Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Triangles

Short Answer Type

41. In figure, X and Y are two points on equal sides AB and AC of a ∆ABC such that AX = AY. Prove that XC = YB.

3187 Views

42. Prove that the angles opposite to equal sides of a triangle are equal. Is the converse true?

1183 Views

43. In figure, AB = AC, D is the point in the interior of ∆ABC such that ∠DBC = ∠DCB. Prove that AD bisects ∠BAC of ∆ABC.

4123 Views

Long Answer Type

44.
In figure, ABCD is a square and ∠DEC is an equilateral triangle. Prove that

(i) ∆ADE ≅ ∆BCE
(ii) AE = BE
(iii) ∠DAE = 15°

Given: ABCD is a square and ∆DEC is an equilateral triangle.

To Prove:

(i) ∆ADE ≅ ∆BCE

(ii) AE = BE

(iii) ∠DAE = 15°

Proof: (i) In ∆ADE and ∆BCE,
AD = BC

$left enclose table row cell because space ABCD thin space is space straight a space square end cell row cell therefore AB equals BC equals CD equals DA end cell end table end enclose$

DE=CE

$left enclose table row cell because space increment EDC space is space equilateral end cell row cell therefore space ED space equals space DC space equals space CE end cell end table end enclose$

$angle EDA space equals space angle ECB space space space space space space space space space space space left enclose table row cell because space increment EDC space is space equilateral end cell row cell therefore angle EDC space equals space angle ECD space left parenthesis equals 60 degree right parenthesis space space space.... left parenthesis 1 right parenthesis end cell row cell because space ABCD space is space straight a space square end cell row cell therefore space angle ADC space equals space angle BCD space equals left parenthesis 90 degree right parenthesis space space.... left parenthesis 2 right parenthesis end cell row cell Adding space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma end cell row cell table row cell angle EDC space plus space angle ADC space equals space angle ECD plus angle BCD end cell row cell rightwards double arrow space angle EDA space equals space angle ECB end cell end table end cell end table end enclose$

∴ ∆ADE ≅ ∆BCE | SAS congruence rule
(ii)    ∵ ∆ADE ≅ ∆BCE | Proved in (1)
∴ AE = BE    | CPCT
(iii)    In ∆DAE,
∵ DE = DA    | Given
∴ ∠DAE = ∠DEA    ...(1)
| Angles opposite to equal sides of a triangle are equal Also, ∠ADE + ∠DAE + ∠DEA = 180°
| Angle sum property of a triangle
⇒ (∠ADC + ∠EDC) + ∠DAE + ∠DEA = 180°
⇒ (90° + 60°) + ∠DAE + ∠DEA = 180°
⇒ ∠DAE + ∠DEA = 30°    ...(2)
From (1) and (2),
∠DAE = 15° = ∠DEA

1427 Views