Question

in the adjoining figure, triangle ABC is an isosceles triangle in which AB = AC and AD is the bisector of angle A.
Prove that :
1. Triangle ABC is congruent to Triangle ADC
2. Angle B = Angle C
3. BD = DC
4. AD is perpendicular to BC

Answer 1

Step-by-step explanation:

$Given \: Δ \: is \: an \: isosceles \: triangle \\ ⇒ AB-BC ...(1) \\ and ∠B=∠C... (2) \\ Here \: E \: and \: F \: are \: midpoints \: of \: AC \: and \: AB \: respectively \\ ∴ AF \: = FB \: and \: AE \: = EC know , \: AB \: = BC \\ ⇒ AF \: +FB = AE + EC \\ ⇒ 2AF = 2AE \\ ⇒ AF = AE. \\ </p><p>⇒ AF =FB = AE =EC ...(3) \\ In \: ΔBCF \: and \: Δ CBEBC = BC [common side] \\ </p><p>∠B=∠C [from (2)] \\ BF = EC [from (3)] \\ By \: SAS \: condition \: for \: congruency. \\ </p><p>ΔBCF≅ΔCBE. \\ ∴ since \: ΔBCF≅Δ \: CBE, \: by \: we \: can \: with \: that \: BE = CF.$