PROPERTIES OF EQUALITY AND CONGRUENCE

Name the property illustrated below.

Problem 1 :

If UV = KL and KL = 6, UV = ? 

Solution :

Using Reflexive Property of Equality, we understand that UV and KL are equal.

If KL = 6, then UV = 6.

Problem 2 :

If m∠1 + m∠2 = m∠4 + m∠2, then m∠1 = m∠4

Solution :

m∠1 + m∠2 = m∠4 + m∠2

Using subtraction property, subtract m∠2 on both sides.

m∠1 + m∠2 - m∠2  = m∠4 + m∠2 - m∠2

So, we get

m∠1 = m∠4

Property used : Subtraction Property

Problem 3 :

∠ABC ≅ ∠ABC

Solution :

Property used : Reflexive Property of Congruence

Problem 4 :

If 1/2m∠D = 45, then m∠D = 90

Solution :

Given that, (1/2)m∠D = 45

Multiplying by 2 on both sides.

2 x (1/2)m∠D = 45 x 2

m∠D = 90

Property used : Multiplication Property of Equality

Problem 5 :

If ∠DEF ≅ ∠HJK, then ∠HJK ≅ ∠DEF

Solution :

∠DEF ≅ ∠HJK, then ∠HJK ≅ ∠DEF

If a = b, then b = a, it means symmetric property.

Property used : Symmetric Property of Congruence

Problem 6 :

If y = 12 - x and 2x + 3y = 10, then 2x + 3(12 - x) = 10

Solution :

Let

y = 12 - x  ----(1)

2x + 3y = 10 -----(2)

Given that, 2x + 3(12 - x) = 10

2x + 3(12 - x) = 10

Property used : Substitution Property

Problem 7 :

If x = 5, then x + 3 = 8

Solution :

Given that, x = 5

Adding 5 on both sides,

x + 3 = 5 + 3 

x = 8

Property used : Addition Property

Problem 8 :

If AB = MN and MN = XY, then AB = XY

Solution :

Take three quantities, a, b and c

If a = b, b = c, then a = c

It means transitive property.

Property used : Transitive Property of Equality

Problem 9 :

If 2(AX) = 2(BY), then AX = BY

Solution :

Given that,

2(AX) = 2(BY)

Dividing by 2 on both sides.

2(AX)/2 = 2(BY)/2

AX = BY

Property Used : Division Property of Equality

Problem 10 :

If m∠1 = 40 and m∠2 = m∠1 + 50, then m∠2 = 90

Solution :

Given that, 

m∠1 = 40 ----(1)

m∠2 is defined as m∠1 + 50----(2)

Using substitution property, applying (1) in (2), we get

m∠2 = 40 + 50

m∠2 = 90

Property used : Substitution Property

Use the given property to complete each statement.

Problem 11 :

Reflexive Property of Congruence

∠TRS

Solution :

∠TRS ≅ ∠TRS 

Problem 12 :

Substitution Property

If AB = 2, and AC = AB + BC, then

Solution :

AB = 2 ----(1)

AC = AB + BC------(2)

Applying (1) in (2), we get

AC = 2 + BC

Subtraction Property of Equality

If 25x + 12 = 32, then 25x = 20

Problem 13 :

Transitive Property of Equality

If RM = OP and OP = XT, then

Solution :

RM = XT

Problem 14 :

Symmetric Property of Congruence

If ∠TES ≅ ∠BKC, then

Solution :

Using the symmetric property of congruence, we get

 ∠BKC ≅ ∠TES

Problem 15 :

Division Property of Equality

If 4m∠ABC = 120°, then

Solution :

Given that, 4m∠ABC = 120°

Dividing by 4 on both sides.

m∠ABC = 30°