Question

find angle abd
if ab equal to ac​

Answer 1

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Question :-

Find the measure of ∠ABD .

If AB is equal to AC .

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Answer :-

Given :-

In a ∆ ABC

AB = AC

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Required to find :-

• Find the measurement of ∠ABD

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Conditions used :-

Here conditions refer to the properties of the geometrical figures .

☞ In a Trapezium the sum of any two adjacent sides is supplementary .

☞ In a triangle, the sum of angles is equal to 180°

☞ If corresponding sides are equal corresponding angles are also equal .

☞ In a Trapezium , one pair of opposite sides are parallel .❎

☞ Exterior angle is equal to sum of two opposite interior angles

But here we should not use the 4 condition mentioned above this is due to in the question it is not given which sides are parallel .

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Solution :-

First let's consider Trapezium ACDE .

So,

In a Trapezium ACDE ,

Here let add a new point x in the above figure till which the side AC is extending .

So,

From the above construction we can conclude that CX is a straight line .

Hence,

∠EAC + ∠EAX = 180° ( since CX is a straight line) .

But, we know that

∠EAX = 70°

So,

∠EAC + 70° = 180°

∠EAC = 180° - 70°

∠EAC = 110°

Now ,

We know that ,

In a Trapezium the sum of any two sides is supplementary

So,

∠EAC + ∠ACB = 180

But, ∠EAC = 110°

Then,

110° + ∠ACB = 180°

∠ACB = 180° - 110°

∠ACB = 70°

Hence,

Now let's consider ∆ABC .

In ∆ABC ,

It is given that ,

AB = AC

Now , recall the properties of triangle .

We know that,

If corresponding sides are equal , corresponding angles are equal .

So,

AB = AC

Then, ∠B = ∠C

Hence ,

∠ABC = ∠ACB ( If corresponding sides are equal corresponding angles are equal )

But, ∠ACB = 70°

So,

∠ABC = 70°

Now we should use the most familiar property in a triangle .

That is ,

Sum of all angles in a triangle = 180 ° ( THIS IS ALSO KNOWN AS ANGLE SUM PROPERTY )

so,

∠BAC + ∠ABC + ∠ACB = 180°

But,

• ∠ABC = ∠ACB = 70°

Now substitute these values in the above .

Hence,

∠BAC + 70° + 70° = 180°

∠BAC + 140° = 180°

∠BAC = 180° - 140°

∠BAC = 40°

Now above the figure carefully .

We can find an exterior angle .

So,

It is the time to use the Exterior angle property here,

____Exterior Angle property____

Exterior angle property states that ,

Sum of 2 opposite interior angles is equal to the exterior angle .

So, from the figure let's identity the exterior angle .

The exterior angle is ∠ABD .

So, the two opposite interior are ∠BAC & ∠ACB .

Hence this is written as ,

∠ABD = ∠BAC + ∠ACB ( Exterior angle property )

But ,

∠BAC = 40° & ∠ACB = 70°

Now substitute this in the above one .

So,

∠ABD = 40° + 70°

∠ABD = 110°

Therefore,

Measure of ∠ABD = 110°

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✅ Hence Solved ..

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