Justifying Your Thinking
PART 1
1. Using your piece of patty paper and a straight-edge, draw a line segment that is NOT parallel to
the edge of the patty paper.
2. Construct the perpendicular bisector of your line segment. Write down (in words and diagrams)
your procedure for doing this.
3. Write a convincing mathematical explanation justifying how you know that the object you have
constructed is truly a perpendicular bisector.
4. Pair up with your partner and compare your constructions and your justifications.
5. Mark a point on your perpendicular bisector and use it, along with the endpoints of your original
line segment as the three vertices of a triangle (draw the connecting line segments to show the
triangle). What kind of triangle have you drawn? Are any of its angles congruent to each other?
6. Write a convincing mathematical explanation justifying your answers to #5.
PART 2
1. Using a new piece of patty paper and a straight-edge, draw a pair of intersecting, nonperpendicular line segments.
2. Experiment with paper folding to see if you can develop an argument proving that vertical
angles are congruent.
3. Write a convincing mathematical explanation justifying how you know that two vertical angles
are congruent. Be prepared to share your explanation with the class.
PART 3
1. Using the same piece of patty paper as in part 2, construct a line parallel to one of your line
segments.
2. Write a convincing mathematical explanation justifying how you know that the line you’ve
constructed is in fact parallel.
3. Pair up with your partner and compare your constructions and justifications. Be prepared to
share your explanations with the class.
PART 4
1. Using a new piece of patty paper, draw a line segment, ̅̅̅̅
𝐴𝐵, and a point C not on ̅̅̅̅
𝐴𝐵.
Connect C to A and to B with line segments so that you have a triangle, ∆ABC. By folding,
̅̅̅̅ through C.
construct a line parallel to 𝐴𝐵
2. Write a convincing mathematical explanation justifying how you know that the sum of the
measures of the angles in a triangle are equal to 180°--the equivalent of a straight line.
PART 5
1. Using a new piece of patty paper, draw a scalene triangle (a triangle with all three sides of
different length). By folding, construct the perpendicular bisector of all three sides of the
triangle. Do they meet in a single point? Is this always the case? How can you know for
sure?
2. Write a convincing mathematical explanation justifying how you know if and when the three
perpendicular bisectors meet in a single point.
3. (Extension) Try other sets of constructions (use a new fresh patty paper if needed).
a. Construct all three angle bisectors of the triangle.
b. Construct all three medians of the triangle.
4. (Extension) Explore all three constructions (perpendicular bisectors, angle bisectors,
medians) with measuring tools—ruler, protractor, or dynamic geometry software and write
down any conjectures that you discover.