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Investigation: Construct congruent angles

Lesson

Compass and straightedge construction

As we have seen, congruent means same, so congruent angles are 2 (or more) angles that are exactly the same size. They can be facing in any direction, but if they are the same size then the angles are congruent. You can think of congruent angles as copies of each other.

To construct congruent angles you need a compass and a straight edge (ruler).

Steps for construction

  1. Start with the angle $BAC$BAC ($\angle BAC$BAC) on the page that you want to copy.  
  2. Draw a ray, $\overrightarrow{PQ}$PQ, on the page.  
  3. Position the compass on point $A$A, adjust to any width.
  4. Draw an arc that crosses both legs of $\angle BAC$BAC, name the points where the arc intersects the rays $F$F and $G$G
  5. Position the compass on point on point $P$P, and draw another arc.  Where it crosses $\overrightarrow{PQ}$PQ call it $M$M.
  6. Measure the distance $FG$FG with the compass.
  7. Place the compass at point $M$M, and cross the arc.  Call this intersection $N$N.
  8. Draw in a ray from $P$P, through $N$N.  Call it $\overrightarrow{PR}$PR
  9. Now you have created $\angle RPQ$RPQ.
  10. $\angle RPQ\cong\angle BAC$RPQBAC

 

Dynamic software construction

We can also construct congruent angles using dynamic geometry software.  Press the pause/play button in the applet below to see the steps of the construction in action.  To test the construction, move the points around and test that the angles remain the same size.

Now it's your turn!  Repeat the steps for construction to construct your own set of congruent angles.  Click here to open the applet in a larger web browser window.

Steps for construction using technology

Move Tool Point Tool Line Tool Line Segment Tool Ray Tool Compass Tool
  1. Use ray tool to draw rays $\overrightarrow{AB}$ABand $\overrightarrow{AC}$AC. Place a third ray $\overrightarrow{DE}$DE anywhere else in the window.
  2. Choose the compass tool, then click on points $A$A and $B$B to set the radius length to be the same as $AB$AB, then click on center $A$A to draw the circle.
  3. Use the compass tool again to draw another circle with radius also $AB$AB and center at $D$D.
  4. Use the point tool to mark point $F$F at the intersection of circle $A$A and $\overrightarrow{AC}$AC.
  5. Mark point $G$G at the intersection of circle $D$D and $\overrightarrow{DE}$DE.
  6. Use the compass tool to construct a circle with radius $BF$BF and center at $G$G.
  7. Use the point tool to mark point $H$H as the intersection of circles $D$D and $G$G.
  8. Draw a ray from $D$D through $H$H.
  9. Use the move tool to drag the points and test that the angles stay the same size.

 

Save your work!

Be sure to save your construction often, especially if you would like to come back to it at a later time.  If you refresh this page before saving, your work will be lost.

Outcomes

GEO-G.CO.12

Make, justify, and apply formal geometric constructions.

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