in figure of a perpendicular to ab and perpendicular to AC prove that a is bisector of angle C A B
Answers
Answer:
In quadrilateral ACBD, AC = AD and AB bisects ∠A )see figure). Show that ΔABC ≌ ΔABD. What can you say about BC and BD?
Ans. In quadrilateral ABCD we have
AC = AD
and AB being the bisector of ∠A.
Now, in ΔABC and ΔABD,
AC = AD
[Given]
AB = AB
[Common]
∠CAB = ∠DAB [∴ AB bisects ∠CAD]
∴ Using SAS criteria, we have
ΔABC ≌ ΔABD.
∴ Corresponding parts of congruent triangles (c.p.c.t) are equal.
∴ BC = BD.
2. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Figure). Prove that
(i) ΔABD ≌ ΔBAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.
Ans. (i) In quadrilateral ABCD, we have AD = BC and
∠DAB = ∠CBA.
In ΔABD and ΔBAC,
AD = BC
[Given]
AB = BA
[Common]
∠DAB = ∠CBA
[Given]
∴ Using SAS criteria, we have ΔABD ≌ ΔBAC
(ii) ∵ ΔABD ≌ ΔBAC
∴ Their corresponding parts are equal.
⇒ BD = AC
(ii) Since ΔABD ≌ ΔBAC
∴ Their corresponding parts are equal.
⇒ ∠ABD = ∠BAC.
3. AD and BC are equal perpendiculars to a line segment AB (see figure). Show that CD bisects AB.
Ans. We have ∠ABC = 90° and ∠BAD = 90°
Also AB and CD intersect at O.
∴ Vertically opposite angles are equal.
Now, in ΔOBC and ΔOAD, we have
∠ABC = ∠BAD
[each = 90°]
BC = AD
[Given]
∠BOC = ∠AOD
[vertically opposite angles]
∴ Using ASA criteria, we have
ΔOBC ≌ ΔOAD
⇒ OB = OA
[c.p.c.t]
i.e. O is the mid-point of AB
Thus, CD bisects AB.
4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see figure). Show that ΔABC ≌ ΔCDA.