Question

the figure shows a rhombus ABCD where ACD is equal to 32 degree . AB is produced to E such that AC is equal to CE .

find
1 ABC
2 BCE​

Answer 1

Given:

ABCD is a rhombus

∠ACD = 32°

AB is produced to E such that AC = CE

To find:

(1). ∠ABC

(2). ∠BCE​

Solution:

(1). Finding the value of ∠ABC:

From the properties of a rhombus, we can say that,

• the opposite sides of a rhombus are parallel to each other
• all the 4 sides of a rhombus are equal in length

So,

AD = CD

∴ ∠ACD = ∠CAD = 32° .... [∵ the angles opposite to equal sides are also equal]

Also,

AB // CD and BC // AD

∴ ∠ACD = ∠CAB = 32° .... [alternate angles]

and

∴ ∠ACB = ∠CAD = 32° ....... [alternate angles]

Now, in Δ ABC, we have

∠ACB + ∠CAB + ∠ABC = 180° ..... [Angle sum property]

⇒ 32° + 32° + ∠ABC = 180°

⇒ 64° + + ∠ABC = 180°

⇒ ∠ABC = 180° - 64°

⇒ ∠ABC = 116°

Thus, $\boxed{\bold{\underline{\angle ABC = 116\°}}}$.

(2). Finding the value of ∠BCE:

It is given that,

AC = CE

Then, ∠CAB = ∠CEB = 32° ....... [∵ the angles opposite to equal sides are also equal]

In Δ ACE, we have

∠CAB + ∠CEB + ∠ACE = 180°

⇒ 32° + 32° + ∠ACE = 180°

⇒ 64° + ∠ACE = 180°

⇒ ∠ACE = 180° - 64°

⇒ ∠ACE = 116°

Now, from the figure, we can see

∠ACE = ∠ACB + ∠BCE

⇒ 116° = 32° + ∠BCE

⇒ ∠BCE = 116° - 32°

⇒ ∠BCE = 84°

Thus, $\boxed{\bold{\underline{\angle BCE = 84\°}}}$.

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