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Triangles

    • Question 1
    CBSEENMA9009806
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    In quadrilateral ACBD, AC = AD and AB bisects ∠A (See the given figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD?

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    Solution

    In ΔABC and ΔABD,

    AC = AD (Given)

    ∠CAB = ∠DAB (AB bisects ∠A)

    AB = AB (Common)

    ∴ ΔABC ≅ ΔABD (By SAS congruence rule)

    ∴ BC = BD (By CPCT)

    Therefore, BC and BD are of equal lengths.

    Question 2
    CBSEENMA9009807
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    ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (See the given figure). Prove that

    (i) ΔABD ≅ ΔBAC

    (ii) BD = AC

    (iii) ∠ABD = ∠BAC.

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    Solution

    In ΔABD and ΔBAC,

    AD = BC (Given)

    ∠DAB = ∠CBA (Given)

    AB = BA (Common)

    ∴ ΔABD ≅ ΔBAC (By SAS congruence rule)

    ∴ BD = AC (By CPCT)

    And, ∠ABD = ∠BAC (By CPCT)

    Question 3
    CBSEENMA9009808
    Wired Faculty App

    AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.

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    Solution

    In ΔBOC and ΔAOD,

    ∠BOC = ∠AOD (Vertically opposite angles)

    ∠CBO = ∠DAO (Each 90º)

    BC = AD (Given)

    ∴ ΔBOC ≅ ΔAOD (AAS congruence rule)

    ∴ BO = AO (By CPCT)

    ⇒ CD bisects AB.

    Question 4
    CBSEENMA9009809
    Wired Faculty App

    l and m are two parallel lines intersected by another pair of parallel lines p and q (see the given figure). Show that ΔABC ≅ ΔCDA.

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    Solution

    In ΔABC and ΔCDA,

    ∠BAC = ∠DCA (Alternate interior angles, as p || q)

    AC = CA (Common)

    ∠BCA = ∠DAC (Alternate interior angles, as l || m)

    ∴ ΔABC ≅ ΔCDA (By ASA congruence rule)

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