Advertisements
Advertisements
प्रश्न
In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O.
Prove that : (i) BO = CO
(ii) AO bisects angle BAC.
Advertisements
उत्तर
In ΔABC,
AB = AC
⇒ ∠B = ∠C ...( angles opposite to equal sides are equal )
⇒ `1/2 ∠"B" = 1/2∠"C"`
⇒ ∠OBC = ∠OCB ...[ ∵ OB and OC are bisectors of ∠B and ∠C respectively, ∠OBC = `1/2∠"B" and ∠"OCB" = 1/2∠"C"` ] ...(i)
⇒ OB = OC ...( Sides opposite to equal angles are equal ) ...(ii)
Now, in ΔABO and ΔACO,
AB = AC ...( given )
∠OBC = ∠OCB ...[ from(i) ]
OB = OC ...[ from(ii) ] ...( proved )
∴ ΔABO ≅ ΔACO ...( by SAS congruence criterion )
⇒ ∠BAO = ∠CAO ...( c.p.c.t. )
⇒ AO bisects ∠BAC ...(proved)
APPEARS IN
संबंधित प्रश्न
In quadrilateral ACBD, AC = AD and AB bisects ∠A (See the given figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD?
In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:
- ΔAMC ≅ ΔBMD
- ∠DBC is a right angle.
- ΔDBC ≅ ΔACB
- CM = `1/2` AB
You want to show that ΔART ≅ ΔPEN,
If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have
1) RT = and
2) PN =
In triangles ABC and CDE, if AC = CE, BC = CD, ∠A = 60°, ∠C = 30° and ∠D = 90°. Are two triangles congruent?
The given figure shows a circle with center O. P is mid-point of chord AB.
Show that OP is perpendicular to AB.
In the given figure: AB//FD, AC//GE and BD = CE;
prove that:
- BG = DF
- CF = EG
In ∆ABC, AB = AC. Show that the altitude AD is median also.
In a ΔABC, BD is the median to the side AC, BD is produced to E such that BD = DE.
Prove that: AE is parallel to BC.
AD and BC are equal perpendiculars to a line segment AB. If AD and BC are on different sides of AB prove that CD bisects AB.
PQRS is a parallelogram. L and M are points on PQ and SR respectively such that PL = MR.
Show that LM and QS bisect each other.
