A true statement that follows as a result of other statements is called a theorem. All theorems must be proved. We can prove a theorem using a two-column proof. A two-column proof has numbered statements and reasons that show the logical order of an argument.

## Theorem : Properties of Segment Congruence

REFLEXIVE

SYMMETRIC

TRANSITIVE

For any segment AB, AB ≅ AB

If AB ≅ CD, then CD ≅ AB

If AB ≅ CD, and CD ≅ EF, then AB ≅ EF

A proof which is written in paragraph form is called as paragraph proof.

Here is a paragraph proof for the Symmetric Property of Segment Congruence.

Paragraph Proof :

We are given that PQ ≅ XY. By the definition of congruent segments, PQ = XY. By the symmetric property of equality, XY = PQ. Therefore, by the definition of congruent segments, it follows that XY ≅ PQ.

## Proving statements about segments - Examples

Example 1 :

In the diagram given above, PQ ≅ XY. Prove XY ≅ PQ

Solution :

PQ ≅ XY

PQ = XY

XY = PQ

XY ≅ PQ

Given

Definition of congruence statements

Symmetric property of equality

Definition of congruence segments

Example 2 :

Use the diagram and the given information to complete the missing steps and reasons in the proof.

Given : LK = 5, JK = 5, JK ≅ JL

Prove : LK ≅ JL

A

B

LK = JK

LK ≅ JK

JK ≅ JL

D

Given

Given

Transitive property of equality

C

Given

Transitive property of congruence

Solution :

A.  LK = 5

B.  JK = 5

C.  Definition of congruence segments

D.  LK ≅ JL

Example 3 :

In the diagram given below, Q is the midpoint of PR.

Show that PQ and QR are each equal to 1/2 ⋅ PR.

Solution :

Given : Q is the midpoint of PR

Prove : PQ  =  1/2 ⋅ PR and QR  =  1/2 ⋅ PR

Q is the midpoint of PR

Given

PQ = QR

PQ + QR = PR

PQ + PQ = PR

⋅ PQ = PR

PQ = 1/2 ⋅ PR

QR = 1/2 ⋅ PR

Definition of midpoint

Substitution property of equality

Distributive property

Division property of equality

Substitution property of equality

After having gone through the stuff given above, we hope that the students would have understood, how to prove statements about segments".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6