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Question
your friend asks you for help on a geometry exercise. your friend’s paper is shown to the right. what error did your friend make? explain choose the correct answer below.
a. your friend’s paper shows that the sas postulate should be used to show congruence because a pair of congruent angles is included between two pairs of congruent sides.
b. your friend’s paper does not name the triangles correctly for them to be congruent.
c. the asa postulate cannot be used to prove the congruence of the two triangles as shown because the triangles do not have two pairs of congruent angles.
d. the asa postulate cannot be used to prove the congruence of the two triangles as shown because the included sides between the two pairs of congruent angles are not marked as congruent in both triangles.
To determine the error, we analyze the ASA (Angle - Side - Angle) postulate. The ASA postulate requires two pairs of congruent angles and the included side (the side between the two angles) to be congruent. If the included side between the two pairs of congruent angles is not marked as congruent in both triangles, then the ASA postulate cannot be used to prove congruence. Let's analyze each option:
- Option A: Talks about SAS (Side - Angle - Side), but the question is about an error in using ASA, so A is incorrect.
- Option B: The error is not about naming the triangles, so B is incorrect.
- Option C: The ASA postulate requires two pairs of congruent angles and the included side. If the triangles do not have two pairs of congruent angles, ASA can't be used. But from the diagram (implied by the ASA claim), the main issue is about the included side. However, re - evaluating, if the triangles don't have two pairs of congruent angles, ASA is inapplicable. But the more precise error related to ASA is about the included side, but let's check option D.
- Option D: The ASA postulate needs the included side between the two pairs of congruent angles to be congruent. If that side is not marked as congruent in both triangles, ASA can't be used. But the key for ASA is two angles and the included side. If the triangles don't have two pairs of congruent angles (option C), then ASA is wrong. Wait, maybe the diagram shows that the friend tried to use ASA, but the triangles don't have two pairs of congruent angles. So the friend's error is that the ASA postulate cannot be used because the triangles do not have two pairs of congruent angles (option C) or about the included side (option D). Wait, let's re - read the options. Option D says "The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the included sides between the two pairs of congruent angles are not marked as congruent in both triangles". Option C says "The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the triangles do not have two pairs of congruent angles". If the friend is using ASA, ASA requires two angles and the included side. So first, we need two pairs of congruent angles and the included side. If the triangles don't have two pairs of congruent angles, ASA is invalid. But maybe the diagram has one pair of angles, so option C is correct? Wait, no. Let's think again. The ASA postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. So for ASA, we need two pairs of congruent angles (so two angles in each triangle) and the included side. If the triangles do not have two pairs of congruent angles (i.e., only one pair or none), then ASA can't be used. So option C is correct because the friend tried to use ASA, but the triangles don't have two pairs of congruent angles, so ASA is inapplicable. Wait, but maybe the diagram shows that there is one pair of angles, so the friend's error is that the triangles don't have two pairs of congruent angles, so ASA can't be used. So the correct option is C? Wait, no, let's check the options again. Option D: "the included sides between the two pairs of congruent angles are not marked as congruent in both triangles". If there are two pairs of congruent angles, but the included side is not congruent, then ASA fails. But if there aren't two pairs of congruent angles, then ASA is not applicable. So the error is that the ASA postulate cannot be used because…
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C. The ASA Postulate cannot be used to prove the congruence of the two triangles as shown because the triangles do not have two pairs of congruent angles.