L = Angle 1 ABCD is a IIgm in which LADC = 75 and side AB is produced to - Maths - Quadrilaterals

Question

L = Angle
1 ABCD is a IIgm in which LADC = 75 and side AB is produced to E
and LC = X and LCBE = Y Find (X + Y)
2 ABCis a triangle in which L is the Mid point of AB and N is
the point on AC such that AN = 2CN Aline through L parallel to BN
meets AC at M Prove that AM = CN
3 the probability of guessing the correct answer to a certain
question is x / 2 If the probability of not guessing the correct
answer is 2/3 then find the value of X

Utsav · Accepted Answer

1)

Since ABCD is a parallelogramHence opposite angles are equal ∴∠ADC =∠ABC =75°Now ∠ABC+∠CBE =180°(linear pair of angles are supplementary)⇒75°+y =180°⇒y=105°Now in ∥gm ABCD ∠DAB =∠DCB=x(opposite angles of a parallelogram)Now sum of interior angles of a ∥gm =360°∴∠ADC+∠DCB+∠CBA+∠BAD=360°⇒75°+x+75°+x=360°⇒2x+150°=360°⇒x=105°∴x+y =105°+105°=210° 2)

Given ABC is a triangle L is the midpoint of AB AN =2CN To prove AM = CN In ∆ ABN, we haveLM∥BN and L is the midpoint of AB Hence M is the midpoint of AN (converse of midpoint theorem)∴AM=MNNow it is given AN = 2 CN⇒AM+MN =2CN⇒AM+AM =2CN ⇒2AM=2CN⇒AM =CN proved 3) Probability of guessing the correct answer = 1- Probability of not guessing the correct answer∴Probability of guessing the correct answer=1-23=13But in the question Probability of guessing the correct answer is given as x2∴x2=13⇒x=23