Question

In the adjoining figure , AB = BC and AD-CD. Prove that ZA= 2C​

Answer 1

Answer: In triangle ABD and CBD,

AB = BC (Given)

AD = CD (Given)

BD = BD (Common side)

Hence, Triangle ABD and CBD are congruent by SSS criteria.

∠ADB = ∠CDB (CPCT)

Also,

∠ADB + ∠CDB = 180° (Linear Pair)

=> ∠ADB + ∠ADB = 180°

=> 2∠ADB = 180°  (∠ADB = ∠CDB)

=> ∠ADB = 90°

∠ADB + ∠ADE = 180° (Linear pair)

=> 90° + ∠ADE = 180°

=> ∠ADE = 90°

Hence, ∠ADE is a right angle

Similarly, as ∠ADE and ∠EDC form a linear pair, ∠EDC is 90°

In triangles ADE and EDC,

AD = DC (Given)

∠ADE = ∠EDC = 90° (Proved above)

ED = ED (Common side)

Hence, triangles ADE and EDC are congruent by SAS criteria

Therefore, AE = EC (CPCT)

Answer 2

The (a - b)3 formula is also known as one of the important algebraic identities.

It is read as a minus b whole cube.

Its (a - b)3 formula is expressed as (a - b)3 = a3 - 3a2b + 3ab2 - b3

How To Simplify Numbers Using the (a - b)3 Formula?

Let us understand the use of the (a - b)3 formula with the help of the following example.

Example: Find the value of (20 - 5)3 using the (a - b)3 formula.

To find: (20 - 5)3

Let us assume that a = 20 and b = 5

We will substitute these in the formula of (a - b)3.

(a - b)3 = a3 - 3a2b + 3ab2 - b3

(20-5)3 = 203 - 3(20)2(5) + 3(20)(5)2 - 53

= 8000 - 6000 + 1500 - 125

= 3375

Answer: (20 - 5)3 = 3375.