Question

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★Solve the question given in the attachment.


✴ Correct answers will be appreciated!
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Answer 1

Answer :-

★ Concept :-

Here the concept of Congruency and Angle Sum property of a triangle is used. According to this, if sides, angles or both are equal to each other of two triangles, both the triangles are said to be congruent to each other. Even, angle sum property of a triangle states that sum of all the angles of triangle measure equal to 180°.

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★ Question :-

ABC is an equilateral triangle and BCDE is a square.

i) Show that ; AE = ED

ii) Find < ADC

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★ Solution :-

Let us derive some information fro. the question itself.

Given that,

• ABC is an equilateral triangle.

Then, AB = BC = AC

• Since, ABC is equilateral triangle then, <BAC = <ABC = <CBD = 60°

• BCDE is square.

Then, BC = CD = ED = BE

• Since, BCDE is a square.

Then, <EBC = <BCD = <CDE = <DEB = 90°

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i.) In order to prove, AE = AD, we must prove

∆ABE ≅ ∆ACD.

So, let us consider ∆ABE and ∆ACD

=> BE = CD (given)

=> AB = AC (given)

Since, <EBC = <BCD = 90°

And, <ABC = <ACB = 60°

So, adding both the like terms, we get,

=> <ABE = <ACD = 150°

So by SAS (Side - Angle - Side) Congruency,

» ∆ABE ≅ ∆ACD

Then, by CPCT (common part of congruent triangle),

☞ AE = AD

Hence, proved.

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ii) Now let us find <ADC.

Let us consider ∆ADC. Then,

→ <ACD = 150° (proved above)

→ AC = CD (proved above) [Since ∆ABE ≅ ∆ACD , CD = AB = AC]

→ <CAD = <ADC (angles opposite to equal sides are equal. Since, AC = CD)

By angle sum property of triangle, we know that,

✏ <ACD + <CAD + <ADC = 180°

✏ 150° + <ADC + <ADC = 180°

✏ 2 <ADC = 180° - 150°

✏ 2 <ADC = 30°

✏ <ADC = ½ × 30 = 15°

✳ Hence, we get, <ADC = 15°

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★ More to know :-

• RHS Congruency rule states that if hypotenuse and right angle of two triangles are same, then both triangles are congruent.

• AAS Congruency rule states that if two angles and one side of two triangles are equal then both the triangles are congruent.

• SSS Congruency rule states that if all three sides of two triangles are equal then both the triangles are congruent.

• Square is the figure which is closed by four sides and all the sides are perpendicular to each other. Its a two dimensional figure.

• Triangle is the two dimensional figure formed by three sides which can have any angle but the total of all three angles should be 180°.

• Congruent Triangles are definitely equal to each other but similar triangles may or may not be equal to each other.

Answer 2

Step-by-step explanation:

Refer to Attachment for confusion related to angles.

Answer:-

We need :-

AE = AD

Concept:-

Congruency of Triangles,

Chapter Triangles.

Let's Do!

It is given that ABC is equilateral, so it's all sides are equal and 60°.

Now, one side is also the side of the corresponding square.

It means the measure of side of square and Triangle is equal.

So, we can say

AB = BC = AC = CD = DE = EB.

Now, in Triangle ABE and Triangle ACD

AB = AC

EB = CD

Angle ABE = Angle ACD ( Cuz 90+60 in both angles I.e 150)

By SAS or Side angle Side Congruency,

Triangle ABC = Triangle ACD

And hence

AE = AD ( CPCT)

2nd Part:-

I already proved how AB = BE.

Now, they are a part of Traingle ABE.

We know:-

Two sides of isosceles triangle are equal, so are their corresponding angles.

Let us consider the angle BAE and BEA as x.

Now,

2x + 150 = 180

$2x = 180 - 150$

$x = 15$

So, BEA = Angle ADC

Hence, 2nd answer is 15°.