Prove that segment BC is congruent to segment AD?

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  • 1 month ago

    The given conditions are not clear. I understand your problem as that

    AB // CD

    <CBD=<ADB

    are given, & prove that BC=AD. If so, then

    Proof:

    <CBD=<ADB (given)

    AB//CD=><ADB=<BDC (alternate angles equal)

    BD=BD (common sides equal)

    Thus, Tri. BCD is congruent to tri. DAB (A.S.A)

    =>

    BC=AD (the corresponding sides of 2 congruent triangles equal)

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  • sepia
    Lv 7
    1 month ago

    Obviously, BC = AD, AB = DC,

    and ∠CBD = ∠ADB.

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  • 1 month ago

    In the figure itself it is seen that the line segment BC 

    is equal to the line segment AD.

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  • 1 month ago

    Well, they are already given as congruent in the diagram so technically a proof is unnecessary.

    If they weren't marked as congruent, you would say:

    AB || CD --- Given

    ∠ADB ≅ ∠CBD --- Given

    DB ≅ BD --- Reflexive property

    ∠ABD ≅ ∠CDB --- When a transversal (DB) cuts parallel lines, alternate interior angles are congruent.

    △ADB ≅ △CBD --- ASA (angle-side-angle) triangle congruence postulate

    BC ≅ AD --- Corresponding parts of congruent triangles are congruent (CPCTC)

    Q.E.D.

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  • Anonymous
    1 month ago

    You have a parallelogram since by the alternate angle theorem, BC is parallel to AD. 

    Property of a parallelogram is that opposite sides are equal, so BC = AD

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  • 1 month ago

    Something is not right with the question, "congruent" is not used for line segments, only triangles (or other polygons), as far as I know. In the diagram it is given that the segments BC and AD are equal.

    I'm assuming you meant "prove triangle ABD is congruent to triangle CDB", is that right? If so, use the parallel lines AB and DC to get another pair of equal angles, then use AAS (angle, angle, side) to prove congruence.

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    • qspeechc1 month agoReport

      Must be an American thing, this use of the word "congruent" does not appear in any of the textbooks I have, nor in Euclid's "Elements".

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