Question

If in △ABC, m∠A=38° and ​m∠B=67°, and in △DEF, m∠D=38° and m∠E=67°, which statements are true?

Select all correct answer choices. If there is only one correct answer choice, select "only" and the correct answer choice.

m∠F=67°

∠C≅∠F

m∠C=75°

only

∠C is not necessarily congruent to ∠F.

It is not possible to determine m∠F.

It is not possible to determine m∠C.

Answer 1

The correct answer is: [B]: " ∠C ≅ ∠F " .
______________________________________________________
Note: We are given that:

"In △ABC, m∠A=38° and m∠B=67 " .

Since a triangle has three (3) angles (by definition) ; and since all 3 angles in any triangle must add up to 180° (by definition); we can solve for:
m∠C ;

m∠C = 180 − (38 + 67) = 180 − 105 = 75 .

m∠C = 75° ;
________________________________________________________
Likewise, given △DEF, m∠D=38° and m∠E=67 ; we find m∠F as follows:

m∠F = 180 - (38 + 67) = 180 − 105 = 75 .

m∠F = 75° .
__________________________________________________
So, let us consider the answer choices given:
__________________________________________________
Choice [A]: "m∠F = 67° " ; is incorrect. We know that "m∠F = 75° " .
__________________________________________________
Choice [B]: " ∠C ≅ ∠F " is correct. We know that ΔABC ≅ ΔDEF ; and that;

m∠A = m∠D = 38° ; and that: m∠B = m∠E = 67° ; and that:
m∠C = m∠F = 75° ; and as such:

∠A ≅ D ; and ∠B ≅ ∠E ; and ∠C ≅ ∠F .
___________________________________________________
Choice [C]: "m∠C = 75° only" ; is incorrect. While it is correct that:
"m∠C = 75° " ; this answer choice is INCORRECT; since this answer choice is not the ONLY answer choice provided that holds true. In fact, we have already determined that answer choice: [B] (see above) is a correct statement.
___________________________________________________
Choice [D]: "∠C is not necessarily congruent to ∠F. " ; This answer choice is INCORRECT; as explained in Answer choice: [B]: (see above).
___________________________________________________
Choice [E]: "It is not possible to determine m∠F." ; This answer choice is INCORRECT. We know that " m∠F = 75° " .
___________________________________________________
Choice [F]: " It is not possible to determine m∠C." ; This answer choice is
INCORRECT. We know that " m∠C = 75° " .
___________________________________________________
The correct answer is: [B]: " ∠C ≅ ∠F " .
___________________________________________________