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
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
Angles A, B, C of a triangle ABC are equal to each other. Prove that ΔABC is equilateral.
Fill the blank in the following so that the following statement is true.
In an equilateral triangle all angles are .....
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than twice the median drawn to the third side.
Fill in the blank to make the following statement true.
If two angles of a triangle are unequal, then the smaller angle has the........ side opposite to it.
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
If the angles A, B and C of ΔABC satisfy the relation B − A = C − B, then find the measure of ∠B.
If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?
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 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 = ______.
It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true?
In ∆PQR, if ∠R > ∠Q, then ______.
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
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].
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
