This document discusses teaching students how to write mathematical proofs through engaging activities. It provides examples of proofs involving geometry topics like congruent triangles. Students work in groups to solve proofs and then defend their arguments in a mock courtroom setting. This allows students to build confidence in supporting their logical reasoning with statements and evidence while practicing the process of writing formal proofs.

3. What is the biggest weight?
1kg of stone or 1kg of
cotton?
VS.

4. Yes! same

5. How can you
get the
answer? Can
you proof it?

6. In this unit, the process of solving
proofs is practiced using the
comparison and framework of a
courtroom setting. Students will work in
groups to solve a proof and then
defend it in “court.” This unit
challenges and engages students,
while building their confidence as
they learn to support their arguments
with sound, logical statements and
reasons. Students will have both
individual and group assessments
during these lessons

7. Activity
1) Given: CD is the
altitude to AB
AE is the altitude to BC
CD ≅ AE
Prove: Δ ABC is
isosceles
B
D E
A C
Statements Reasons

10. In writing proofs,the properties of equality and
congruence are used as bases for reasoning.
Properties of Equality
Addition Property Of Equality
(APE)
For all real numbers a,b,c
and d,if a =b and c=d,then
a+c=b+d

11. Subtraction Property of Equality (SPE)
If a=b and c=d,then a-c=b-d.
Multiplication Property of Equality
(MPE)
If a=b,then ac=bc

12. Division Property of Equality (DPE)
If a=b and c=0,then a/c =b/c
Substitution Property of Equality
If a=b,then “a” may be replaced
with “b” at any time.

13. Distributive Property
a(b+c) = ab+ac.
Reflexive Property
a=a (Anything is
equal to itself.)

14. Symmetric Property
If a=b,then b-a.
Transitive Property
If a=b and b=c,then
a=c.

15. Properties of Congruence
Reflexive Property
AB = AB, (An angle or a segment is
congruent to itself.)
Smymetric Property
If A = B and B= A.

16. Transitive Property
If A = B and B = C,then A =
C.

17. Justify each statement by giving the Property of
Equality or Property of Congruence
Used.
1. If TX = BK,then BK=TX
Transitive Property

18. 2. 8(m+n) = 8m+8n
Distributive Property

19. 3. If CT= 12 and PR+CT
=20,then PR + 12 =20
Addition Property of
Equality (APE)

20. 4. m /_ HIT = m /_HIT
Reflexive Property

21. 5. If G=H,then H=G
Symmetric Property

22. 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.
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:

23. Example
If BD is a perpendicular bisector of AC, prove
that ΔABC isosceles. B
A C
D
Paragraph proof
To prove that ΔABC is isosceles, show
that
BA ! BC . We can
do this by showing that the two
segments are corresponding
parts of congruent triangles.

24. Two column-proof
Given: BD is a bisector of AC.
BD is perpendicular to AC.
Statement Reason
BDbisects AC . Given
BD ! AC Given
AD ! CD Def. of bisector
∠ADB and ∠BDC Def. of perpendicular
are right angles
∠ADB ≅ ∠BDC All right angles are ≅.
BD ! BD Reflexive property
ΔABD ≅ ΔCBD S.A.S.
AB ! CB ≅ Δ's have ≅ parts
∴ΔABC is isosceles Def. of isoscelelarl

25. Exercises
Two-Column Form Given : m/_SEP=m/_TER
Prove: m/_ = m/_3
E
S
T
P
R
1
2
3
Write the missing reasons
Statement Reason
1. m/_SEP = m/_TER
2. m/_SEP=m/_1+m/_2
3. m/_TER=m/_2+m/_3
4. m/_2=m/_2
5. m/_1=m/_3

26. Check
1. Symmetric Property
2. Angle Addition Postulate
3. Angle Addition Postulate
4. Reflexive Property
5. Subtraction Property

27. Quiz
DECODER: Is a logical argument in which each statement is
supported/justified by given
information,definition,axioms,postulates,theorems, and previously
proven statements. Complete the correct statement or reason below.
Given:
Segment AD bisects segment BC.
Segment BC bisects segment AD.
Prove:
Triangles ABM and DCM are
congruent.

28. Statement Reason
Segment AD bisects segment BC 1.
2. When a segment is bisected,the
resulting segments are congruent.
3. Given
Segments BM and CM are
4.
Congruent
5. Vertical angles are congruent.
o P o R F
Segment BC
bisects
segment AD
Given When a segment is
a bisected,the two
resulting segments
are congruent.
Segments
AM and MD
are
Congruent.
Angles AMB
and DMC are
congruent.

29. Answer
P R O O F
1 2 3 4 5