Question

E is mid-point of side BC of parallelogram ABCD such that CD : AD = 1 : 2. If ∠BAE = 40°, then ∠ADE equals ____ ​

Answer 1

Answer:

50

Step-by-step explanation:

Parallelogram ABCD => AD = BC, AB = CD

CD : AD = 1 : 2 => CD= 1/2AD = 1/2BC

E is mid-point of side BC of parallelogram ABCD => EC = EB = 1/2BC = CD

=> CD = EC => ΔDCE is isosceles triangles  

=>  CD = BE = BA=> ΔABE is isosceles triangles

ΔABE and ΔDCE are isosceles triangles. An isosceles triangle is a triangle with (at least) two equal sides. This property is equivalent to two angles of the triangle being equal.  (1)

=> ∠BAE = ∠AEB  = 40°

A triangle's angles add up to 180 degrees  (2)

∠BAE + ∠AEB + ∠ABE = 180°

40° + 40° + ∠ABE = 180°

∠ABE = 180° - (40° + 40°) = 100°

=> ∠ABC= ∠ABE =100°

Consecutive angles are supplementary ∠BAD + ∠ABC = 180°

∠BAD = 180° - ∠ABC

∠BAD = 180° - 100° = 80°

Opposite angels are congruent

∠ABC = ∠ADC = 100° (3)

∠BAD = ∠DCB = ∠DCE = 80°

(2) = > ΔDCE = ∠CDE + ∠CED + ∠DCE  = 180°

∠CDE + ∠CED  + 80° = 180°

∠CDE + ∠CED = 100°

( 1 )  => ∠CDE = ∠CED  = 50°

(3) ∠ADC  =  ∠ADE + ∠CDE

          100 = ∠ADE + 50° => ∠ADE = 50°