There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle. Show
AA (Angle-Angle)If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear. Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar. SAS (Side-Angle-Side)If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional. ConclusionThese are the main techniques for proving congruence and similarity. With these tools, we can now do two things.
Did you know you can highlight text to take a note? x Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. We denote the similarity of triangles here by ‘~’ symbol. DefinitionTwo triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles. If two or more figures have the same shape, but their sizes are different, then such objects are called similar figures. Consider a hula hoop and wheel of a cycle, the shapes of both these objects are similar to each other as their shapes are the same. In the figure given above, two circles C1 and C2 with radius R and r respectively are similar as they have the same shape, but necessarily not the same size. Thus, we can say that C1~ C2. It is to be noted that, two circles always have the same shape, irrespective of their diameter. Thus, two circles are always similar. Triangle is the three-sided polygon. The condition for the similarity of triangles is; i) Corresponding angles of both the triangles are equal, and ii) Corresponding sides of both the triangles are in proportion to each other. Similar Triangle ExampleIn the given figure, two triangles ΔABC and ΔXYZ are similar only if, i) ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z Hence, if the above-mentioned conditions are satisfied, then we can say that ΔABC ~ ΔXYZ It is interesting to know that if the corresponding angles of two triangles are equal, then such triangles are known as equiangular triangles. For two equiangular triangles we can state the Basic Proportionality Theorem (better known as Thales Theorem) as follows:
Properties
FormulasAccording to the definition, two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional. Hence, we can find the dimensions of one triangle with the help of another triangle. If ABC and XYZ are two similar triangles, then by the help of below-given formulas, we can find the relevant angles and side lengths.
Once we have known all the dimensions and angles of triangles, it is easy to find the area of similar triangles. Similar Triangles and Congruent TrianglesThe comparison of similar triangles and congruent triangles is given below in the table.
To Know how to Find the Area Of Similar Triangles, Watch The Below Video:
Similar triangles Theorems with ProofsLet us learn here the theorems used to solve the problems based on similar triangles along with the proofs for each. AA (or AAA) or Angle-Angle SimilarityIf any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other. From the figure given above, if ∠ A = ∠X and ∠C = ∠Z then ΔABC ~ΔXYZ. From the result obtained, we can easily say that, AB/XY = BC/YZ = AC/XZ and ∠B = ∠Y SAS or Side-Angle-Side SimilarityIf the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar. Thus, if ∠A = ∠X and AB/XY = AC/XZ then ΔABC ~ΔXYZ. From the congruency, AB/XY = BC/YZ = AC/XZ and ∠B = ∠Y and ∠C = ∠Z SSS or Side-Side-Side SimilarityIf all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar. Thus, if AB/XY = BC/YZ = AC/XZ then ΔABC ~ΔXYZ. From this result, we can infer that- ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z Problem and SolutionsLet us go through an example to understand it better. Solution: In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles) ⇒ ΔABC ~ ΔAPQ (AA criterion for similar triangles) ⇒ AP/AB = PQ/BC ⇒ 5/15 = PQ/20 ⇒ PQ = 20/3 cm Q.2: Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that AO/OC = OB/OD. Solution: ABCD is a trapezium and O is the intersection of diagonals AC and BD. In ΔDOC and ΔBOA, AB || CD, thus alternate interior angles will be equal, ∴∠CDO = ∠ABO Similarly, ∠DCO = ∠BAO Also, for the two triangles ΔDOC and ΔBOA, vertically opposite angles will be equal; ∴∠DOC = ∠BOA Hence, by AAA similarity criterion, ΔDOC ~ ΔBOA Thus, the corresponding sides are proportional. DO/BO = OC/OA ⇒OA/OC = OB/OD Hence, proved. Q.3: Check if the two triangles are similar. Solution: In triangle PQR, by angle sum property; ∠P + ∠Q + ∠R = 180° 60° + 70° + ∠R = 180° 130° + ∠R = 180° Subtract both sides by 130°. ∠ R= 50° Again in triangle XYZ, by angle sum property; ∠X + ∠Y + ∠Z = 180° ∠60° + ∠Y + ∠50°= 180° ∠ 110° + ∠Y = 180 ° Subtract both sides by 110° ∠ Y = 70° Since,∠Q = ∠ Y = 70° and ∠Z = ∠ R= 50° Therefore, by Angle-Angle (AA) rule, ΔPQR~ΔXYZ. Similar Triangles Video LessonThis video will help you visualize basic criteria for the similarity of triangles. To learn more about similar triangles and properties of similar triangles, download BYJU’S- The Learning App. Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles. If ABC and PQR are two similar triangles, then they are represented by: Similar triangles have the same shape but sizes may vary but congruent triangles have the same shape and size. Congruent triangles are represented by symbol ‘≅’. The three similarities theorem are: Angle-angle (AA) Side-angle-side (SAS) Side-side-side (SSS) If two triangles are similar and have sides A,B,C and a,b,c, respectively, then the pair of corresponding sides are proportional, i.e., |