB\angle ABC \cong \angle DCB
  • ∠CDE\angle CDE and ∠DEB\angle DEB are supplementary
  • ∠DEB\angle DEB and ∠DEF\angle DEF form a linear pair
  • The measures of ∠DCB\angle DCB and ∠CDE\angle CDE sum to 203∘203^\circ
Use this information to find the measure of angle ∠DEF\angle DEF.