Project in MathBy Joseph Molina

8-Mango

What is the biggest weight?

1kg of stone or 1kg of cotton?

VS.

Yes! same

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

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

Activity

1) Given: CD is the altitude to ABAE is the altitude to BCCD ≅ AEProve: ∆ ABC is isosceles

B

D E

A C

Statements Reasons

Writing a

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

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

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.

Distributive Property

a(b+c) = ab+ac.

Reflexive Property

a=a (Anything is equal to itself.)

Symmetric Property

If a=b,then b-a.

Transitive Property

If a=b and b=c,then a=c.

Properties of Congruence

Reflexive Property

AB = AB, (An angle or a segment is congruent to itself.)

Smymetric Property

If A = B and B= A.

Transitive Property

If A = B and B = C,then A = C.

ExerciseJustify each statement by giving the Property of Equality or Property of CongruenceUsed.

1. If TX = BK,then BK=TX

Transitive Property

2. 8(m+n) = 8m+8n

Distributive Property

3. If CT= 12 and PR+CT =20,then PR + 12 =20

Addition Property of Equality (APE)

4. m /_ HIT = m /_HIT

Reflexive Property

5. If G=H,then H=G

Symmetric Property

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:

Example

If BD is a perpendicular bisector of AC, prove that ∆ABC isosceles. B

A CD

Paragraph proofTo prove that ∆ABC is isosceles, show thatBA ! BC . We can do this by showing that the two segments are corresponding parts of congruent triangles.

Given: BD is a bisector of AC.BD is perpendicular to AC.

Two column-proof

Prove: ∆ ABC is

isoscelesStatement Reason

BDbisects AC . GivenBD ! AC GivenAD ! CD Def. of bisector∠ADB and ∠BDC Def. of perpendicular are right angles∠ADB ≅ ∠BDC All right angles are ≅.BD ! BD Reflexive property∆ABD ≅ ∆CBD S.A.S.AB ! CB ≅ ∆'s have ≅ parts∴∆ABC is isosceles Def. of isoscelelarl

Exercises

Two-Column Form Given : m/_SEP=m/_TERProve: m/_ = m/_3

E

S

T

P

R

12

3

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

Write the missing reasons

Answer Check

1. Symmetric Property

2. Angle Addition Postulate3. Angle Addition Postulate

4. Reflexive Property

5. Subtraction Property

QuizDECODER: 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.

Segment AD bisects segment BC.Segment BC bisects segment AD.

Given:

Prove:

Triangles ABM and DCM are congruent.

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 Congruent

4.

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.

P R O O F

1 2 3 4 5

Answer