The document provides instructions for writing two-column geometric proofs. It explains that a two-column proof consists of statements in the left column and reasons for those statements in the right column. Each step of the proof is a row. It then gives examples of properties that can be used as reasons, such as angle addition postulate, congruent supplements theorem, and triangle congruence postulates. Sample proofs are also provided to illustrate the two-column format.

1. GUIDE
CARD

3. GUIDE CARD
A two-column geometric proof consists of a list of statements, and
the reasons that we know those statements are true. The
statements are listed in a column on the left, and the reasons for
which the statements can be made are listed in the right column.
Every step of the proof (that is, every conclusion that is made) is a
row in the two-column proof.
Writing a proof consists of a few different steps.
1. Draw the figure that illustrates what is to be proved. The figure
may already be drawn for you, or you may have to draw it
yourself.
2. List the given statements, and then list the conclusion to be
proved. Now you have a beginning and an end to the proof.

4. 3. Mark the figure according to what you can deduce about it from the
information given. This is the step of the proof in which you actually
find out how the proof is to be made, and whether or not you are able
to prove what is asked. Congruent sides, angles, etc. should all be
marked so that you can see for yourself what must be written in the
proof to convince the reader that you are right in your conclusion.
4. Write the steps down carefully, without skipping even the simplest
one. Some of the first steps are often the given statements (but not
always), and the last step is the conclusion that you set out to prove.
A sample proof looks like this:

5. Given:
Segment AD bisects segment BC.
Segment BC bisects segment AD.
Prove:
Triangles ABM and DCM are congruent.
Notice that when the SAS postulate was used, the numbers in parentheses correspond to the numbers of
the statements in which each side and angle was shown to be congruent. Anytime it is helpful to refer to
certain parts of a proof, you can include the numbers of the appropriate statements in parentheses after
the reason.

6. Here are some properties you can use as reasons in a proof:
General:
Reflexive Property A quantity is congruent (equal) to itself. a = a
Symmetric Property If a = b, then b = a.
Transitive Property If a = b and b = c, then a = c.
Addition Postulate If equal quantities are added to equal quantities, the
sums are equal.
Subtraction Postulate
If equal quantities are subtracted from equal
quantities, the differences are equal.
Multiplication
Postulate
If equal quantities are multiplied by equal quantities,
the products are equal. (also Doubles of equal
quantities are equal.)

7. Division Postulate
If equal quantities are divided by equal nonzero
quantities, the quotients are equal. (also Halves
of equal quantities are equal.)
Substitution
Postulate
A quantity may be substituted for its equal in
any expression.
Partition Postulate
The whole is equal to the sum of its parts.
Also: Betweeness of Points: AB + BC = AC
Angle Addition Postulate: m<ABC + m<CBD =
m<ABD
Construction
Two points determine a straight line.
Construction
From a given point on (or not on) a line, one
and only one perpendicular can be drawn to

8. Angles:
Right Angles
All right angles are congruent.
Straight Angles
All straight angles are congruent.
Congruent Supplements
Supplements of the same angle, or congruent angles, are
congruent.
Congruent Complements
Complements of the same angle, or congruent angles, are
congruent.
Linear Pair
If two angles form a linear pair, they are supplementary.
Vertical Angles
Vertical angles are congruent.
Triangle Sum
The sum of the interior angles of a triangle is 180º.
Exterior Angle
The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two non-adjacent interior angles.
The measure of an exterior angle of a triangle is greater than
either non-adjacent interior angle.
Base Angle Theorem
(Isosceles Triangle)
If two sides of a triangle are congruent, the angles opposite these
sides are congruent.
Base Angle Converse
(Isosceles Triangle)
If two angles of a triangle are congruent, the sides opposite these
angles are congruent.

9. Triangles:
Side-Side-Side (SSS)
Congruence
If three sides of one triangle are congruent to three
sides of another triangle, then the triangles are
congruent.
Side-Angle-Side (SAS)
Congruence
If two sides and the included angle of one triangle are
congruent to the corresponding parts of another
triangle, the triangles are congruent.
Angle-Side-Angle (ASA)
Congruence
If two angles and the included side of one triangle are
congruent to the corresponding parts of another
triangle, the triangles are congruent.
Angle-Angle-Side (AAS)
Congruence
If two angles and the non-included side of one triangle
are congruent to the corresponding parts of another
triangle, the triangles are congruent.
Hypotenuse-Leg (HL)
Congruence (right
triangle)
If the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right
triangle, the two right triangles are congruent.
CPCTC
Corresponding parts of congruent triangles are
congruent.
Angle-Angle (AA)
Similarity
If two angles of one triangle are congruent to two
angles of another triangle, the triangles are similar.

