Transcript  A deductive argument that contains statements and reasons organized in two columns.


A deductive argument that contains
statements and reasons organized in two
columns.
The point of writing proofs is to PROVE that
two triangles are congruent.
 There isn’t a set number of proofs that you
have to have to prove that two triangles are
congruent.
 The more proofs, the better.
 There isn’t a set order, just make sure that
the proofs make sense – sometimes one
thing can’t happen before another.
 One of the 5 Congruence Postulates must
be proven


Set up a two-column proof format.
› The left-hand column should be labeled
"Statements" and the right-hand column labeled
"Reasons."
STATEMENT
REASON
 List
your givens as the first step
in your statements
column. Under the reasons
column, you should write
"given."
 Translate
the givens into useful
information.
› For each given, you'll want to think
what you could do with that
knowledge. At this point in the
proof, most of your reasons will be
definitions.
› If one of the givens is "Point C is the
midpoint of AB," your statement
would read "AC = CB" and your
reason would be "Definition of
midpoint," or the full theorem: If a
segment has a midpoint, then it
divides the segment into two
congruent parts.
› If segment XY bisects segment NM,
and P is the point where XY intersects
NM, your statement would read "NP =
PM" and your reason would be
"Definition of bisector," or the full
theorem: If a segment is bisected,
then it is divided into two congruent
parts.
› If the givens say "KL is perpendicular to HJ,"
your statement would be "Angle KLH is a
right angle." Your reason would then be: If
two perpendicular lines intersect, then they
form right angles. After this, you would
reference the step that says "KL is
perpendicular to HJ," and write the number
in parentheses next to the reason. The
letters will depend on the diagram, which is
one reason why the diagram is important.
 Look
for any isosceles triangles in
the diagram based on your givens.
› Isosceles triangles are triangles that
have two legs of equal length. Write
the two legs are congruent as a
statement citing "Definition of
isosceles" as the reason.

Look for parallel lines.
› If a third line runs through both of them, it will form
alternate interior angles and corresponding
angles. These are quite common in triangle proofs.
The overall shape formed by the angles looks like a
the letter "z.“ Write in your statements column that
those angles are congruent to each other with the
reason being "Alternate interior angles are
congruent" or "Corresponding angles are
congruent."
› Do not use reasons such as "Definition of parallel"
or "Definition of alternate interior angles." These are
not valid proof reasons.

Look in the diagram for vertical angles.
› Vertical angles will form an X and are the angles
across from each other, touching at the center
but not along their sides. These vertical angles are
congruent. An example would be to use the
statement "Angle P is congruent to angle R" with
the reason "Vertical angles are congruent."

Look in the diagram for any lines or angles
shared by triangles.
› This is where the reflexive property comes into
play. Your statement should read "XY = XY" or
"Angle A = angle A" and the reason would be
"Reflexive property of congruence."

Transfer every congruency statement you've
found so far, including the givens, into the
diagram.
› Congruent sides get marked with hash marks;
congruent angles are marked with arcs.

Look at one of the triangles in the picture.
› Note all the markings you just made and the ones that were given
›
›
›
›
to you. How many angles are congruent? How many sides?
Match this information with one of the triangle congruency
theorems. Your statement should read "Triangle ABC is congruent
to triangle XYZ" and the reason would be the appropriate choice
between AAS, ASA, SAS, SSS, HL, etc.
SSS stands for side, side, side. All three sides must be congruent.
Two sides and an angle must be congruent for it to be SAS (side,
angle, side).
See if your triangles have congruent right angles, and two
congruent sides, one being the hypotenuse (which is the side
directly across from the right angle). If they do, then state them as
being congruent with the reason HL (Hypotenuse Leg Theorem).
It can be ASA or AAS if two angles are congruent, but only one
side.

Take another look at the "Prove" line of your
problem.
› If it's the same as your last statement, then you are
finished.