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    CAT Important Theorem of Triangles (Part 2) - Practice Questions & MCQ

    Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

    Quick Facts

    • 5 Questions around this concept.

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    QuestionEASY

    Two sides AB and BC, and the median AD of $\triangle$ABC are correspondingly equal to the two sides PQ and QR, and the median PM of $\triangle$PQR. Then:

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    In the given figure,

    ABCD is a square, M is the midpoint of AB, and PQ $\perp$ CM meets AD at P and BC produced at Q. Then:

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    In the given figure,

    ABC is a triangle, right angled at B. If BCDE is a square on side BC and ACFG is a square on AC. Then:

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    Let P be a point equidistant from two intersecting lines L and M at point A. Then:

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    Concepts Covered - 1

    Important Theorem of Triangles (Part 2)

    Theorem 1: The angles opposite to two equal sides of a triangle are equal.

    Given: Let a \triangle A B C\text{ in which }A B=A C

    To prove: \angle B=\angle C

    Construction: Draw AD, the bisector of \angle A, to meet BC in D.

    Proof: 

    {\color{Blue} \text{In } \triangle A B D\text{ and }\triangle A C D,\text{ we have,}}

    {\color{Blue} A B=A C \quad(\text { given })}

    {\color{Blue} AD=AD \quad(\text { common })}

    {\color{Blue} \angle B A D=\angle C A D \quad(\text { by construction })}

    {\color{Blue} \therefore \quad \triangle A B D \cong \triangle A C D \quad \text { (SAS-criteria) } }

    {\color{Blue} \text{Hence}, \;\;\angle B=\angle C \quad ( \text{ Corresponding parts of Congruent triangles }) }

     

    Theorem 2 (Converse of Theorem 2): If two angles of a triangle are equal then the sides opposite to them are also equal..

    Given: Let a \triangle A B C\text{ in which }\angle B=\angle C

    To prove: A B=A C

    Construction: Draw AD, the bisector of \angle A, to meet BC at D.

    Proof: 

    {\color{Blue} \text{In } \triangle A B D\text{ and }\triangle A C D,\text{ we have,}}

    {\color{Blue} \angle B=\angle C \quad(\text { given })}

    {\color{Blue} \angle B A D=\angle C A D \quad(\text { by construction })}

    {\color{Blue} AD=AD \quad(\text { common })}

    {\color{Blue} \therefore \quad \triangle A B D \cong \triangle A C D \quad \text { (AAS-criteria) } }

    {\color{Blue} \text{Hence}, \;\;AB=AC \quad ( \text{ Corresponding parts of Congruent triangles }) }

    Study other Related Concepts

    CAT Important Theorem of Triangles (Part 2) Current Topic

    CAT Basic Terms and Definitions of Lines and Angles
    CAT Various Types of Angle
    CAT Linear Pair of Angles
    CAT Vertically Opposite Angles
    CAT Corresponding Angle Axioms
    CAT Alternate Interior Angle Theorem
    CAT Interior Angle Theorem
    CAT Parallel and a Transversal Lines
    CAT Introduction to Triangles
    CAT Congruence of Triangles

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