Using Proofs to Create Theorems

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.

The “is a” chartTrapezoid

Isosceles Trapezoid

Parallelogram

Kite

Rhombus

Square

Rectangle

BUT the “Is A” Chart flows from specific to general. A Parallelogram “Is A” Trapezoid

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

Trapezoid

Parallelogram

Proves chartTrapezoid

Isosceles Trapezoid

Parallelogram

Kite

Rhombus

Square

Rectangle

Proof Tool KitBefore we go further, let’s take a moment to remember what we know about congruent

triangles.

SSS

SAS ASA

AAS

CPCTCDon’t forget that Corresponding Parts of

Congruent Triangles are Congruent.

means

means

means

means

Trapezoids

Trapezoids have one pair of parallel sides.

What can you say about consecutive interior angles when the lines are parallel?

ParallelogramConjectures 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.

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?

So now we have congruent triangles… It is time for CPCTC!

With CPCTC, we have opposite angles and sides.

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!

Pick a pair of triangles, you do not need all four.

We are ready to prove bisecting diagonals.Bisect: Cut into congruent parts.

Again, CPCTC finishes the proof.

Remember, parallel lines mean flip the triangles.

Remember, rectangles have 4 congruent angles. By the way, the angles are _______?

Is a rectangle a parallelogram?

What do we know about lines when the consecutive angles are supplementary?

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!

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

C

B

EXTRA CREDIT: Use this proof to show that congruent opposite sides mean parallelogram.

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

A

D

C

B

kite.

Come on guys, we’ve been through this…

First we have to prove that kites have one pair of congruent triangles.

Here’s a twist, there are the little Δ’s inside the big Δ’s.

Do kites have perpendicular diagonals?

How would you prove it?

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.

Draw the diagonals and pick which triangles you will use. There are two pairs from which to choose.

Do kites have perpendicular diagonals?

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?

Draw the diagonals and pick which triangles you will use. There are two pairs from which to choose.

Isosceles TrapezoidConjectures 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.

Isosceles TrapezoidThe 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.

Isosceles TrapezoidProve 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!

Isosceles Trapezoid

Are the diagonals congruent?

You betcha! Use the same proof you used for the diagonals of a rectangle.