We will go through the blank spaces one-by-one starting with a.
Blank a.
To fill in blank a., we have to provide a reason why ∠ 3 and ∠ 6 are congruent. Examining the diagram, we see that ∠ 3 and ∠ 6 are vertical angles. By the Vertical Angles Theorem, we know these angles are congruent.
Therefore, the missing information in a. is "Vertical Angles Theorem"
2)& ∠ 3≅ ∠ 6
2)& a. Vertical Angles Theorem
Blank b
The Transitive Property of Congruence tells us that if two angles are both congruent to a third angle, the two angles are congruent as well. Since ∠ 1 ≅ ∠ 3 and ∠ 3 ≅ ∠ 6, we can claim that ∠ 1 and ∠ 6 are congruent as well. This is the missing information in b.
3)& b. ∠ 1 ≅ ∠ 6
3)& Transitive Property of Congruence
Blank c
Similar to a., ∠ 1 and ∠ 4 are vertical angles. We can therefore claim that these are congruent by the Vertical Angles Congruence theorem.
Therefore, the missing information in c. is "Vertical Angles Theorem"
4)& ∠ 1≅ ∠ 4
4)& c. Vertical Angles Theorem
Blank d
Similar to d., we can use the Transitive Property of Congruence since both ∠ 4 and ∠ 6 are congruent to ∠ 1. This is the missing information in d.
5)& ∠ 6 ≅ ∠ 4
5)& d. Transitive Property of Congruence
Now we can complete the table.
