Unit 3: Geometry
th
5 Grade
Name____________________ Teacher_________ School_________
HOW TO BUILD A TWO COLUMN PROOF
A two-column geometric proof consists of list of statements (on the left
side of the page), and a matched list of reasons or explanations that
show how we know those statements are true. Again, the statements are
listed in a column on the left, and the reasons or explanations for which
the statements can be made are listed in the right column.
Writing a proof consists of a few different steps.
1. Draw the figure (the geometric shape) so you can illustrate what
you will prove. Sometimes the figure is already drawn for you.
Sometimes you will have to draw it yourself.
2. List the given statements, and then list the last reason to be
proved….a conclusion. Now you have a beginning and an end to
the proof. (NOTE: There is a list of easy abbreviations that you
can use in this step.)
3. Use a pencil to mark up the figure according to what you can
figure out about it from the information you were given. Congruent
sides, angles, etc. should all be marked so that you can see what
must be written in the proof to convince the reader that you are
right in your conclusion.
4. Write the steps carefully, without skipping even the simplest one.
Some of the first steps are often the given statements, and the last
step is the conclusion that you set out to prove.
Elementary Geometry
Here is everything you need to know to solve the following problems. Most of it
will be a review for you. Some information might be shown in a way that looks
new at first, but it is really connected to the properties of triangles that you have
been learning for years.
Lines and Angles: When two lines intersect, opposite angles are equal and the
sum of adjacent angles is 180 degrees. When two parallel lines are intersected by
a third line, the corresponding angles of the two intersections are equal.
Triangles:
The sum of the interior angles of a triangle is 180 degrees.
An isosceles triangle has two equal sides and the two angles opposite those sides
are equal.
An equilateral triangle has all sides equal and all angles equal.
A right triangle has one angle equal to 90 degrees.
Two triangles are called similar if they have the same angles (same shape).
Two triangles are called congruent if they have the same angles and the same
sides (same shape and size).
Basic list of abbreviations: Used in proofs about triangles.



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SAS (Side-Angle-Side): Two triangles are congruent if two pairs of sides of
the two triangles are equal, and the included angles are equal.
SSS (Side-Side-Side): Two triangles are congruent if three pairs of sides of
the two triangles are equal.
ASA (Angle-Side-Angle): Two triangles are congruent if two pairs of angles
of the two triangles are equal, and the included sides are equal.
AAS (Angle-Angle-Side): Two triangles are congruent if two pairs of angles
of the two triangles are equal, and a pair of corresponding non-included
sides are equal.
Sample proof:
First, figure out the “GIVEN”information: What do you know?
 Segment AD bisects segment BC.
 Segment BC bisects segment AD.
Then prove that triangles ABM and DCM are congruent. Move step
by step through the proof. Start with the information that is given.
STATEMENTS
REASONS (Explanations)
1. Segment AD bisects segment BC
1. Given.
2. Segments AM and MD are congruent.
2. When a line segment is bisected, the
two resulting segments are congruent.
3. Segment BC bisects segment AD.
3. Given.
4. Segments BM and CM are congruent.
4. When a line segment is bisected, the
two resulting segments are congruent.
5. Angles AMB and DMC are congruent. 5. Vertical angles are congruent.
6. Triangles ABM and DCM are
congruent.
6. SAS postulate (2, 4, 5)
Problem :
Given:
 Circle C with triangles ABC and DEC.
 Chord AB is congruent to chord DE.
Prove:
Triangles ABC and DEC are congruent.
STATEMENTS
1 Chord AB is congruent to chord DE.
REASONS (Explanations)
1 Given.
2 Segment CA is congruent to segment
2 Radii of a circle are congruent.
CD
3 Segment BC is congruent to segment
3 Radii of a circle are congruent.
EC
4 Triangle ABC is congruent to triangle
4 SSS (1, 2, 3)
DEC
See the abbreviation in the last step? SSS is one of the abbreviations
mentioned earlier. Turn to the next page to read about these
abbreviations.
Problem :
Given: Quadrilateral DEFG is a parallelogram.
Prove:
Angles G and E are congruent
Hint: Use the definition of a parallelogram.
STATEMENTS
REASONS (Explanations)
1 Quadrilateral DEFB is a parallelogram.
1 Given.
2 Segment DE is parallel to segment GF.
2 Definition of a parallelogram.
3 Angle GDF is congruent to angle EFD.
3 When a transversal cuts parallel lines, the
alternate interior angles are congruent.
4 Angle GFD is congruent to angle EDF
4 When a transversal cuts parallel lines, the
alternate interior angles are congruent.
5 Segment DF is congruent to itself
5 Reflexive property.
6 Triangle GDF is congruent to triangle EFD
6 ASA postulate (3, 4, 50
7 Angles G and E are congruent.
7 Corresponding parts of congruent triangles
are congruent.
Using only elementary geometry, figure
out the measurement of angle x. Then do
a step-by-step proof to back up your
answer. You may work together as a
team with another fifth grader to figure
out the steps to take.
If you can even begin to write a proof
for this one, you are just amazing.
However, can you figure out all or most
of the measurements of the angles drawn
inside the main triangle?
Good luck!
CROP CIRCLE FUN
After you complete this puzzle of a proof – or have
given it your best try, go to the Geometry of Crop
Circles page on Coolmath.com and begin to figure out
how to make at least two designs.
Write a set of directions for both of your designs.
Keep it fairly simple, please.
Sketch your designs or paste in a small photocopy.
If you have steady hands, use a compass and draw your
crop circle design.