The Angle Bisectors of a Triangle
Materials Needed: Patty paper, ruler, protractor, colored pencils, compass
Lesson Opener Activity
1. On a sheet of patty paper, use a straightedge to draw any angle (not a straight
angle, but also not less than 50ο). Use most of the space on the patty paper.
2. Fold the paper through the vertex so that one side of the angle lies on top of the
other side. Hint: Pinch the vertex and then crease the fold.
3. Open the paper and draw a dashed line on the crease. Verify that this is the angle
bisector by using a protractor to measure each angle.
4. Draw a point anywhere inside the angle on the angle bisector and label the point A.
5. This is tricky!. Fold a perpendicular line from point A to one side of the angle by
pinching a crease at A and adjusting the ray so that it folds on top of itself. Make
sure to line up the point and the folded ray at the same time.
Alternative Method: Slide the edge of a second patty paper along one side of the
angle until an adjacent perpendicular side of the patty paper passes through point A.
Draw a perpendicular line from A to the side of the angle.
6. Repeat for other side of the angle. The distance from Point A to a side is the
shortest distance when the segment is perpendicular.
7. Measure the distances from Point A to each side of the triangle. What do you notice
about these measurements?
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8. Mark another point on the angle bisector and label it Point B. Repeat steps 5-7.
9. Based on your observations, make a conjecture about the distances from any point
on the angle bisector to the sides of an angle.
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Group Activity (groups of three or four)
1. One student in the group will use a straightedge to draw an acute triangle, one will
draw a right triangle, and one will draw an obtuse triangle on a piece of patty paper.
The fourth student can draw any triangle of his/her choice. Label it ∆ABC.
2. As in the previous activity, fold and crease the angle bisector of A (pinch vertex
A and fold one side of the angle on top of the other).
3. Repeat for the other two vertices. What do you notice about the three creased
lines?
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4. Label the point of intersection of the creased lines D.
5. Using a colored pencil, draw a line on one crease from the vertex to point D. Repeat
for the other two vertices. Verify and label your diagram showing that the angles
are bisected.
6. Using a different colored pencil, draw a perpendicular line from point D to each side
of the triangle (slide a second sheet of patty paper along one side of the triangle
until it intersects with point D as in the previous activity).
7. Measure the distance from point D to each side of the triangle (in centimeters).
8. Compare your results with the other members of your group. What can you conclude
about the point of intersection of the angle bisectors of a triangle?
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9. How is this conjecture related to the Opening Activity?
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10. Compare your triangle with the same type in other groups. Describe the location of
the point of intersection of the angle bisectors on all three triangles.
Acute Triangle __________Right Triangle ________Obtuse Triangle_________
11. How is the point of concurrency of the perpendicular bisectors of a triangle similar
to the point of concurrency of the angle bisectors of a triangle? How are they
different?
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Extension Activity
Use a compass to construct a circle centered at point D with a radius equal to the
distance from intersection of the perpendicular segments to the side of the
triangle. Compare the circle you constructed with your group members. What do you
notice about the relationship between the triangle and the circle you constructed?
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 The point of concurrency of the angle bisectors of a triangle is called
the incenter because it is the center of the circle that is inscribed
within the triangle.