Answer
424.8k+ views
Hint: We prove that the perpendicular bisects the base and the vertex angle by showing the two triangles formed as congruent by Side angle side congruence rule.
* Isosceles triangle is a triangle with at least two equal sides. This also means corresponding two angles are also equal.
Complete step-by-step answer:
Here ABC is the isosceles triangle where the double line on sides AB and AC represent equal sides. That is AB=AC . The point D lies on the side BC and we need to prove AD bisects the angle A and side BC . That is BD=DC.
We need to prove two statements: AD bisect angle A and AD bisect side BC .
Step 1: To prove the statements, we prove the congruence of needed triangles which is $\vartriangle ABD$ and $\vartriangle ADC$
Considering these two triangles, the sides AB=AC (since $\vartriangle ABC$ is an isosceles triangle)
$\angle ADB = \angle ADC = {90^0}$ (Since AD is the perpendicular drawn to base BC)
AD is a common side for both the triangles.
Thus two sides and an angle of $\vartriangle ABD$ and $\vartriangle ADC$ are equal. By SAS(side angle side) congruence theorem, when two sides and an angle of two triangles are equal, those triangles are congruent.
Here $\vartriangle ABD$ and $\vartriangle ADC$ are congruent.
As BD and DC are sides of congruent triangles, BD=DC
And as $\angle BAD$ and $\angle DAC$ are angles of congruent triangles, $\angle BAD = \angle DAC$ which means that AD bisects angle A.
Hence proved.
Note: Other congruence theorems which can be used are SSS(Side Side Side) congruence theorem for equilateral triangle, ASA(Angle Side Angle) congruence theorem, AAS(Angle Angle Side) congruence theorem.
Students might get confused while writing which triangle are congruent, they should keep in mind the name of triangles is always written in the same sequence of angles and sides to which they are convergent, Example triangle ABC is congruent to triangle PQR means AB=PQ, BC=QR and CA=RP.
* Isosceles triangle is a triangle with at least two equal sides. This also means corresponding two angles are also equal.
![seo images](https://www.vedantu.com/question-sets/7f90c646-b8cc-4e62-8b86-e03f462194041602472227588732217.png)
Complete step-by-step answer:
Here ABC is the isosceles triangle where the double line on sides AB and AC represent equal sides. That is AB=AC . The point D lies on the side BC and we need to prove AD bisects the angle A and side BC . That is BD=DC.
We need to prove two statements: AD bisect angle A and AD bisect side BC .
Step 1: To prove the statements, we prove the congruence of needed triangles which is $\vartriangle ABD$ and $\vartriangle ADC$
Considering these two triangles, the sides AB=AC (since $\vartriangle ABC$ is an isosceles triangle)
$\angle ADB = \angle ADC = {90^0}$ (Since AD is the perpendicular drawn to base BC)
AD is a common side for both the triangles.
Thus two sides and an angle of $\vartriangle ABD$ and $\vartriangle ADC$ are equal. By SAS(side angle side) congruence theorem, when two sides and an angle of two triangles are equal, those triangles are congruent.
Here $\vartriangle ABD$ and $\vartriangle ADC$ are congruent.
As BD and DC are sides of congruent triangles, BD=DC
And as $\angle BAD$ and $\angle DAC$ are angles of congruent triangles, $\angle BAD = \angle DAC$ which means that AD bisects angle A.
Hence proved.
Note: Other congruence theorems which can be used are SSS(Side Side Side) congruence theorem for equilateral triangle, ASA(Angle Side Angle) congruence theorem, AAS(Angle Angle Side) congruence theorem.
Students might get confused while writing which triangle are congruent, they should keep in mind the name of triangles is always written in the same sequence of angles and sides to which they are convergent, Example triangle ABC is congruent to triangle PQR means AB=PQ, BC=QR and CA=RP.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)