Advertisements
Advertisements
प्रश्न
If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Advertisements
उत्तर
Given that, if perpendicular from any point within, an angle on its arms is congruent, prove that it lies on the bisector of that angle
Now,
Let us consider an angle ABC and let BP be one of the arm within the angle
Draw perpendicular PN and PM on the arms BC and BA such that they meet BC and BA in N and M respectively.
Now, in ΔBPM and ΔBPN
We have ∠BMP= BNP = 90° [given]
BP=BP [Common side]
And MP=NP [given]
So, by RHS congruence criterion, we have
ΔBPM≅ΔBPN
Now,
∠MBP=∠NBP [ Corresponding parts of congruent triangles are equal]
⇒ BP is the angular bisector of ∠ABC
∴ Hence proved
APPEARS IN
संबंधित प्रश्न
Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see the given figure). Show that:
- ΔAPB ≅ ΔAQB
- BP = BQ or B is equidistant from the arms of ∠A.
In the given figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE.
Explain, why ΔABC ≅ ΔFED.
In Δ ABC, ∠B = 35°, ∠C = 65° and the bisector of ∠BAC meets BC in P. Arrange AP, BP and CP in descending order.
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
In a triangle ABC, D is mid-point of BC; AD is produced up to E so that DE = AD.
Prove that :
(i) ΔABD and ΔECD are congruent.
(ii) AB = CE.
(iii) AB is parallel to EC
A line segment AB is bisected at point P and through point P another line segment PQ, which is perpendicular to AB, is drawn. Show that: QA = QB.
In the following figures, the sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and median PS of the triangle PQR.
Prove that ΔABC and ΔPQR are congruent.
 |
 |
ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such produced to E and F respectively, such that AB = BE and AD = DF.
Prove that: ΔBEC ≅ ΔDCF.
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.
