Download What is Congruent? - Miami Beach Senior High School

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” charT
Isosceles
Trapezoid
Trapezoid
Parallelogram
Kite
Rectangle
Rhombus
Square
BUT the “Is A” Chart
flows from specific to
general. A Parallelogram
“Is A” Trapezoid
Trapezoid
Parallelogram
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.
Proves chart
Isosceles
Trapezoid
Trapezoid
Parallelogram
Kite
Rectangle
Rhombus
Square
Proof Tool Kit
Before we go further, let’s take a moment to
remember what we know about congruent triangles.
SSS
SAS
AAS
ASA
CPCTC
Don’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?
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.
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.
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!
Again, CPCTC finishes
the proof.
Remember,
parallel lines
mean flip the
triangles.
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?
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
EXTRA CREDIT: Use this proof to show that
congruent opposite sides mean parallelogram.
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 kite.
A
D
B
C
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.
Do kites have perpendicular diagonals?
Draw the diagonals and pick which triangles you will use.
There are two pairs from which to choose.
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.
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?
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.
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.
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!
Isosceles Trapezoid
Are the diagonals congruent?
You betcha! Use the
same proof you used for
the diagonals of a
rectangle.