Question

given that ef || hj and ef≠hj, prove that △egf ~ △jgh.
statement reason

1. ∠e = ∠j alternate interior angle theorem
2. ef || hj given
3. ∠egf = ∠jgh vertical angle theorem
4. △egf ~ △jgh aas
5. ∠e and ∠j are alternate interior angles definition of alternate interior angles
6. ∠egf and ∠jgh are vertical angles definition of vertical angles
7. ef ≠ hj given

the proof shown is written in an incorrect order. what is a possible order that the statements and reasons could be written in?
7, 6, 5, 2, 1, 3, 4
3, 6, 7, 2, 1, 5, 4
6, 7, 1, 2, 5, 3, 4
two of these

Explanation:

Step1: Start with vertical - angle def

We first establish the definition of vertical angles (Statement 6) as it is a basic geometric property related to the angles at the intersection point.

Step2: State given side - length

Then, state the given side - length condition (Statement 7) as it is one of the initial given facts.

Step3: Use alternate - interior angles

Next, use the alternate - interior angles theorem (Statement 1) since we know EF || HJ (which we state right after as the given parallel - line condition in Statement 2).

Step4: More on alternate - interior

After that, use the definition of alternate interior angles (Statement 5) for further clarification of angle relationships.

Step5: Vertical - angle theorem

Then, use the vertical - angle theorem (Statement 3) to get another pair of equal angles.

Step6: Prove similarity

Finally, use the AAS (Angle - Angle - Side) criterion (Statement 4) to prove the similarity of the two triangles.

Answer:

6, 7, 1, 2, 5, 3, 4