Question

"Question 4 ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that (i) ABE ≅ ACF (ii) AB = AC, i.e., ABC is an isosceles triangle.

Class 9 - Math - Triangles Page 124"

Answer 1

Congruence of triangles:
Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:
There are 4 criteria for congruence of triangles.

Here we use ASA Congruence.

ASA(angle side angle):
Two Triangles are congruent if two angles and the included side of One triangle are equal to two angles & the included side of the other trian gle.

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Given:
ΔABC in which BE perpendicular to AC & CF perpendicular to AB, such that BE=CF.

To Prove:
i) ΔABE ≅ ΔACF
ii) AB=AC

Proof:
(i) In ΔABE & ΔACF,
∠A = ∠A (Common)
∠AEB = ∠AFC (each 90°)
BE = CF (Given)

Therefore, ΔABE ≅ ΔACF (by ASA congruence rule)

(ii) since ΔABE ≅ ΔACF
Thus, AB = AC (by CPCT)

Therefore ∆ABC is an isosceles triangle.

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Hope this will help you..

Answer 2

Hello!!✋

here is your answer

Proof :-
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(I) In ∆ABE and ∆ACF

BE = CF ( Given )

< BAE = < CAF ( general sides )

< AEB = < AFC ( Each 90° )

.°. ∆ABE =~ ∆ ACF ( by AAS congruence rule )

ii ) ∆ABE =~ ∆ACF [ from (I) , already proved )

.°. AB = AC ( CPCT )

.°. ABC is an isoscele triangle

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