Question

jerry is working on the following geometric proof. what is the missing justification? given: ∠cef is supplementary to ∠hab. prove: △abc ~ △edc ∠cef is supplementary to ∠hab. given ∠hab and ∠hai form a linear pair. definition of a linear pair ∠hab and ∠hai are supplementary. angles in a linear pair are supplementary. ∠hai ≅ ∠cef two angles that are supplementary to the same angle are congruent. ∠hai ≅ ∠bac ; ∠gef ≅ ∠ced ∠bac ≅ ∠ced substitution property of congruence △abc ~ △edc aa similarity postulate 1 of 7 questions adjacent angles are congruent. vertical angles are congruent. corresponding angles are congruent. supplementary angles are congruent.

Explanation:

Step1: Analyze each option

• Adjacent angles are congruent: Adjacent angles are not necessarily congruent. This is a wrong property.
• Vertical angles are congruent: $\angle HAI$ and $\angle BAC$ are vertical angles (formed by the intersection of two lines \(HB\) and \(HI\)). Also, $\angle GEF$ and $\angle CED$ are vertical angles (formed by the intersection of two lines \(DG\) and \(EF\)). Vertical angles have the property of being congruent.
• Corresponding angles are congruent: Corresponding angles are related to parallel - line transversal situations. There is no indication of parallel lines in the given proof setup for this step.
• Supplementary angles are congruent: Supplementary angles (sum to \(180^{\circ}\)) are not necessarily congruent. For example, \(30^{\circ}\) and \(150^{\circ}\) are supplementary but not congruent.

Answer:

Vertical angles are congruent.