Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.
Find the measure of each exterior angle of an equilateral triangle.
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°.
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.
Which of the following statements are true (T) and which are false (F):
Sides opposite to equal angles of a triangle may be unequal
Which of the following statements are true (T) and which are false (F) :
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
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)
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?
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 = ______.
D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then ______.
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2AD? Give reason for your answer.
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
