For Exercises 1-4, if the two triangles are similar, state why they are similar. If not, state that they are not similar.

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19-12-2022
Mathematics

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For Exercises 1-4, if the two triangles are similar, state why they are similar. If not, state that they are not similar.

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zaibfatimarizvi zaibfatimarizvi · Answer 1 · 2022-12-23T11:28:13-05:00

Based on the given figures similar triangles are (i), (ii), (iv), and (vi) while (iii) and (v) are not similar.

Similar triangles refer to triangles that have the same shape, but their sizes may vary. If two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. Similar triangles can be identified by applying three theorems namely AAA, SAS and SSS.

Angle - Angle (AAA) theorem states that if any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.

Side - Angle - Side (SAS) theorem states if 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.

Side - Side - Side (SSS) theorem states that if all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.

i) Triangles are similar based on AAA similarity as corresponding angles are equal in both triangles. (∠A = ∠P = 60°, ∠B = ∠Q = 80°, ∠C = ∠R = 40°)

ii) Triangles are similar based on SSS similarity as three corresponding sides of both triangles are proportional. (AB/QR = 2/4 BC/RP = 2.5 / 5= CA/PQ = 3/6 =1/2)

iii) Triangle is not similar, as the given corresponding sides are not proportional (MP/EF = 2/5, LP/DF = 3/6 = ½, and LM/DE= 2.7/4)

iv) Triangles are similar based on SAS similarity as two corresponding sides are proportional (MN/QP = ML/QR = ½) and the inclusive angle is equal.

v) Triangles are not similar as the inclusive angle is given but the included side is not given.

vi) Triangles are similar based on AAA criteria, as the corresponding angles are equal. (The missing angles can be determined as sum of all angles of a triangle is 180, hence the ∠F = 30° and ∠ P = 70°, hence ∠D = ∠P = 70°, ∠F = ∠Q = 80°, and ∠F = ∠R = 30°)

Note: The question is incomplete. The complete question probably is: State which pairs of triangles in the following figure are similar? State why they are similar. If not, state that they are not similar.

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