The given figure shows two isosceles triangles: ΔABC in which AB =
AC and ΔDBC in which DB = DC. What is the measure of ∠BAC?
Answers
Answer:
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:
SAS( side angle side):
Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.
SSS(side side side):
Three sides of One triangle are equal to the three sides of another triangle then the two Triangles are congruent.
__________________________________
Solution;
Given,
ΔABC and ΔDBC are two isosceles triangles in which AB=AC & BD=DC.
To Prove:
(i) ΔABD ≅ ΔACD
(ii) ΔABP ≅ ΔACP
(iii) AP bisects ∠A as well as ∠D.
(iv) AP is the perpendicular bisector of BC.
Proof:
(i) In ΔABD and ΔACD,
AD = AD (Common)
AB = AC (given)
BD = CD (given)
Therefore, ΔABD ≅ ΔACD (by SSS congruence rule)
∠BAD = ∠CAD (CPCT)
∠BAP = ∠CAP
(ii) In ΔABP & ΔACP,
AP = AP (Common)
∠BAP = ∠CAP
(Proved above)
AB = AC (given)
Therefore,
ΔABP ≅ ΔACP
(by SAS congruence rule).
(iii)
∠BAD = ∠CAD (proved in part i)
Hence, AP bisects ∠A.
also,
In ΔBPD and ΔCPD,
PD = PD (Common)
BD = CD (given)
BP = CP (ΔABP ≅ ΔACP so by CPCT.)
Therefore, ΔBPD ≅ ΔCPD (by SSS congruence rule.)
Thus,
∠BDP = ∠CDP( by CPCT.)
Hence, we can say that AP bisects ∠A as well as ∠D.
(iv)
∠BPD = ∠CPD
(by CPCT as ΔBPD ≅ ΔCPD)
& BP = CP (CPCT)
also,
∠BPD + ∠CPD = 180° (BC is a straight line.)
⇒ 2∠BPD = 180°
⇒ ∠BPD = 90°
Hence,
AP is the perpendicular bisector of BC.
Answer:
123121231221222221212111