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
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.
The vertical angle of an isosceles triangle is 100°. Find its base angles.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
In a ΔABC, it is given that AB = AC and the bisectors of ∠B and ∠C intersect at O. If M is a point on BO produced, prove that ∠MOC = ∠ABC.
Which of the following statements are true (T) and which are false (F):
The measure of each angle of an equilateral triangle is 60°
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.
Fill the blank in the following so that the following statement is true.
In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……
Is it possible to draw a triangle with sides of length 2 cm, 3 cm and 7 cm?
O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
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.
Fill in the blank to make the following statement true.
If two sides of a triangle are unequal, then the larger side has .... angle opposite to it.
The side BC of ΔABC is produced to a point D. The bisector of ∠A meets side BC in L. If ∠ABC = 30° and ∠ACD = 115°, then ∠ALC = ______.
In the given figure, if l1 || l2, the value of x is
The angles of a right angled triangle are
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].
In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.
