If two triangles are similar then their equivalent anglesare equal and corresponding sides space proportional.
You are watching: Triangles pqr and xyz are similar
Here, the two triangles XYZ and also PQR room similar.
So, ∠X = ∠P, ∠Y = ∠Q, ∠Z = ∠R and (fracXYPQ) = (fracYZQR) = (fracXZPR).
∆XYZ is comparable to ∆PQR. We write ∆XYZ ∼ ∆PQR (the prize ‘∼’ means ‘similar to‘.)
Corrosponding Sides:
Sides opposite come equal angle in similar triangles are known as matching sides and also they room proportional.
Here indigenous the given numbers ∠X = ∠P, ∠Y = ∠Q and also ∠Z = ∠R.
Therefore, XY and also PQ are corresponding sides together they room opposite come ∠Z and ∠R respectively.
Similarly indigenous the offered figure, YZ and also QR are a pair of equivalent sides. XZ and PR are likewise a pair of matching sides.
Thus, (fracXYPQ) = (fracYZQR) = (fracXZPR), as corresponding sides of similar triangles are proportional.
Corrosponding Angles:
Angles opposite come proportional sides in similar triangles are known as matching angles.
If ∆XYZ ∼ ∆PQR and also (fracXYPQ) = (fracYZQR) = (fracXZPR) climate ∠X = ∠P as they space opposite to equivalent sides YZ and QR respectively.
Similarly native the offered figure, ∠Y = ∠Q and ∠Z = ∠R.
Congruency and Similarity of Triangles:
Congruency is a certain case of similarity. In both the cases, three angles that one triangle room equal come the three equivalent angles that the other triangle. However in similar triangles the matching sides are proportional, when in congruent triangles the corresponding sides room equal.
∆XYZ ∼ ∆TUV.
Therefore, (fracXYTU) = (fracYZUV) = (fracXZTV)= k, where k is the consistent of proportionality or the scale element of sizetransformation.
∆XYZ ≅ ∆PQR.
Here, (fracXYPQ) = (fracYZQR) = (fracXZPR)= 1.
Therefore, in congruent triangle the consistent ofproportionality in between the equivalent sides is same to one. Thus,congruent triangles have the very same shape and size while comparable triangles havethe same shape however not have to the very same size.
Congruent triangle are always similar, but comparable trianglesare no necessarily congruent.
Note: Triangles the are comparable to the same triangle aresimilar to every other.
Here, ∆XYZ ∼ ∆PQR and also ∆ABC ∼ ∆PQR.
Therefore, ∆XYZ ∼ ∆ABC.
9th class Math
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