# PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Proving statements about segments and angles :

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

## Theorem : Properties of Angle Congruence

REFLEXIVE

SYMMETRIC

TRANSITIVE

For any angle A, ∠A ≅ ∠A

If ∠A ≅ ∠B, then ∠B ≅ ∠A

If ∠A ≅ ∠B∠B ≅ ∠C, then ∠A ≅ ∠C

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.

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

Paragraph Proof :

We are given that ∠A ≅ ∠B. By the definition of congruent angles, A = B. By the symmetric property of equality, B = A. Therefore, by the definition of congruent angles, it follows that ∠B ≅ ∠A.

## Proving statements about segments and angles - Examples

Example 1 :

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

Solution :

 Statements PQ ≅ XYPQ = XYXY = PQXY ≅ PQ ReasonsGivenDefinition of congruence statementsSymmetric property of equalityDefinition 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

 StatementsABLK = JKLK ≅ JKJK ≅ JLD ReasonsGivenGivenTransitive property of equalityCGivenTransitive 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

 Statements aaaa Q is the aaa  aamidpoint of PRPQ = QRPQ + QR = PRPQ + PQ = PR2 ⋅ PQ = PRPQ = 1/2 ⋅ PRQR = 1/2 ⋅ PR ReasonsGiven aaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaDefinition of midpointSegment Addition PostulateSubstitution property of equalityDistributive propertyDivision property of equalitySubstitution property of equality

Example 4 :

Prove the Transitive Property of Congruence for angles.

Solution :

To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles.

Label the vertices as A, B and C.

Given :

∠A ≅ ∠B

∠B ≅ ∠C

Prove :

∠A ≅ ∠C

Statements

∠A ≅ ∠B, ∠B ≅ ∠C

m∠A = m∠B

m∠B = m∠C

m∠A = m∠C

∠A ≅ ∠C

Reasons

Given

Definition of congruent angles

Definition of congruent angles

Transitive property of equality

Definition of congruent angles

Example 5 :

In the diagram shown below,

m∠3 = 40°, ∠1 ≅ ∠2, ∠2 ≅ ∠3

Prove m∠1 = 40°

Solution :

Statements

m∠3 = 40°

∠1 ≅ ∠2

∠2 ≅ ∠3

∠1 ≅ ∠3

m∠1 = m∠3

m∠1 = 40°

Reasons

Given

Transitive Property of Congruence

Definition of congruent angles

Substitution property of equality

Example 6 :

In the diagram shown below,

∠1 and ∠2 are right angles

Prove ∠1 ≅ ∠2

Statements

aaaa ∠1 and ∠2 are aa aaaaa right angles

m∠1 = 90°,  m∠2 = 90°

m∠1 = m∠2

∠1 ≅ ∠2

Reasons

Given aaaaaaaaaaaaaaaaaaaaa aaaaaaaaa

Definition of right angle

Transitive property of equality

Definition of congruent angles

Example 7 :

In the diagram shown below,

∠1 and ∠2 are supplements,

∠3 and ∠4 are supplements,

∠1 ≅ ∠4

Prove ∠2 ≅ ∠3

Statements

∠1 and ∠2 are supplements

∠3 and ∠4 are supplements

∠1 ≅ ∠4

m∠1 + m∠2 = 180°   m∠3 + m∠4 = 180°

m∠1  =  m∠4

a ∠1 + ∠2 = ∠3 + ∠1 aaaaaa

m∠2 = m∠3

∠2 ≅ ∠3

Reasons

aaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa

Given aaaaaaaaaaaaaaaaaaaaaa aaaaaa

Definition of Supplementary angles aaaaaaaaaaaaaaaaaaaa

Definition of congruent angles

Substitution property of equality aaaaaaaaaaaaaaaaaa

Subtraction property of equality

Definition of congruent angles

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