ADVANCED GEOMETRY LAB: A BEGINNING PROOF!
INTRODUCTION
We all likely recall that the sum of the three angle measures in any triangle is 180 degrees. Let’s confirm
this with Sketchpad.
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Using the segment tool, draw a triangle in Sketchpad.
Using the text tool, label the three points and then click on the arrow tool and then in the open
background.
Measure one of three angles. To do this, recall, you highlight the three points of a given angle
in the same order that you would name the angle. Then go to the Measure menu to measure
the angle. Once you see the angle measure, drag it near to the vertex of your angle. Then click
on the arrow tool and the open background. Repeat for the other two angles.
Now, we’ll use Sketchpad to calculate the sum of the three angle measures. To do this, first
make sure nothing is highlighted by clicking on the arrow tool and the open background. Then,
go to the Measure menu and choose Calculate. You’ll see a calculator pop-up on your screen.
Click on the measurement box for one of your angles, then click “+” on the calculator. Next
highlight a second measurement box, then click “+” on the calculator, and then highlight the
third measurement box and then click “OK” on the calculator. You will see the angle sum
appear on the screen.
Finally, drag any of the three vertices to observe the varying angle measures. However, not
surprisingly, the sum of the three angle measures is indeed 180.
Based on our observations, it appears that indeed we can make the following conjecture:
“The sum of the measures of the three angles in a triangle is always 180
degrees.”
But now, our goal is to write a proof that will EXPLAIN why that is indeed the case!
In other words,
let’s prove this conjecture!
LEADING TO PROOF
Now, we are ready to create a construction in Sketchpad that will help to explain WHY the sum of the
three angle measures in a triangle is always 180 degrees.
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Open a new sketch and create triangle ABC as shown. (This will be considered our GIVEN in the
proof later on).
B
A
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C
Mark a vector from A to C and then translate the entire triangle along the marked vector. Label
it as shown.
B
A
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D
C
E
Next, rotate triangle ABC by 180 degrees about point C to create triangle CEF shown below.
B
A
D
E
C
F
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Next, we wish to take triangle CEF and translate it along vector FC in order to fit it into the space
between triangle ABC and CDE. To do this, mark a vector from point F to point C. Then,
highlight all points and segments of triangle CEF and translate it along the marked vector. Your
image should look as follows:
B
D
E
C
A
F
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Now, all four of these triangles will be congruent, as each is simply an image of the original
triangle (recall that translations and rotations are rigid motion transformations). Drag any
vertex to confirm that each triangle indeed remains congruent to the original. Return to the
orientation shown above. Then, choose the arrow tool and click in the open background so that
nothing is highlighted.
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While triangle CEF is an important piece of this puzzle, we would like to hide it (not delete it).
(While we may feel sorry for triangle CEF for being hidden from here forward, recall we have not
deleted it. Triangle CEF is still there. In fact, it is not clear whether triangle CEF even knows that
it has been hidden. And, actually, part of it will still be seen, as described next.) Highlight
segment CF by clicking on the segment itself, then highlight point F, and then highlight segment
FE (but do NOT highlight points E and C and segment EC – we still want to see those.) Go to the
Display menu and select Hide Objects. The image should look as below.
B
A
D
C
E
Believe it or not, we are now ready to explain why the sum of the three angle measures in a
triangle is 180 degrees! In other words, on to the proof!
ADVANCED GEOMETRY: CONSTRUCTING A PROOF TO PROVE A CONJECTURE
STEP ONE: STATING A CONJECTURE
We already know our conjecture based on our work so far. Namely,
“The sum of the three angle measures of any triangle is 180 degrees.”
STEP TWO: IDENTIFYING THE GIVENS AND WHAT WE NEED TO PROVE.
Given: An arbitrary triangle ABC. (In other words, “Given any triangle ABC that you can
imagine”)
Prove: The sum of the three angle measures in triangle ABC is 180 degrees.
STEP THREE: THE PROOF
See if you can fill in the blanks of this proof below.
Given triangle ABC, we translated the triangle along a vector from point ___ to point ___ to create
triangle ____ as shown.
B
A
D
C
E
We just performed a rigid motion transformation. Hence, we know the two triangles are congruent.
Use good symbolism to write the triangle congruence statement here:
Next, let’s rotate the original triangle ABC about point ____ by _______ degrees to create triangle
____ as shown below.
B
D
E
C
A
F
A rotation is also a rigid motion transformation. Hence, we know the newest triangle in the image
above is congruent to the original triangle ABC. In fact, we can now say all three triangles are
congruent. Write a congruence statement using good symbolism to say that all three triangles are
congruent:
What property allows us to say that all three triangles are congruent? (See Proof Toolbox if
necessary)
Next, let’s translate triangle EFC along a vector from point ___ to point ____ in order to form
triangle _____ as shown below.
B
A
D
E
C
F
Note that we just performed another rigid motion transformation. We now have four congruent
triangles above. Write a congruence statement using good symbolism to say that all four triangles
are congruent.
We will hide triangle EFC. But, be careful not to hide any part of the other three triangles.
Now, take a moment to mark the congruent angles in the diagram below. Since all three triangles
have been shown to be congruent above, every angle in each triangle should be marked in some
way before moving on! There is no need to mark the side congruencies.
B
A
D
C
E
We know angle _____ is a straight angle in the diagram above (make sure that you are using a 3letter system to name the straight angle). Notice that the straight angle is composed of three
angles. Name the three angles here that make up the straight angle:
Let’s write an equation now that is a direct result of the fact that the three angles together make a
straight angle:
 ______ +  _____+  _____ = 180 degrees.
Now, if we can match up each of the three angles that make up the straight angle with each of the
three angles that make up triangle ABC, then we can argue that the sum of the measures of the
angles in triangle ABC is also 180 degrees!
Now, let’s indeed try to match each angle in triangle ABC with a corresponding part of the straight
angle.
 ____ in triangle ABC is congruent to  ______ that is part of the straight angle.
 ____ in triangle ABC is congruent to  ______ that is part of the straight angle.
 ____ in triangle ABC is congruent to  ______ that is part of the straight angle.
Since we know the three angles of the straight angle add to 180, that means we ALSO know
that  _____ +  _____ +  _____ = 180 degrees!
And hence, we have just completed our proof! We started with an arbitrary triangle ABC and then
argued, through a step by step process (each step justified along the way), that the sum of the angle
measures of triangle ABC must equal 180 degrees. How exciting! This proof serves as an
explanation as to WHY we can indeed make this conjecture (rather than just saying “I know this
because my teacher said so” or “It sure looked like it when we dragged stuff around in Sketchpad.”)
Now, let’s use our new proof to help us solve a couple of sample problems:
Sample #1:
Show your work in solving the problem below.
Given: Diagram as shown, and A=41.6
Find mB and mC.
A
B
C
Sample #2:
Show your work in finding the values of m, n, x, y and z below.
80
100
n
y
55
m
x
z
60
Sample Problem #3: Show your work for the problem below:
Given: BE bisects ABC
CE bisects ACB
A=80
Find: mBEA
A
80
E
B
C
Hint: At some point it might be helpful to make use of the real number property: ab + ac = a(b + c).