Lesson 1

Relationships of Angles

Let’s examine some special angles.

1.1: Visualizing Angles

Use the applet to answer the questions.

  1. Which angle is bigger, \(a\) or \(b\)?
  2. Identify an obtuse angle in the diagram. 

1.2: Pattern Block Angles

  1. Look at the different pattern blocks inside the applet. Each block contains either 1 or 2 angles with different degree measures. Which blocks have only 1 unique angle? Which have 2?

  2. If you place three copies of the hexagon together so that one vertex from each hexagon touches the same point, as shown, they fit together without any gaps or overlaps. Use this to figure out the degree measure of the angle inside the hexagon pattern block.

    Three copies of a hexagon drawn so one vertex from each touches the same point. They fit together without gaps or overlaps.
  3. Figure out the degree measure of all of the other angles inside the pattern blocks. (Hint: turn on the grid to help align the pieces.)  Be prepared to explain your reasoning. 

 


We saw that it is possible to fit three copies of a regular hexagon snugly around a point.

Each interior angle of a regular pentagon measures \(108^\circ\). Is it possible to fit copies of a regular pentagon snugly around a point? If yes, how many copies does it take? If not, why not?

A regular pentagon, one angle labeled 108 degrees.

 

1.3: More Pattern Block Angles

  1. Use pattern blocks to determine the measure of each of these angles.

    Two obtuse angles and a straight angle.  Please ask for further assistance.
  2. If an angle has a measure of \(180^\circ\) then the two legs form a straight line. An angle that forms a straight line is called a straight angle. Find as many different combinations of pattern blocks as you can that make a straight angle.

Use the applet if you choose. (Hint: turn on the grid to help align the pieces.) 

 

1.4: Measuring Like This or That

Tyler and Priya were both measuring angle \(TUS\).

A protractor is pictured. An acute angle has T at 0 or 180 degrees, vertex at the center of the protractor, and S at the end of a line through 40 or 140 degrees.
Priya thinks the angle measures 40 degrees. Tyler thinks the angle measures 140 degrees. Do you agree with either of them? Explain your reasoning.

Summary

When two lines intersect and form four equal angles, we call each one a right angle. A right angle measures \(90^\circ\). You can think of a right angle as a quarter turn in one direction or the other.

Two lines meet to form four angles. Each angle is marked 90 degrees, and with the symbol for perpendicular lines.

An angle in which the two sides form a straight line is called a straight angle. A straight angle measures \(180^\circ\). A straight angle can be made by putting right angles together. You can think of a straight angle as a half turn, so that you are facing in the opposite direction after you are done.

A straight line is indicated to be an angle with measure 180 degrees.

If you put two straight angles together, you get an angle that is \(360^\circ\). You can think of this angle as turning all the way around so that you are facing the same direction as when you started the turn.

A circle from a segment, around one endpoint, indicates that the measure is 360 degrees.

When two angles share a side and a vertex, and they don't overlap, we call them adjacent angles.

Glossary Entries

  • adjacent angles

    Adjacent angles share a side and a vertex.

    In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

    Three segments all joined at endpoint B. Point A is to the left of B and segment A B is drawn. Point C is above B and segment C B is drawn. Point D is to the right of B and segment B D is drawn.
  • right angle

    A right angle is half of a straight angle. It measures 90 degrees.

    a right angle
  • straight angle

    A straight angle is an angle that forms a straight line. It measures 180 degrees.

    a 180 degree angle