Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.
In Figure AB = AC and ∠ACD =105°, find ∠BAC.
The vertical angle of an isosceles triangle is 100°. Find its base angles.
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.
BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.
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.
Which of the following statements are true (T) and which are false (F):
The bisectors of two equal angles of a triangle are equal
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
In Fig. 10.131, prove that: (i) CD + DA + AB + BC > 2AC (ii) CD + DA + AB > BC
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.
The sum of three altitudes of a triangle is ..... than its perimeter.
Fill in the blank to make the following statement true.
The sum of any two sides of a triangle is .... than the third side.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
The base BC of triangle ABC is produced both ways and the measure of exterior angles formed are 94° and 126°. Then, ∠BAC =
It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true?
