Advertisements
Advertisements
Question
In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:
- OB = OC
- AO bisects ∠A
Advertisements
Solution
(i) ABC is an isosceles triangle in which AB = AC
∠C = ∠B ...[Angles opposite to equal sides in a triangle are equal.]
⇒ ∠OCA + ∠OCB = ∠OBA + ∠OBC
⇒ ∠OCB + ∠OCB = ∠OBC + ∠OBC
∵ OB bisects ∠B.
∴ ∠OBA = ∠OBC
And OC bisects ∠C.
∴ ∠OCA = ∠OCB
⇒ 2∠OCB = 2∠OBC
⇒ ∠OCB = ∠OBC
Now, in △OBC,
∠OCB = ∠OBC ...[Proved above]
∴ OB = OC ...[Sides opposite to equal angles]
(ii) Now, in △AOB and △AOC,
AB = AC ...[Given]
∠OBA = ∠OCA
∠B = ∠C
BO bisects ∠B and CO bisects ∠C.
∠OBA = ∠OCA
OB = OC
∴ △AOB ≌ △AOC ...[By SAS congruence rule]
⇒ ∠OAB = ∠OAC ...[Corresponding parts of congruent triangles]
So, AO bisects ∠A.
APPEARS IN
RELATED QUESTIONS
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
In a ΔABC, if AB = AC and ∠B = 70°, find ∠A.
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
ABC is a triangle in which ∠B = 2 ∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
Prove that ∠BAC = 72°.
In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that LN = MN.
Which of the following statements are true (T) and which are false (F):
Angles opposite to equal sides of a triangle are equal
Which of the following statements are true (T) and which are false (F):
If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.
Fill the blank in the following so that the following statement is true.
In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …… CE.
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
In ΔABC, side AB is produced to D so that BD = BC. If ∠B = 60° and ∠A = 70°, prove that: (i) AD > CD (ii) AD > AC
Which of the following statements are true (T) and which are false (F)?
Sum of the three sides of a triangle is less than the sum of its three altitudes.
Which of the following statements are true (T) and which are false (F)?
If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.
Write the sum of the angles of an obtuse triangle.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
In the given figure, what is z in terms of x and y?
In the given figure, what is y in terms of x?
ABC is an isosceles triangle with AB = AC and D is a point on BC such that AD ⊥ BC (Figure). To prove that ∠BAD = ∠CAD, a student proceeded as follows:
In ∆ABD and ∆ACD,
AB = AC (Given)
∠B = ∠C (Because AB = AC)
and ∠ADB = ∠ADC
Therefore, ∆ABD ≅ ∆ACD (AAS)
So, ∠BAD = ∠CAD (CPCT)
What is the defect in the above arguments?
[Hint: Recall how ∠B = ∠C is proved when AB = AC].