10. SSS for Similarity
If the three sets of corresponding sides of two triangles are
in proportion, the triangles are similar.
SAS for Similarity
If an angle of one triangle is congruent to the
corresponding angle of another triangle and the lengths of
the sides including these angles are in proportion, the
triangles are similar.
Side Proportionality
If two triangles are similar, the corresponding sides are in
proportion.
Mid-segment Theorem
(also called mid-line)
The segment connecting the midpoints of two sides of a
triangle is parallel to the third side and is half as long.
Sum of Two Sides
The sum of the lengths of any two sides of a triangle must
be greater than the third side
Longest Side
In a triangle, the longest side is across from the largest
angle.
In a triangle, the largest angle is across from the longest
side.
Altitude Rule
The altitude to the hypotenuse of a right triangle is the
mean proportional between the segments into which it
divides the hypotenuse.
Leg Rule
Each leg of a right triangle is the mean proportional
between the hypotenuse and the projection of the leg on
the hypotenuse.

11. The Same Angle Supplements Theorem states that if two angles are
supplementary to the same angle then the two angles are congruent. Prove this
theorem.
Given : and are supplementary angles. and are supplementary
angles.
Prove :
Statement Reason
1. and are supplementary
and are supplementary
1. Given
2.
2. Definition of supplementary angles
3. 3. Substitution
4. 4. Subtraction
5. 5. angles have = measures

12. The Vertical Angles Theorem states that vertical angles are congruent.
Prove this theorem.
Given : Lines and intersect.
Prove:
Statement Reason
1. Lines and intersect 1. Given
2. and are a linear pair
and are a linear pair
2. Definition of a Linear Pair
3. and are supplementary
and are supplementary
3. Linear Pair Postulate

13. 4. 4. Definition of
Supplementary Angles
5. 5. Substitution
6. 6. Subtraction
7. 7. angles have = measures

14. ACTIVITY
CARD

15. If you’re ready, let’s start the
adventure!
GENERAL INSTRUCTIONS: Since our entire topic is all about writing statement
and its reason so, first, in this following activity let us connect it in real life
wherein we are going to distinguish its cause and effect.
ACTIVITY 1
Instructions: Complete the table below.
Cause Effect
1.The girl played in the mud
2.I never brush my teeth
3. The doctor put a cast on his arm.
4.I flipped the light switch on
5. We were late for work
6.Julia’s cat died
7. Laura eat too much candies
8.she’s healthy
9.Mila is sad
10. we had to cancel the match

16. Are you ready for the next activity?
Okay, if you are, let’s now proceed to the
next.
ACTIVITY 2
Instructions: Match the following phrases to complete a sentence.
CAUSE EFFECT
People killed too many of them so now he have lung cancer
We ran out of milk we went to the grocery store
Powerful winds blew the roof she is well loved by her pupils
off the house
You should brush your teeth a star explodes
James is smoking cigarettes he was hungry
When nuclear fusion stop or starts and a thief broke into the room
Mrs. Ramos is a kind teacher you won’t get cavities
The room was left unlocked alligator almost became extinct
Keegan skipped lunch the whole family had nowhere to
live
Heavy rain traffic

17. ACTIVITY 3
Instructions: Fill in the blanks
1. There was flooding because of
____________________________________________
2. The air get cold so
______________________________________________________
3. Jim is scared because
____________________________________________________
4. The baby was crying, so
__________________________________________________
5. I learned to play the drums quickly because
__________________________________
6. Mara didn’t follow the recipe correctly, so
___________________________________
7. Some believe dinosaurs died out because
____________________________________
8. I had to mop since
______________________________________________________
9. John made a rude comment, so
____________________________________________
10. We went to the mall since,
_______________________________________________

18. Assessme
nt
CARD

19. Make a two-column proof
for the following
problems. Choose your
answers from the box
below each problem.
The Right Angle Theorem states that if two angles are right angles, then the angles are congruent. Prove this theorem.
To prove this theorem, set up your own drawing and name some angles so that you have specific angles to talk
about.
Given : and are right angles
Prove :
Statement Reason
1. and are right angles 1. Given
2. and 2.
3. 3. Transitive
4. 4. angles have = measures
Definition of right angles
angles have = measures

20. Write a two-column proof for the following:
Given: and
Prove:
Statement Reason
1. 1. Given
2. and are right angles 2.
3. 3. Right Angle Theorem
4. 4.
5. 5. Subtraction
Definition of Perpendicular Lines
Given

21. Write a two-column proof for the following:
Given: is supplementary to , is supplementary to ,
Prove:
Statement Reason
1. is supplementary to , is supplementary to 1. Given
2. , 2.
3. 3. Substitution
4. 4.
5. 5. Substitution
6. 6.
Definition of Supplementary Angles
Given
Subtraction

