Question - Geometric Proof for Parallel Line Segments
mathgeometric proofcorresponding angles postulateparallel line segmentsalternate interior angles theorem
Solution:
To prove $$ ABCD \parallel EFG $$, we look for angle relationships that would imply parallelism through the Converse of the Corresponding Angles Postulate or the Converse of the Alternate Interior Angles Theorem.The correct statement that would allow Steve to prove $$ ABCD \parallel EFG $$ is:$$ \angle CBF \cong \angle GFC $$Using the Converse of the Alternate Interior Angles Theorem, if $$ \angle CBF \cong \angle GFC $$, then line segment $$ ABCD $$ is parallel to line segment $$ EFG $$.
Tutor Solution:
In order to prove that line segments $$ABCD$$ is parallel to $$EFG$$, we must use theorems or postulates related to parallel lines and their corresponding angles.Looking at the options provided, we need to find a statement that involves angle relationships which imply that $$ABCD$$ || $$EFG$$.1) $$\angle CFG \cong \angle FCB$$: This would suggest the Alternate Interior Angles Theorem, which says if two alternate interior angles are congruent, then the lines are parallel. Thus, if $$\angle CFG$$ is congruent to $$\angle FCB$$, it implies that $$ABCD$$ is parallel to $$EFG$$. This is a valid statement to prove parallel lines.2) $$\angle ABF \cong \angle BFC$$: These angles are not alternate interior angles; they are adjacent to each other and form a linear pair. Congruence between these angles does not imply that $$ABCD$$ is parallel to $$EFG$$.3) $$\angle EFB \cong \angle CFB$$: These angles are corresponding angles, and the Corresponding Angles Postulate states that if a transversal intersects two lines such that a pair of corresponding angles is congruent, then the lines are parallel. Therefore, if $$\angle EFB$$ is congruent to $$\angle CFB$$, $$ABCD$$ would be parallel to $$EFG$$. This is also a valid statement to prove parallel lines.4) $$\angle CBF \cong \angle GFC$$: These angles do not provide direct evidence about alternate interior, corresponding, or alternate exterior angles that are related to parallel lines. Thus, the congruence between these angles does not imply that $$ABCD$$ is parallel to $$EFG$$.The statements that can prove $$ABCD || EFG$$ are the statements that involve alternate interior angles (option 1) or corresponding angles (option 3). Hence, options 1 and 3 are correct, and each of them would allow Steve to prove that $$ABCD$$ is parallel to $$EFG$$. However, as the image only allows us to choose one option, typically the one that is directly observed in the given diagram, option 1 would be the preferred choice here if the angles mentioned are indeed congruent as per the given diagram.
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