Advertisements
Advertisements
Question
In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O.
Prove that : (i) BO = CO
(ii) AO bisects angle BAC.
Advertisements
Solution
In ΔABC,
AB = AC
⇒ ∠B = ∠C ...( angles opposite to equal sides are equal )
⇒ `1/2 ∠"B" = 1/2∠"C"`
⇒ ∠OBC = ∠OCB ...[ ∵ OB and OC are bisectors of ∠B and ∠C respectively, ∠OBC = `1/2∠"B" and ∠"OCB" = 1/2∠"C"` ] ...(i)
⇒ OB = OC ...( Sides opposite to equal angles are equal ) ...(ii)
Now, in ΔABO and ΔACO,
AB = AC ...( given )
∠OBC = ∠OCB ...[ from(i) ]
OB = OC ...[ from(ii) ] ...( proved )
∴ ΔABO ≅ ΔACO ...( by SAS congruence criterion )
⇒ ∠BAO = ∠CAO ...( c.p.c.t. )
⇒ AO bisects ∠BAC ...(proved)
APPEARS IN
RELATED QUESTIONS
AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.
In Fig. 10.92, it is given that AB = CD and AD = BC. Prove that ΔADC ≅ ΔCBA.
In the given figure, prove that:
CD + DA + AB > BC
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ∠C = 75°, DE = 2.5 cm, DF = 5cm and ∠D = 75°. Are two triangles congruent?
In the given figure, AB = DB and Ac = DC.
If ∠ ABD = 58o,
∠ DBC = (2x - 4)o,
∠ ACB = y + 15o and
∠ DCB = 63o ; find the values of x and y.
In the following figure, AB = AC and AD is perpendicular to BC. BE bisects angle B and EF is perpendicular to AB.
Prove that : ED = EF
In the following figures, the sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and median PS of the triangle PQR.
Prove that ΔABC and ΔPQR are congruent.
 |
 |
In the following figure, OA = OC and AB = BC.
Prove that: ΔAOD≅ ΔCOD
In the following figure, ABC is an equilateral triangle in which QP is parallel to AC. Side AC is produced up to point R so that CR = BP.
Prove that QR bisects PC.
Hint: ( Show that ∆ QBP is equilateral
⇒ BP = PQ, but BP = CR
⇒ PQ = CR ⇒ ∆ QPM ≅ ∆ RCM ).
ABC is a right triangle with AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.
