Write the conditions for two triangles to be similar.
Answer
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Hint:
Here, we will use the concept of similarity of triangles. We will write the four conditions that make two triangles similar, that is AAA, AA, SSS and SAS similarity criterion. We will state the criterion, explain them and provide an example if required.
Complete step by step solution:
Two triangles are similar if they have the same shape, irrespective of the size of the two triangles. The size may be equal or may not be equal.
Now, we will observe the conditions for two triangles to be similar. In other words, these are the conditions we can observe in two triangles with the same shape.
There are four criteria that a pair of triangles must follow to be similar. If any of the conditions proves to be true, then the two triangles are similar.
1) AAA Similarity criterion
According to this criterion, if the corresponding angles of two triangles are equal, then the two triangles are similar.
2) AA Similarity criterion
According to this criterion, if any two corresponding angles of two triangles are equal, then the two triangles are similar.
The third angle of the two triangles will also be equal since the sum of angles of a triangle is always 180 degrees.
Hence, if two of the angles of the triangles are equal, then the third angle of the two triangles will also be equal.
3) SSS Similarity criterion
According to this criterion, if the corresponding sides of the two triangles are in the same ratio, then the two triangles are similar.
This is true because if the corresponding sides of the two triangles are in the same ratio, then their corresponding angles will be equal.
For example: Suppose there is a triangle ABC with sides AB, BC and CA and another triangle PQR with sides PQ, QR and RP. If \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{CA}}{{RP}}\], then the triangles ABC and PQR are similar.
4) SAS Similarity criterion
According to this criterion, if one corresponding angle of the two triangles is equal, and the corresponding sides of the two triangles that form the equal angle are in the same ratio, then the two triangles are similar.
For example: Suppose there is a triangle ABC with sides AB, BC and CA and another triangle PQR with sides PQ, QR and RP. If \[\angle B = \angle Q\], and \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}}\], then the triangles ABC and PQR are similar.
These are the conditions for two triangles to be similar.
Note:
In this problem, it is important to remember that two triangles are similar if the two triangles have the same shape, irrespective of the size. We should also remember different criteria because two triangles are equal. Whereas, two triangles are said to be congruent, if they have the same shape and size. Thus, all congruent triangles are similar, but all similar triangles are not congruent.
Here, we will use the concept of similarity of triangles. We will write the four conditions that make two triangles similar, that is AAA, AA, SSS and SAS similarity criterion. We will state the criterion, explain them and provide an example if required.
Complete step by step solution:
Two triangles are similar if they have the same shape, irrespective of the size of the two triangles. The size may be equal or may not be equal.
Now, we will observe the conditions for two triangles to be similar. In other words, these are the conditions we can observe in two triangles with the same shape.
There are four criteria that a pair of triangles must follow to be similar. If any of the conditions proves to be true, then the two triangles are similar.
1) AAA Similarity criterion
According to this criterion, if the corresponding angles of two triangles are equal, then the two triangles are similar.
2) AA Similarity criterion
According to this criterion, if any two corresponding angles of two triangles are equal, then the two triangles are similar.
The third angle of the two triangles will also be equal since the sum of angles of a triangle is always 180 degrees.
Hence, if two of the angles of the triangles are equal, then the third angle of the two triangles will also be equal.
3) SSS Similarity criterion
According to this criterion, if the corresponding sides of the two triangles are in the same ratio, then the two triangles are similar.
This is true because if the corresponding sides of the two triangles are in the same ratio, then their corresponding angles will be equal.
For example: Suppose there is a triangle ABC with sides AB, BC and CA and another triangle PQR with sides PQ, QR and RP. If \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{CA}}{{RP}}\], then the triangles ABC and PQR are similar.
4) SAS Similarity criterion
According to this criterion, if one corresponding angle of the two triangles is equal, and the corresponding sides of the two triangles that form the equal angle are in the same ratio, then the two triangles are similar.
For example: Suppose there is a triangle ABC with sides AB, BC and CA and another triangle PQR with sides PQ, QR and RP. If \[\angle B = \angle Q\], and \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}}\], then the triangles ABC and PQR are similar.
These are the conditions for two triangles to be similar.
Note:
In this problem, it is important to remember that two triangles are similar if the two triangles have the same shape, irrespective of the size. We should also remember different criteria because two triangles are equal. Whereas, two triangles are said to be congruent, if they have the same shape and size. Thus, all congruent triangles are similar, but all similar triangles are not congruent.
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