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
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.
In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles
Fill the blank in the following so that the following statement is true.
If altitudes CE and BF of a triangle ABC are equal, then AB = ....
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.
In ΔABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.
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)?
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, the sides BC, CA and AB of a Δ ABC have been produced to D, E and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the Δ ABC.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
In the given figure, if BP || CQ and AC = BC, then the measure of x is
In the given figure, AB and CD are parallel lines and transversal EF intersects them at Pand Q respectively. If ∠APR = 25°, ∠RQC = 30° and ∠CQF = 65°, then
In a ΔABC, ∠A = 50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACDmeet at E, then ∠E =
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
In the following figure, D and E are points on side BC of a ∆ABC such that BD = CE and AD = AE. Show that ∆ABD ≅ ∆ACE.
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].
