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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

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  • 5 Questions around this concept.

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

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In the given figure,

ABCD is a square, M is the midpoint of AB, and PQ ⊥ 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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