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Kuzas, Richmond, Tocchi – Team 6 – Angle Bisector
Activity I. Definition and Properties of an Angle Bisector
Lesson 1:
(Introduction) Angle Bisector
Paper Folding
Level:
High School Geometry
Strand:
(4) Geometry; Standard 7 – Geometry from a synthetic perspective
Learning Objectives:
1. To draw an angle bisector and to discover its basic properties
Materials:
1 square piece of paper, ruler, protractor, scissors, and a pencil.
Procedure:
1. Fold square piece of paper diagonally:
2. Open the piece of paper and cut along the newly folded diagonal.
3. Take the two corners A & B that make up the new side and fold the paper so that A & B
lie on top of each other to create another fold CQ.
4. Using a ruler, draw a line segment CQ along the new folded line.
5. Taking the protractor, measure the two new angles formed. That is, measure BCQ and
ACQ . What do you notice about the two angles?
The two angles are both 45o.
(With this, you have just formed the angle bisector of CQ of BCA !)
6. At this point, how would you define an angle bisector?
I would define an angle bisector as a line or ray that bisects an angle creating two equal
angles on either side of the bisector.
Kuzas, Richmond, Tocchi – Team 6 – Angle Bisector
7. Draw a point anywhere on line CQ and name it point P.
8. Using the protractor and ruler, form perpendicular line segments from point P to each of
the legs of the right triangle.
9. Measure the new line segments PX and PY .
10. What do you notice about these measurements?
The lines PX and PY have the same length.
11. Chose a different point R on CQ and repeat steps 8-10 for point R. Does your previous
observation remain true?
The newly created lines also have the same length so yes, my observation remained true.
12. What property seems to be true for all the points on the angle bisector of any angle?
Any two perpendicular lines created off of an angle bisector to the rays of an angle will have
the same length.
Kuzas, Richmond, Tocchi – Team 6 – Angle Bisector
Activity II. Definition and Properties of an Angle Bisector
Lab goals: To draw an angle bisector and to discover its basic properties using Cabri II.
Investigation:
1. Draw two rays CQ and CQ that form angle ABC .
[use the ray tool]
2. Draw the angle bisector of ABC .
[use angle bisector tool]
3. Mark a point P anywhere on the bisector in the interior of ABC .
[use point tool]
4. Measure ABP and CBP and label the measurements.
What do you notice about these angles?
[use angle measure tool]
Both angles are 25.5 degrees.
Drag P along the angle bisector. Do the angle measurements change?
The angle measurements do not change.
Grab and move ray AB or BC up and down. What do you notice about the angles?
The angle measurements change, but they always remain equal.
5. Create perpendicular lines from point P to rays AB and BC . Call the intersection point
on ray AB point X and call the intersection point on ray BC point Y.
[use perpendicular tool]
Kuzas, Richmond, Tocchi – Team 6 – Angle Bisector
6. Now make segments from point P to X, and another from P to Y. Hide lines XP and
YP .
[using segment tool and hide/show tool]
They are equal.
7. Grab point P and move it further along the angle bisector line. What do you notice about
the measures of ABP and CRP ?
The angle measurements do not change.
What do you notice about the measure of segments XP and YP ?
The measurements change, but they remain equal to one another.
Grab and move ray AB and BC up and down. What do you notice about the measures
of ABP and CRP ? Do they remain the same? Do they remain equal to each other?
They do not remain the same, but they do remain equal to one another.
What can you conclude about the properties that characterize an angle bisector?
An angle bisector is a line that travels through the vertex and splits the angle into two
equal parts.
Kuzas, Richmond, Tocchi – Team 6 – Angle Bisector
Journal Activity
Angle Bisector
1. List all definitions and properties that you have learned in this activity.
Angle bisector – A line that splits an angle into two equal angles
Perpendicular – a line that forms a right angle with the line that it is drawn to.
2. Can you think of any applications of this topic?
Cutting cake/pizza/anything that is circular really.
3. Can you relate this topic/concepts with other(s) previously studied? Explain your answer.
Not at this point.