Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given: D and E are the points on side BC of a ∆ABC such that BD = CE and AD = AE.
To show: ∆ABD ≅ ∆ACE
Proof: We have, AD = AE ...[Given]
⇒ ∠ADE = ∠AED ...(i) [Since, angles opposite to equal sides are equal]
We have, ∠ADB + ∠ADE = 180° ...[Linear pair axiom]
⇒ ∠ADB = 180° – ∠ADE
= 180° – ∠AED ...[From equation (i)]
In ∆ABD and ∆ACE,
∠ADB = ∠AEC ...[∵ ∠AEC + ∠AED = 180°, linear pair axiom]
BD = CE ...[Given]
And AD = AE ...[Given]
∴ ∆ABD ≅ ∆ACE ...[By SAS congruence rule]
APPEARS IN
संबंधित प्रश्न
Find the measure of each exterior angle of an equilateral triangle.
In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that
(i) PT = QT (ii) ∠TQR = 15°
BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Fill the blank in the following so that the following statement is true.
In an equilateral triangle all angles are .....
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.
Fill the blank in the following so that the following statement is true.
In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……
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
O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
In Fig. 10.131, prove that: (i) CD + DA + AB + BC > 2AC (ii) CD + DA + AB > BC
Fill in the blank to make the following statement true.
The sum of three altitudes of a triangle is ..... than its perimeter.
In the given figure, if AB ⊥ BC. then x =
In the given figure, what is the value of x?
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
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 = ______.
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2AD? Give reason for your answer.
M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.
