Answer
405k+ views
Hint: We prove two triangles similar by SAS theorem of similarity. Use the given set of equal angles to prove the triangle PQR an isosceles triangle. Use the proportional sides and common angle to show triangles similar.
SAS theorem of similarity states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are equal, then the two triangles are similar.
Two triangles are equilateral if they have two sides equal. Also, angles opposite to equal sides are equal.
Complete step-by-step solution:
We are given \[\angle 1 = \angle 2\]
In \[\vartriangle PQR\], \[\angle 1 = \angle 2\]
Since we know the sides opposite to equal angles are equal in any triangle.
\[ \Rightarrow PQ = PR\]...............… (1)
So\[\vartriangle PQR\] is an isosceles triangle.
Now we are given \[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PR}}\]
Substitute the value of\[PR = PQ\] in the proportion
\[ \Rightarrow \dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PQ}}\].................… (2)
Now in \[\vartriangle PQS\] and \[\vartriangle TQR\];
\[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PQ}}\]{From equation (2)}
\[\angle A = \angle A\](Common angle in both triangles)
We have two sides in one triangle that are proportional to two sides in another triangle and included angles equal in both the triangles.
Then by SAS similarity theorem, we can say that \[\vartriangle PQS \sim \vartriangle TQR\]
\[\therefore \vartriangle PQS \sim \vartriangle TQR\]
Note: Students many times confuse SAS similarity theorem with SAS congruence. They try to prove sides of triangles congruent and equal, but keep in mind here we have to prove triangles similar and the rules are different for proving congruence and different for proving similarity. Also, focus on the pair of equal angles in both triangles as they should be between the proportional sides, i.e. the angle should be included and not any of the angles.
SAS theorem of similarity states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are equal, then the two triangles are similar.
Two triangles are equilateral if they have two sides equal. Also, angles opposite to equal sides are equal.
Complete step-by-step solution:
We are given \[\angle 1 = \angle 2\]
In \[\vartriangle PQR\], \[\angle 1 = \angle 2\]
Since we know the sides opposite to equal angles are equal in any triangle.
\[ \Rightarrow PQ = PR\]...............… (1)
So\[\vartriangle PQR\] is an isosceles triangle.
Now we are given \[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PR}}\]
Substitute the value of\[PR = PQ\] in the proportion
\[ \Rightarrow \dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PQ}}\].................… (2)
Now in \[\vartriangle PQS\] and \[\vartriangle TQR\];
\[\dfrac{{QR}}{{QS}} = \dfrac{{QT}}{{PQ}}\]{From equation (2)}
\[\angle A = \angle A\](Common angle in both triangles)
We have two sides in one triangle that are proportional to two sides in another triangle and included angles equal in both the triangles.
Then by SAS similarity theorem, we can say that \[\vartriangle PQS \sim \vartriangle TQR\]
\[\therefore \vartriangle PQS \sim \vartriangle TQR\]
Note: Students many times confuse SAS similarity theorem with SAS congruence. They try to prove sides of triangles congruent and equal, but keep in mind here we have to prove triangles similar and the rules are different for proving congruence and different for proving similarity. Also, focus on the pair of equal angles in both triangles as they should be between the proportional sides, i.e. the angle should be included and not any of the angles.
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)