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.
REFLEXIVE
SYMMETRIC
TRANSITIVE
For any segment AB, AB ≅ AB
If AB ≅ CD, then CD ≅ AB
If AB ≅ CD, and CD ≅ EF, then AB ≅ EF
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.
Example 1 :
In the diagram given above, PQ ≅ XY. Prove XY ≅ PQ
Solution :
Statements PQ ≅ XY PQ = XY XY = PQ XY ≅ PQ |
Reasons 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
Statements A B LK = JK LK ≅ JK JK ≅ JL D |
Reasons 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
Statements aaaa Q is the aaa aamidpoint of PR PQ = QR PQ + QR = PR PQ + PQ = PR 2 ⋅ PQ = PR PQ = 1/2 ⋅ PR QR = 1/2 ⋅ PR |
Reasons Given aaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaa Definition of midpoint Segment Addition Postulate Substitution property of equality Distributive property Division property of equality Substitution 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 ≅ ∠2
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
∠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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