Mathematics
Assertion (A): ΔABD ≅ ΔACE
Reason (R): ∠ADE + ∠ADB = ∠AEC + ∠AED
But AD = AE
⇒ ∠ADE = ∠AED
∴ ∠ADB = ∠AEC
⇒ ∠ABD ≅ ∠AEC
A is true, but R is false.
A is false, but R is true.
Both A and R are true and R is the correct reason for A.
Both A and R are true and R is the incorrect reason for A.
Answer
In ΔADE,
⇒ AD = AE (Given)
⇒ ∠ADE = ∠AED = x ………………(1) [Angles opposite to equal sides of the triangle are also equal]
As we know ∠ADE, ∠ADB and ∠AEC, ∠AED forms linear pairs.
So, ∠ADE + ∠ADB = ∠AEC + ∠AED
Using equation (1), we get
⇒ x + ∠ADB = ∠AEC + x
⇒ ∠ADB = ∠AEC
So, reason (R) is true.
In ΔABD and ΔACE,
⇒ ∠ADB = ∠AEC (Proved above)
⇒ AD = AE (Given)
⇒ BD = EC (Given)
∴ ΔABD ≅ ΔACE (By SAS congruency criterion)
∴ Both A and R are true, and R is the correct reason for A.
Hence, option 3 is the correct option.
Related Questions
Statement 1: If two angles and a side of one triangle are equal to two angles and a side of some another triangle, the triangles are congruent.
Statement 2: The two triangle will be congruent, if corresponding sides of the two triangles are equal.
Both the statements are true.
Both the statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
Assertion (A): If PQ = PR, ΔPQS ≅ ΔPRT
Reason (R): PQ = PR, ∠P = ∠P and ∠Q = ∠R
A is true, but R is false.
A is false, but R is true.
Both A and R are true and R is the correct reason for A.
Both A and R are true and R is the incorrect reason for A.
Which of the following pairs of triangles are congruent ? In each case, state the condition of congruency :
(a) In △ ABC and △ DEF, AB = DE, BC = EF and ∠B = ∠E.
(b) In △ ABC and △ DEF, ∠B = ∠E = 90°; AC = DF and BC = EF.
(c) In △ ABC and △ QRP, AB = QR, ∠B = ∠R and ∠C = ∠P.
(d) In △ ABC and △ PQR, AB = PQ, AC = PR and BC = QR.
(e) In △ ABC and △ PQR, BC = QR, ∠A = 90°, ∠C = ∠R = 40° and ∠Q = 50°.
In quadrilateral ABCD, AB = AD and CB = CD. Prove that AC is perpendicular bisector of BD.