Which rotation would NOT map the octagon onto itself?

(C) Base angle theorem

Step-by-step explanation:

Given: JKLM is an isosceles trapezoid, KL ∥ JM  

Prove: KM ≅ JL

Proof:

Statements                                                                 Reasons

1. JKLM is an isosceles trapezoid, KL ∥ JM               Given

2. JK ≅ LM                                                definition of isosceles trapezoid

3.  KL ≅ KL                                                          reflexive property

4. ∠JKL ≅ ∠MLK                                          Base angle theorem (base=LK)

5. △JKL ≅ △MLK                                                       SAS rule

6. KM ≅ JL                                                                  CPCTC  

Hence proved.

therefore, option C is correct.