Criteria for Similarity of Triangles - Triangles, CBSE, Class 10, Mathematics | Extra Documents, Videos & Tests for Class 10 PDF Download

AXIOMS OF SIMILARITY OF TRIANGLES

1. AA (Angle-Angle) Axiom of Similarity :
If two triangles have two pairs of corresponding angles equal, then the triangles are similar. In the given figure, ΔABC and ΔDEF are such that

2. SAS (Side-Angle-Side) Axiom of Similarity :
If two triangles have a pair of corresponding angles equal and the sides including them proportional, then the triangles are similar. In the given fig, ΔABC and ΔDEF are such that

3. SSS (Side-Side-Side) Axiom of Similarity :
If two triangles have three pairs of corresponding sides proportional, then the triangles are similar.
If in ΔABC and ΔDEF we have :
AB/DE= AC/DF= BC/EF, then ΔABC ~ ΔDEF.

Ex.4 In figure, find ∠L. 
 Sol. In ΔABC and ΔLMN,

Ex.5 In the figure, 

Sol.

Ex.6

Ex.7 It is given that  ΔABC ~  ΔPQR, area ( ΔABC) = 36 cm² and area ( ΔPQR) = 25 cm². If QR = 6 cm, find the length of BC. 

Sol. We know that the areas of similar triangles are proportional to the squares of their corresponding sides.

Ex.8  P and Q are points on the sides AB and AC respectively of ΔABC such that PQ || BC and divides ΔABC into two parts, equal in area. Find PB : AB.

Sol. Area (ΔAPQ) = Area (trap. PBCQ) [Given]
⇒Area (ΔAPQ) = [Area (ΔABC) – Area (ΔAPQ)]
⇒ 2 Area (ΔAPQ) = Area (ΔABC)

⇒Area of( APQ) / Area of( ABC) =1/2 ...(i)
Now, inΔAPQ and ΔABC, we have
∠PAQ = ∠BAC [Common ∠A]
∠APQ = ∠ABC [PQ║BC, corresponding ∠s are equal]

  Δ APQ ~ ΔABC.
We known that the areas of similar Δs are proportional to the squares of their corresponding sides.

Ex.9 Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.
 Sol. Let ΔABC and ΔDEF be the given triangles in which AB = AC, DE = DF, ∠A = ∠D and Area of( ΔABC) /Area of( ΔDEF) = 16/25  Draw AL ⊥ BC and DM ⊥ EF

But, the ratio of the areas of two similar Δs is the same as the ratio of the squares of their corresponding heights.

AL : DM = 4 : 5, i.e., the ratio of their corresponding heights = 4 : 5.

Ex.10 If the areas of two similar triangles are equal, prove that they are congruent.

Sol. Let ΔABC ~ΔDEF and area (ΔABC) = area (ΔDEF).

Since the ratio of the areas of two similar Δs is equal to the ratio of the squares on their corresponding sides, we have