Advertisements
Advertisements
प्रश्न
In a ΔABC, ∠A = 50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACDmeet at E, then ∠E =
पर्याय
25°
50°
100°
75°
Advertisements
उत्तर
In the given figure, bisectors of ∠ABCand ∠ACDmeet at E and ∠BAC = 50°
We need to find ∠BEC
Here, using the property, “an exterior angle of the triangle is equal to the sum of the opposite interior angles”, we get,
In ΔABC with ∠ACD as its exterior angle
ext . ∠ACD = ∠A + ∠ABC ........(1)
Similarly, in Δ BECwith ∠ECDas its exterior angle
ext . ∠ECD = ∠EBC + ∠BEC
`1/2 ∠ACD = 1/2 ∠ABC + BEC`
(CE and BE are the bisectors of ∠ACD and ∠ABC)
∠BEC = 1/2 ∠ACD - 1/2 ∠ABC .......(2)
Now, multiplying both sides of (1) by 1/2
We get,
`1/2 ∠ACD = 1/2 ∠A +1/2 ∠ABC`
`1/2 ∠A = 1/2 ∠ACD - 1/2 ∠ABC` ......(3)
From (2) and (3) we get,
`∠BEC =1/2 ∠A`
`∠BEC =1/2 (50°)`
`∠BEC = 25°`
Thus, `∠BEC = 25°`
APPEARS IN
संबंधित 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
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
- ΔABE ≅ ΔACF
- AB = AC, i.e., ABC is an isosceles triangle.
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
Prove that the medians of an equilateral triangle are equal.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.
Prove that each angle of an equilateral triangle is 60°.
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 = ....
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.
In a right triangle the hypotenuse is the .... 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.
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.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
In the given figure, if AB ⊥ BC. then x =
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 = ______.
Which of the following correctly describes the given triangle?
D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then ______.
