In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. 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. 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. Figure %: Three pairs of congruent angles determine similar triangles In the above figure, angles A, B, and C are vertices of a triangle. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape. Hence, any triangles with three pairs of congruent angles will be similar. Also, note that if the three vertices are exactly the same distance from each other, then the triangle will be congruent. In other words, congruent triangles are a subset of similar triangles.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. Figure %: If all three pairs of sides of corresponding triangles are in proportion, the triangles are similarSAS (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. Figure %: Two pairs of proportional sides and a pair of equal included angles determines similar trianglesConclusionThese are the main techniques for proving congruence and similarity. With these tools, we can now do two things.
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All congruent figures are similar, but it does not mean that all similar figures are congruent. Two polygons of the same number of sides are similar, if:
Two triangles are similar, if:
According to Greek mathematician Thales, “The ratio of any two corresponding sides in two equiangular triangles is always the same.”
According to the Indian mathematician Budhayan, “The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth).” In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. THEOREM 1:If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Construction: ABC is a triangle. DE || BC and DE intersects AB at D and AC at E. Join B to E and C to D. Draw DN ⊥ AB and EM ⊥ AC. To prove: `(AD)/(DB)=(AE)/(EC)` Proof: `text(ar AEM)=1/2xx(AD)xx(EM)` Similarly; `text(ar BDE)=1/2xx(DB)xx(EM)` `text(ar ADE)=1/2xx(AE)xx(DN)` `text(ar DEC)=1/2xx(EC)xx(DN)` Hence; `text(ar ADE)/text(ar BDE)=(1/2xx(AD)xx(EM))/(1/2xx(DB)xx(EM))=(AD)/(DB)` Similarly; `text(ar ADE)/text(ar DEC)=(AE)/(EC)` Triangles BDE and DEC are on the same base, i.e. DE and between same parallels, i.e. DE and BC. Hence, ar(BDE) = ar(DEC) From above equations, it is clear that; `(AD)/(DB)=(AE)/(EC)` proved THEOREM 2:If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Construction: ABC is a triangle in which line DE divides AB and AC in the same ratio. This means: `(AD)/(DB)=(AE)/(EC)` To Prove: DE || BC Let us assume that DE is not parallel to BC. Let us draw another line DE’ which is parallel to BC. Proof: If DE’ || BC, then we have; `(AB)/(DB)=(AE’)/(E’C)` According to the theorem; `(AB)/(DB)=(AE)/(EC)` Then according to the first theorem; E and E’ must be coincident. This proves: DE || BC THEOREM 3:If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. This is also called AAA (Angle-Angle-Angle) criterion. Construction: Two triangles ABC and DEF are drawn so that their corresponding angles are equal. This means: ∠ A =∠ D, ∠ B = ∠ E and ∠ C = ∠ F To prove: `(AB)/(DE)=(AC)/(DF)=(BC)/(EF)` Draw a line PQ in the second triangle so that DP = AB and PQ = AC Proof: `ΔABC≅ΔDPQ` Because corresponding sides of these two triangles are equal This means; ∠ B = ∠ P = ∠ E and PQ || EF This means; `(DP)/(PE)=(DQ)/(QF)` Hence; `(AB)/(DE)=(AC)/(DF)` `(AB)/(DE)=(BC)/(EF)` Hence; `(AB)/(DE)=(AC)/(DF)=(BC)/(EF)` proved THEOREM 4:If in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. This is also called SSS (Side-Side-Side) criterion. Construction: Two triangles ABC and DEF are drawn so that their corresponding sides are proportional. This means: `(AB)/(DE)=(AC)/(DF)=(BC)/(EF)` To Prove: ∠ A = ∠ D, ∠ B = ∠ E and ∠ C = ∠ F And hence; Δ ABC ∼ Δ DEF In triangle DEF, draw a line PQ so that DP = AB and DQ = AC Proof: `ΔABC≅ΔDPQ` Because corresponding sides of these two triangles are equal This means; `(DP)/(PE)=(DQ)/(QF)=(PQ)/(EF)` This also means; ∠ P = ∠ E and ∠ Q = ∠ F We have taken; ∠ A = ∠ D, ∠ B = ∠ P and ∠ C = ∠ Q Hence; ∠ A = ∠ D, ∠ B = ∠ E and ∠ C = ∠ F From AAA criterion; Δ ABC ∼ Δ DEF proved Copyright © excellup 2014 |