Vertical Angles and Linear Pairs Using Geometer Sketchpad

Purpose:
To discover the relationships between pairs of vertical angles and between linear pairs.

Procedure:
1. Select the Line tool and construct Line AB.
2. Construct another line, making sure it intersects Line AB between Point A and Point B, and that Point C and Point D are on opposite sides of Line AB.
3. Select the Arrow tool and highlight both lines.
4. Select Construct, then Point of Intersection.
5. Your figure should now resemble the figure shown above.
6. We now want to measure some angles. To measure AEC, you select the Arrow tool, then select those three points in that order- A, E, C, - with the vertex letter of the angle in the middle. Once all three points are highlighted, select Measure, then Angle. Record this measurement in the space provided for that angle for intersection 1 in the chart on the next page.
7. Repeat this procedure to measure the following angles: <CEB, <BED, and <DEA. Record these measurements in the chart for intersection 1.
8. Click and hold on either Point A, Point B, Point C, or Point D and drag it to a new location in order to create a new intersection. The only restriction to your movement is that Points A and B must remain on opposite sides of line CD and Points C and D must remain on opposite sides of Line AB, and that Point E remain between the other four points.
9. Your measurements will change as you move the point. Record these new measurements for intersection 2.
10. Repeat this process to complete the chart.

Exploration:
1. There are two pairs of vertical angles showing on the screen. Please identify these two pairs.
________________ and _________________
2. What do you notice about each pair of vertical angles for each intersection?_______________________________________________________________
3. There are four linear pairs of angles on the screen. Please identify all four.
__________________ and ___________________
__________________ and ___________________
__________________ and ___________________
__________________ and ___________________
4. What do you notice about the sum of the measures for each linear pair at each new intersection?
_______________________________________________________________________________

Conjectures:
1. If two angles are vertical angles, then _________________________________________________ _______________________________________________________________________________.
2. If two angles form a linear pair, then __________________________________________________ _______________________________________________________________________________.

Developed by Jim Austin, a member of the Geometry for All Network for the 1998 Kentucky Christa McAuliffe Fellowhship, Lisa Willian, fellow. Permission granted for classroom use only.
©Geometer Sketchpad is copyrighted software available from Key Curriculum Press, Berkeley, CA.