Question

hk || ij and ∠hij ≅ ∠ijk. complete the proof that ∠gkh ≅ ∠ghk.
statement reason
1 hk || ij given
2 ∠hij ≅ ∠ijk given
3 ∠ijk ≅ ∠gkh
4 ∠hij ≅ ∠ghk corresponding angles theorem
5 ∠hij ≅ ∠gkh
6 ∠gkh ≅ ∠ghk

Explanation:

Step1: Identify corresponding - angle relationship

Since $\overleftrightarrow{HK}\parallel\overleftrightarrow{IJ}$, by the Corresponding Angles Theorem, $\angle IJK\cong\angle GKH$ because they are corresponding angles.

Step2: Use transitive property

We know that $\angle HIJ\cong\angle IJK$ (given) and $\angle HIJ\cong\angle GHK$ (by the Corresponding Angles Theorem) and $\angle IJK\cong\angle GKH$. So, by the Transitive Property of Congruence, if $\angle HIJ\cong\angle IJK$ and $\angle IJK\cong\angle GKH$, then $\angle HIJ\cong\angle GKH$.

Step3: Conclude the proof

Since $\angle HIJ\cong\angle GKH$ and $\angle HIJ\cong\angle GHK$, by the Transitive Property of Congruence, $\angle GKH\cong\angle GHK$.

Answer:

1. Corresponding Angles Theorem
2. Transitive Property of Congruence
3. Transitive Property of Congruence