∠ 2 and ∠ CBD are corresponding, ∠ 3 and ∠ CDB are corresponding
3.
Definition of corresponding angles.
4.
∠ 2 ≅ ∠ CBD, ∠ 3 ≅ ∠ CDB
4.
Corresponding Angles Theorem
5.
∠ 1 ≅ ∠ CBD ≅ ∠ CDB
5.
Substitution Property of Congruence
6.
△ BCD is equiangular
6.
Definition of an equiangular triangle
Practice makes perfect
Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left and the justification on the right. Each statement must follow logically from the steps before it.
In this case, we are given that △ ACE is equiangular and that BD ∥ AD. This how we will begin our proof!
Statement1)& △ ACE is equiangular, & BD ∥ AD Reason1)& Given
By the definition of an equiangular triangle, we know that △ ACE has three congruent angles.
Statement2) & ∠ 1 ≅ ∠ 2 ≅ ∠ 3 Reason2)& Definition of an equiangular & triangle
Similarly, ∠ 3 and ∠ CDB are corresponding.
Statement3) & ∠ 2 and ∠ CBD are & corresponding, ∠ 3 and ∠ CDB & are corresponding Reason3)& Definition of corresponding & angles
By the Corresponding Angles Theorem, we can conclude that ∠ 2 ≅ ∠ CBD and ∠ 3 ≅ ∠ CDB.
Statement4)& ∠ 2 ≅ ∠ CBD, ∠ 3 ≅ ∠ CDB
Reason4) & Corresponding Angles Theorem
Recall that we know that ∠ 1 ≅ ∠ 2 ≅ ∠ 3. Therefore, by the Substitution Property of Congruence, we can substitute ∠ CBD for ∠ 2 and ∠ CDB for ∠ 3, so ∠ 1 ≅ ∠ CBD ≅ ∠ CDB.
Statement5)& ∠ 1 ≅ ∠ CBD ≅ ∠ CDB
Reason5) & Substitution Prop. of & Congruence
Finally, by the definition of an equilateral triangle, we can conclude that △ BCD is equiangular.
Statement6)& △ BCD is equiangular
Reason6) & Definition of an equiangular & triangle
Now we can complete our two-column table!
Statements
Reasons
1.
△ ACE is equiangular, BD ∥ AD
1.
Given
2.
∠ 1 ≅ ∠ 2 ≅ ∠ 3
2.
Definition of an equiangular triangle
3.
∠ 2 and ∠ CBD are corresponding, ∠ 3 and ∠ CDB are corresponding
