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Iteru [2.4K]
4 years ago
14

Given: m∠ADB = m∠CDB AD ≅ DC Prove: m∠BAC = m∠BCA BD⊥ AC

Mathematics
1 answer:
puteri [66]4 years ago
3 0

Answer:

The proof is explained below.

Step-by-step explanation:

Given m∠ADB = m∠CDB and AD ≅ DC

we have to prove that m∠BAC = m∠BCA  and BD⊥ AC

In ΔADO and ΔCDO

∠OAD=∠OCD          (∵ADC is an isosceles triangle)

AD=DC                     (∵Given)

∠ADO=∠CDO          (∵Given)

By ASA rule, ΔADO≅ΔCDO

In ΔBAD and ΔBCD

AD=DC                (∵ABC is an isosceles triangle)

∠ADB=∠CDB      (∵Given)

DB=DB                (∵common)

By ASA rule, ΔADB≅ΔCDB

Now, ΔADB≅ΔCDB and  ΔADO≅ΔCDO

⇒  ΔADB-ΔADO≅ΔCDB-ΔCDO

⇒  ΔABO≅ΔCBO

Hence, by CPCT,  m∠BAC = m∠BCA

Now, we have to prove that BD⊥ AC i.e we have to prove m∠BOA=90°

Now, ΔABO≅ΔCBO therefore by CPCT,  m∠BOA = m∠BOC

But, m∠BOA + m∠BOC=180°   (linear pair)

⇒  m∠BOA + m∠BOA=180°

⇒  2m∠BOA=180°  ⇒  m∠BOA=90°

Hence,  BD⊥ AC

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Enter a problem...

Algebra Examples

Popular Problems Algebra Solve for x 64^(4x)=16^(x+1)

64

4

x

=

16

x

+

1

Create equivalent expressions in the equation that all have equal bases.

4

3

(

4

x

)

=

4

2

(

x

+

1

)

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

3

(

4

x

)

=

2

(

x

+

1

)

Solve for

x

.

Tap for fewer steps...

Multiply

4

by

3

.

12

x

=

2

(

x

+

1

)

Simplify

2

(

x

+

1

)

.

Tap for fewer steps...

Apply the distributive property.

12

x

=

2

x

+

2

⋅

1

Multiply

2

by

1

.

12

x

=

2

x

+

2

Move all terms containing

x

to the left side of the equation.

Tap for fewer steps...

Subtract

2

x

from both sides of the equation.

12

x

−

2

x

=

2

Subtract

2

x

from

12

x

.

10

x

=

2

Divide each term by

10

and simplify.

Tap for fewer steps...

Divide each term in

10

x

=

2

by

10

.

10

x

10

=

2

10

Cancel the common factor of

10

.

Tap for fewer steps...

Cancel the common factor.

10

x

10

=

2

10

Divide

x

by

1

.

x

=

2

10

Cancel the common factor of

2

and

10

.

Tap for fewer steps...

Factor

2

out of

2

.

x

=

2

(

1

)

10

Cancel the common factors.

Tap for fewer steps...

Factor

2

out of

10

.

x

=

2

⋅

1

2

⋅

5

Cancel the common factor.

x

=

2

⋅

1

2

⋅

5

Rewrite the expression.

x

=

1

5

The result can be shown in multiple forms.

Exact Form:

x

=

1

5

Decimal Form:

x

=

0.2

4 0
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