22. Enrichmen
t
CARD

23. Identify which pair of triangles
below does NOT illustrate an angle
angle side (AAS) relationship.

24. 4.
Proof
5.
Proof

25. 6.
Proof
7.
Proof

26. Reference
CARD

27. examples.yourdictionary.com/cause-and-effect-
examples.html#comment-1741397603
www.grammar-quizzes.com/19-2.html
edhelper.com/cause_and_effect.html
www.grammar-quizzes.com/8-7.html
edhelper.com/language/reading_comprehension_skills202.html
http://www.ck12.org/geometry/Two-Column-Proofs/lesson/Two-
Column-Proofs-Intermediate/
http://m.sparknotes.com/math/geometry3/geometricproofs/proble
ms_1.html
http://www.regentsprep.org/regents/math/geometry/gp4/PracCon
gTri.html

28. Answer Card
Activity 1
1. She got dirty
2. I got cavities
3. He broke his arm
4. The light turned on
5. The alarm didn’t work
6. Julia forgot to feed her cat
7. She got tooth cavity
8. She eats fruits and vegetables
9. She got a bad day
10. It rained so hard

29. Activity 2
1.Alligator almost become extinct
2.We went to grocery
3.The family gad nowhere to live
4.You wont get cavities
5.So now he have lung cancer
6.A star explodes
7.She is well loved by her pupils
8.And a thief broke into the room
9.He was hungry
10. Traffic

30. Activity 3
1. Heavy rains
2. She put on her jacket
3. Its dark
4. The father gave him milk
5. I took advance lesson
6. The food didn’t taste good
7. A meteor hit the earth
8. My juice spilled on the floor
9. Elise got mad
10. We don’t have classes

31. Statement Reason
1. and are right
angles
1. Given
2. and
2. Definition of right
angles
3. 3. Transitive
4.
4. angles have =
measures
ASSESSMENT 1

32. Statement Reason
1. 1. Given
2. and are
right angles
2. Definition of
a Perpendicular Lines
3. 3. Right Angle Theorem
4. 4. Given
5. 5. Subtraction
ASSESSMENT 2

33. Statement Reason
1. is supplementary to
, is supplementary to
1. Given
2. ,
2. Definition of
Supplementary
Angles
3. 3. Substitution
4. 4. Given
5. 5. Substitution
6. 6. Subtraction
ASSESSMENT 3

34. ENRICHMENT 1

35. Enrichment 2
Statements Reasons
1.
1. Given
2. 2. Perpendicular lines meet to form right angles.
3. 3. Right angles are congruent.
4.
4. (ASA) If two angles and the included side of one triangle are
congruent to the corresponding parts of a second triangle, the
triangles are congruent.
5.
5. (CPCTC) Corresponding parts of congruent triangles are
congruent.

36. Statements Reasons
1. 1. Given
2. 2. Two angles that form a linear pair are
supplementary.
3. 3. Supplements of the same angle, or
congruent angles, are congruent.

37. 4. 4. Reflexive property: A quantity
is congruent to itself.
5. 5. (SAS) If two sides and the
included angle of one triangle are
congruent to the corresponding
parts of a second triangle, the
triangles are congruent.
6. 6. (CPCTC) Corresponding parts
of congruent triangles are
congruent.
7. 7. An angle bisector is a ray whose
endpoint is the vertex of the angle
and which divides the angle into
two congruent angles.

38. Enrichment 3
Statements Reasons
1.
;
1. Given
2. 2. Reflexive property - a quantity is
congruent to itself.
3. 3. (ASA) If two angles and the included
side of one triangle are congruent to the
corresponding parts of a second triangle,
the triangles are congruent.

39. 4. 4. (CPCTC) Corresponding
parts of congruent triangles
are congruent.
5. AB = BE ; DB = BC 5. Congruent segments are
segments of equal length.
6.
AB = AD + DB
EB = EC + CB
6. Betweeness of Points: The
whole is equal to the sum of
its parts.
7. AD + DB = EC + CB 7. Substitution.
8. AD = EC 8. Subtraction
9. 9. Congruent segments are
segments of equal length.

40. Statements Reasons
1. ; 1. Given
2. 2. Congruent angles are angles of
equal measure.
3. 3. Reflexive property - a quantity
is equal to itself.
4. 4. Angle Addition Postulate: The
whole is equal to the sum of its
parts.

41. 5. 5. Substitution.
6. 6. Subtraction
7. 7. Congruent angles are angles of
equal measure.
8. 8. Midpoint of a line segment is the
point on that line segment that
divides the segment into two
congruent segments.
9. 9. (SAS) If two sides and the
included angle of one triangle are
congruent to the corresponding parts
of a second triangle, the triangles are
congruent.

42. All in all, if you got…
Perfect Score
7-9
5-6
4 and below