Question

Read the proof.

Given: aeec; bddc
Prove: △aec ~ △bdc
1.aeec;bddc | Given
2.∠aec is a rt. ∠; ∠bdc is a rt. ∠ | Definition of perpendicular
3.∠aec ≅ ∠bdc | All right angles are congruent
4.? | Reflexive property
5.△aec ~ △bdc | AA similarity theorem
What is the missing statement in step 4?
a) ∠ace ≅ ∠ace
b) ∠eab ≅ ∠dbc
c) ∠eac ≅ ∠eac
d) ∠cbd ≅ ∠dbc

Answer 1

The missing statement in step 4 using the Reflexive Property for proving that triangles AEC and BDC are similar is 'c) ∠eac ≅ ∠eac'. option c is the correct answer.

Step-by-step explanation:

The student is given that the lines ae and ec are perpendicular, as well as bd and dc. They have to prove that triangles △aec and △bdc are similar.

The definition of perpendicular lines tells us that ∠aec and ∠bdc are right angles, making them congruent based on the fact that all right angles are congruent. For step 4 of the proof, we need to use the Reflexive Property to identify a side or angle that is equal to itself within the two triangles.

Since ∠eac is an angle in both △aec and △bdc, the correct option in the final answer is 'c) ∠eac ≅ ∠eac'. With two pairs of congruent angles, the two triangles are similar by the AA similarity theorem.