1. Using Proofs to Create Theorems

2. Understand the “Flow” of Logic
If you prove a fact for a category of objects, then you prove something for every object in that category. We will use the “Is A” chart (next slide) to help us know which quadrilaterals are also another kind of quadrilateral.
In this project, we will look at trapezoids and decide what we can prove about them (very little). Then we will prove facts for parallelograms.
Next we will prove which quadrilaterals are also parallelograms and look at what additional facts we can prove.
It will then be time to look at kites and isosceles trapezoids.
And then finally, we can look at symmetry and how to prove it.

3. The “is a” chart Isosceles Trapezoid Trapezoid Parallelogram Kite
Rectangle
Rhombus
Square

4. BUT the “Is A” Chart flows from specific to general
BUT the “Is A” Chart flows from specific to general. A Parallelogram “Is A” Trapezoid
Trapezoid
Parallelogram
The “Proves” Chart flows in the opposite direction, from general to specific. Prove something about Trapezoids and you have proven something about Parallelograms.
Trapezoid
Parallelogram

5. Proves chart Isosceles Trapezoid Trapezoid Parallelogram Kite
Rectangle
Rhombus
Square

6. Proof Tool Kit SSS AAS ASA SAS
Before we go further, let’s take a moment to remember what we know about congruent triangles.
SSS
AAS
ASA
SAS

7. CPCTC means means means means
Don’t forget that Corresponding Parts of Congruent Triangles are Congruent.
means
means
means
means

8. Trapezoids Trapezoids have one pair of parallel sides.
What can you say about consecutive interior angles when the lines are parallel?

9. Parallelogram Conjectures about Parallelograms:
The adjacent angles are supplementary. Hint: What did we just prove about trapezoids?
Opposite sides are congruent.
Opposite angles are congruent.
Diagonals bisect each other.
We can use the same proof to prove opposite sides and angles are congruent with our old friend CPCTC. Then we can prove that the diagonals bisect each other.

10. What is Congruent?
/ADB is congruent to /CBD. Why? Can you do the same trick for /ABD & /CDB?
How does that help prove that ΔADB is congruent ΔCBD?
Remember, parallel lines mean flip the triangles. Why?

11. So now we have congruent triangles… It is time for CPCTC!
With CPCTC, we have opposite angles and sides.

12. We are ready to prove bisecting diagonals.
Bisect: Cut into congruent parts.
Pick a pair of triangles, you do not need all four.
Use what you have just proved! Can you find a pair of congruent sides? Is there a pair of vertical angles? Don’t forget the alternate interior angles!
Remember, parallel lines mean flip the triangles.
Again, CPCTC finishes the proof.

13. Is a rectangle a parallelogram?
Remember, rectangles have 4 congruent angles. By the way, the angles are _______?
What do we know about lines when the consecutive angles are supplementary?

14. Do Rectangles have congruent Diagonals?
First draw the diagonals.
Now separate the triangles.
Use what you know about rectangles (4 congruent angles) and parallelograms (opposite congruent sides). Don’t forget the reflexive property!

15. A D B C Is a rhombus a parallelogram?
A rhombus has 4 congruent sides. By now you should be able to prove that ΔADC is congruent to ΔCBA. Note the flip…
What can you say about lines when the alternate interior angles are congruent? A D B C
EXTRA CREDIT: Use this proof to show that congruent opposite sides mean parallelogram.

16. A D B C What else can we say about rhombi?
If a rhombus is a parallelogram, we ALREADY know everything is true that is true about parallelograms. What about perpendicular diagonals? We need to prove that about kites too. If we prove it about rhombi, then we have to prove it about kites AGAIN. So let’s prove it one time for kites. Remember, a rhombus is a D B C
kite.

17. First we have to prove that kites have one pair of congruent triangles.
Come on guys, we’ve been through this…
Here’s a twist, there are the little Δ’s inside the big Δ’s.

18. Do kites have perpendicular diagonals?
Draw the diagonals and pick which triangles you will use. There are two pairs from which to choose.
First prove the big Δ’s congruent. Which sides are congruent because the triangles are part of a kite? Can you use the reflexive property? Yay! Now we have proved the big Δ’s congruent and can use CPCTC to prove the little triangles congruent.
How would you prove it?

19. Do kites have perpendicular diagonals?
Draw the diagonals and pick which triangles you will use. There are two pairs from which to choose.
Use CPCTC to prove a pair of angles in the little Δ’s congruent, but which pair? Which pair of sides have you already stated as congruent by Def. of Kites? Can you use the reflexive property? With the little Δ’s congruent, you can say that the angles at the diagonals of the kite are congruent. And finally, congruent angles in a linear pair are ____? What does that mean?

20. Isosceles Trapezoid Conjectures about Isosceles Trapezoids:
There are two pairs of adjacent supplementary angles. Hint: What did we day about trapezoids?
The base angles are congruent.
The diagonals are congruent.

21. Isosceles Trapezoid The base angles are congruent.
That proof depends on H-L congruence. H-L congruence is a special case of SSA which works because the angle is a right angle.
Extra Credit: Prove Hypotenuse-Leg Congruence using the Pythagorean Theorem.

22. Isosceles Trapezoid Prove the base angles are congruent.
First draw two right triangles by drawing two altitudes. The altitudes of a trapezoid are congruent because the lines are parallel.
One pair of legs are congruent (they are altitudes), the hypotenuses are congruent because it is an isosceles trapezoid. Therefore the triangles are congruent by HL!
Use CPCTC and we have congruent angles, yay!

23. Isosceles Trapezoid Are the diagonals congruent?
You betcha! Use the same proof you used for the diagonals of a rectangle.