Question

4. ABC is a triangle in which altitudes BE and CF to
sides AC and AB are equal (see Fig. 7.32). Show
that
(i) triangle ABE=CONGRUENT TO TRIANGLE ACF
(ii) AB=AC, i.e., ABC is an isosceles triangle.​

Answer 1

Answer:

hey buddy I have answer for u

Step-by-step explanation:

In,ΔABE and ΔACF,

∠BAE=∠CAF (Common angle)

∠AEB=∠AFC ....(∵BE⊥AC and CF⊥AB)

BE=CF (Given that altitudes are equal)

By AAS criterion of congruence,

ΔABE≅ΔACF

Hence,

AB=AC (by CPCT)

Answer 2

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\tt 4) \:ABC\: is \:a \:triangle \:in \:which\: altitudes\: BE\: and \:CF\\\tt to \:sides \:AC \:and\: AB\: are\: equal.\: Show\: that$

$\tt (i)\: triangle \:ABE \:is\: CONGRUENT\: to\: triangle\: ACF.$

$\tt (ii) \:AB=AC\:, i.e.,\: ABC\: is \:an \:isosceles\: triangle.$

$\sf \bf \huge {\boxed {\mathbb {ANSWER}}}$

$\sf \bf {\boxed {\mathbb {GIVEN}}}$

• $\sf \bf BE=CF$
• $\sf \bf BE\: and \:CF \:are\: altitudes$
• $\sf \bf \angle AFC={90}^{\circ}$
• $\sf \bf \angle AEB={90}^{\circ}$

$\sf \bf {\boxed {\mathbb {TO\:FIND}}}$

• $\sf \bf (i)\: \triangle ABE\: \cong\: \triangle ACF.$
• $\sf \bf (ii) \:AB=AC$

$\sf \bf {\boxed {\mathbb {SOLUTION}}}$

$\sf In \:\triangle ABE \:and \:\triangle ACF$

$\sf \bf \implies \angle A=\angle A \longrightarrow \Big(Common \:Angle \Big)$

$\sf \bf \implies \angle AEB=\angle AFC \longrightarrow \Big(Given\Big)$

$\sf \bf \implies BE=CF \longrightarrow \Big(Given\Big)$

${\orange{\boxed{\boxed{\purple{\sf \therefore \triangle ABE\:\cong\:\triangle ACF \longrightarrow \Big(AAS\:congruence \:rule\Big)}}}}}$

$\implies {\orange{\boxed{\boxed{\purple {\sf \bf AB=AC \longrightarrow \Big(CPCT\Big)}}}}}$

$\sf \bf \huge {\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU}}}$

__________________________________________

$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

${\pink {\mathfrak{ Congruence \:rules}}}$

$\sf ASA=Angle \:Side\:Angle$

$\sf AAS=Angle\:Angle\:Side$

$\sf SAS=Side\:Angle \:Side$

$\sf SSS=Side\:Side \:Side$

$\sf RHS=Right \:angle\:Hypotenuse \:Side$