We can incorporate our exercises' given information and rewrite this property as follows.
Given: & ∠ 1 ≅ ∠ 2 and ∠ 2 ≅ ∠ 3
Prove: & ∠ 1 ≅ ∠ 3
Now, let's prove this using a two-column proof.
Statements
Reasons
1.
∠ 1 ≅ ∠ 2 and ∠ 2 ≅ ∠ 3
1.
Given
2.
m ∠ 1 = m∠ 2 and m∠ 2 = m∠ 3
2.
Definition of Congruent Angles
3.
m ∠ 1 = m∠ 3
3.
Transitive Property of Equality
4.
∠ 1 ≅ ∠ 3
4.
Definition of Congruent Angles
Extra
A Cool Classic Car and the Transitive Property of Equality
First, recall the definition of the property which states that for all real numbers, if two numbers are equal to the same number, then they are equal to each other. Let a, b, and c be real numbers.
Transitive Property of Equality
If a=b and b=c, then a=c.
This property can be used together with other Properties of Equality to solve equations. Consider the following example. x = y-15
y-15 = 10
⇒
x &=10
There is a common thread between the two equations that leads to the conclusion that the other two connecting values also equal one another. Let's consider an example that talks about words used around the world for cars! carro = car
car = voiture
⇒
carro = voiture
This is an example of how a car is still a car across languages! In Spanish, we say carro, which is called a car in English, and a car is called voiture in French. Applying the property, we can conclude that carro is the same thing as a voiture. Take a look at a portion of the world map and make your own example using some of the ways to say car!
Just as a side note, the car image is of a 1936 Auburn 654 Cabriolet, which sold for around $1500 in those days.
