**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.

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.

**Example 1 : **

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 2 : **

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 ≅ ∠2

m∠1 = m∠3

m∠1 = 40°

**Reasons**

Given

Transitive Property of Congruence

Definition of congruent angles

Substitution property of equality

**Example 3 : **

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 4 : **

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

∠2 ≅ ∠3

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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