Advertisements
Advertisements
प्रश्न
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
उत्तर
(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
संबंधित प्रश्न
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
AB is a line seg P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.
The vertical angle of an isosceles triangle is 100°. Find its base angles.
In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC.
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.
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.
Which of the following statements are true (T) and which are false (F):
Sides opposite to equal angles of a triangle may be unequal
Which of the following statements are true (T) and which are false (F):
Angles opposite to equal sides 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
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.
Which of the following statements are true (T) and which are false (F)?
Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.
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.
In the given figure, x + y =
In the given figure, if AB ⊥ BC. then x =
In the given figure, the value of x is ______.
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
In ∆PQR, if ∠R > ∠Q, then ______.
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
