Advertisements
Advertisements
Question
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.
Advertisements
Solution
Given that in , ΔABC,
AB=AC and the bisector of ∠B and ∠C intersect at O and M is
a point on BO produced
We have to prove ∠ MOC=∠ABC
Since,
AB =AC ⇒ΔABC is isosceles ⇒ ∠B=∠C(or) ∠ABC=∠ACB
Now,
BO and CO are bisectors of ∠ABC and ∠ACB respectively
⇒ABO=∠OBC=∠ACO=∠OB=`1/2` ∠ABC=`1/2`∠ACB ............(1)
We have, in ΔOBC
∠OBC +∠OCB +∠BOC =180° .............(2)
And also
∠BOC +∠COM =180° ..................(3)[Straight angle]
Equating (2) and (3)
⇒ ∠OBC+∠OCB+-∠BOC=∠BOC+∠MOC
⇒ ∠OBC+∠OBC=∠MOC [∵ from (1)]
⇒ 2∠OBBC=∠MOC
⇒2`(1/2∠ABC)`=∠MOC [∵from (1)]
⇒ ∠ABC=∠MOC
∴ ∠MOC=∠ABC
APPEARS IN
RELATED QUESTIONS
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that
(i) PT = QT (ii) ∠TQR = 15°
Prove that the medians of an equilateral triangle are equal.
P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.
Fill the blank in the following so that the following statement is true.
In a ΔABC if ∠A = ∠C , then AB = ......
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
Is it possible to draw a triangle with sides of length 2 cm, 3 cm and 7 cm?
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.
Line segments AB and CD intersect at O such that AC || DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =
In the given figure, what is z in terms of x and y?
In the given figure, what is the value of x?
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
The angles of a right angled triangle are
In ∆ABC, AB = AC and ∠B = 50°. Then ∠C is equal to ______.
D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then ______.
Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle cannot be ______.
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2AD? Give reason for your answer.
Is it possible to construct a triangle with lengths of its sides as 9 cm, 7 cm and 17 cm? Give reason for your answer.
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].
