PROVING STATEMENTS ABOUT 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.

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 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

Transitive Property of Angle Congruence

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

Using Transitive Property

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

Right Angle Congruence Theorem

All right angles are congruent.

Proof :

We can prove the theorem as shown below.

Given :  ∠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

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

If m∠1 + m∠2  =  180° and m∠2 + m∠3  =  180°, then,

m∠1  ≅  m∠3

Proof :

We can prove the theorem as shown below.

Given :

∠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

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then the two angles are congruent.

If m∠4 + m∠5  =  90° and m∠5 + m∠6  =  90°, then,

m∠4  ≅  m∠6

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Using Linear Pairs

In the diagram shown below, m∠8 = m∠5 and m∠5 = 125°.

Prove that m∠7  =  55°.

Using the transitive property of equality,

m∠8  =  125°

The diagram shows

m∠7 + m8  =  180°

Substitute 125° for m8.

m∠7 + 125°  =  180°

Subtract 125° from each side.

m∠7  =  55°

Vertical Angles Theorem

Vertical angles are congruent.

Proof :

Given :

m∠5 and m∠6 are a linear pair

m∠6 and m∠7 are a linear pair

Prove :

m∠5  ≅  m∠7

Statements

∠5 and ∠6 are a linear pair

∠6 and ∠7 are a linear pair

∠5 and ∠6 are supplementary

∠6 and ∠7 are supplementary

∠1 ≅ ∠4

Reasons

Givenaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaa

Given aaaaaaaaaaaaaaaaaaaaaa aaaaaa

Linear pair postulate aaaaaaaaa aaaaa

Linear pair postulate aaaaaaaaa aaaaa

Congruent Supplements Theorem 

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