Answer
Verified
479.1k+ views
Hint: In order to solve this question, we will use angle sum property and the property of isosceles triangle according to which angles opposite to equal sides are equal.
Complete step-by-step answer:
Now, given that triangle\[\] is an isosceles triangle.
Also, $AB = AC$ and $AD = AB$.
And we have to prove that $\angle BCD$ is a right angle.
Now,
In triangle $ABC$
It is given that,
$AB = AC$
$ \Rightarrow \angle ACB = \angle ABC - - - - - - \left( i \right)$
(Because angles opposite to equal sides are equal)
Now,
$AB = AC$ (Given)
And $AD = AB$ (Given)
$ \Rightarrow AC = AD$
Now, in triangle $ACD$
$AC = AD$ (Given)
$ \Rightarrow \angle ADC = \angle ACD - - - - - - \left( {ii} \right)$ (Angles opposite to the equal sides are equal)
Angle sum property- In a Euclidean space, the sum of angles of a triangle equals the straight line i.e. \[{180^ \circ }\] ,$\pi $ radians, two right angles, one at each vertex, bounded by a pair of adjacent sides.
In triangle $BCD$
$\angle ABC + \angle BCD + \angle ADC = {180^ \circ }$ (Angle sum property of a triangle)
From $\left( i \right)$ and $\left( {ii} \right)$
$\angle ACB + \angle BCD + \angle ACD = {180^ \circ }$
Now It is clearly visible from the figure that $\angle BCD = \angle ACB + \angle ACD$
Or $\angle BCD + \angle BCD = {180^ \circ }{\text{ }}\left( {\because \angle BCD = \angle ACB + \angle ACD} \right)$
Or $2\angle BCD = {180^ \circ }$
Or $\angle BCD = {90^ \circ }$
Hence Proved.
Note: Whenever we face such types of questions the key concept is that we should write what is given to us then apply the basic properties of triangles like we did in this question. Here, we simply apply the property of the isosceles triangle that angles opposite to equal sides are equal and then we apply Angle sum property of a triangle and we get our answer.
Complete step-by-step answer:
Now, given that triangle\[\] is an isosceles triangle.
Also, $AB = AC$ and $AD = AB$.
And we have to prove that $\angle BCD$ is a right angle.
Now,
In triangle $ABC$
It is given that,
$AB = AC$
$ \Rightarrow \angle ACB = \angle ABC - - - - - - \left( i \right)$
(Because angles opposite to equal sides are equal)
Now,
$AB = AC$ (Given)
And $AD = AB$ (Given)
$ \Rightarrow AC = AD$
Now, in triangle $ACD$
$AC = AD$ (Given)
$ \Rightarrow \angle ADC = \angle ACD - - - - - - \left( {ii} \right)$ (Angles opposite to the equal sides are equal)
Angle sum property- In a Euclidean space, the sum of angles of a triangle equals the straight line i.e. \[{180^ \circ }\] ,$\pi $ radians, two right angles, one at each vertex, bounded by a pair of adjacent sides.
In triangle $BCD$
$\angle ABC + \angle BCD + \angle ADC = {180^ \circ }$ (Angle sum property of a triangle)
From $\left( i \right)$ and $\left( {ii} \right)$
$\angle ACB + \angle BCD + \angle ACD = {180^ \circ }$
Now It is clearly visible from the figure that $\angle BCD = \angle ACB + \angle ACD$
Or $\angle BCD + \angle BCD = {180^ \circ }{\text{ }}\left( {\because \angle BCD = \angle ACB + \angle ACD} \right)$
Or $2\angle BCD = {180^ \circ }$
Or $\angle BCD = {90^ \circ }$
Hence Proved.
Note: Whenever we face such types of questions the key concept is that we should write what is given to us then apply the basic properties of triangles like we did in this question. Here, we simply apply the property of the isosceles triangle that angles opposite to equal sides are equal and then we apply Angle sum property of a triangle and we get our answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE