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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.

Solve by difficulty

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:

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:

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:

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

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 }) }

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