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Firstly, we compare the angles to see if we can use the AA Similarity Test. Using the Triangle Sum Theorem, we can calculate the unknown angles:
So, F=M, E=L, G=N and the triangles are similar
Triangle similarity tests determine if two triangles are similar by comparing their proportionate sides and equal angles, using various methods to evaluate their similarity.
We can see that there are both included angles. We just must check that the sides around the angles are proportional.
Since the ratios are the same ABC=DEF by the SAS Similarity Theorem.
F
AA Similarity (Angle-Angle): If two angles of one triangle are matching up to two angles of another triangle, then the triangles are similar.
The triangles have two pairs of sides and non-included angles, but they are right triangles. The third side can be found using the Pythagoras's Theorem. a²+b²=c²
All three pairs of sides are proportional with a ratio of 3.
45/15 = 3
36/12 = 3
27/9 = 3
Therefore, the triangles are congruent.
SSS Similarity (Side-Side-Side):If all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.
Euclid, the "father of geometry," is renowned for his Elements treatise, which established the foundations of geometry, including Similarity Triangle Tests, over 2,000 years ago.
SAS Similarity (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle and the included angles are corresponding, then the triangles are similar.