If the measures of the corresponding sides of two triangles are proportional then the triangles are similar. Likewise if the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the including angles are congruent then the triangles are similar. Show
$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$ If a line is drawn in a triangle so that it is parallel to one of the sides and it intersects the other two sides then the segments are of proportional lengths: $$\frac{AD}{DB}=\frac{EC}{BE}$$ Parts of two triangles can be proportional; if two triangles are known to be similar then the perimeters are proportional to the measures of corresponding sides. Continuing, if two triangles are known to be similar then the measures of the corresponding altitudes are proportional to the corresponding sides. Lastly, if two triangles are known to be similar then the measures of the corresponding angle bisectors or the corresponding medians are proportional to the measures of the corresponding sides. The bisector of an angle in a triangle separates the opposite side into two segments that have the same ratio as the other two sides: $$\frac{AD}{DC}=\frac{AB}{BC}$$ Video lessonFind the value of x in the triangle
Two triangles are similar if they have:
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough. There are three ways to find if two triangles are similar: AA, SAS and SSS: AAAA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar. So AA could also be called AAA (because when two angles are equal, all three angles must be equal). SASSAS stands for "side, angle, side" and means that we have two triangles where:
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
In this example we can see that:
So there is enough information to tell us that the two triangles are similar. Using TrigonometryWe could also use Trigonometry to calculate the other two sides using the Law of Cosines:
In Triangle ABC:
In Triangle XYZ:
Now let us check the ratio of those two sides: a : x = 22.426... : 14.950... = 3 : 2 the same ratio as before! Note: we can also use the Law of Sines to show that the other two angles are equal. SSSSSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
These ratios are all equal, so the two triangles are similar. Using TrigonometryUsing Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:
In Triangle ABC:
In Triangle XYZ:
So angles A and X are equal! Similarly we can show that angles B and Y are equal, and angles C and Z are equal. Copyright © 2017 MathsIsFun.com 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., |