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I think for the question above, instead of 2 · 3^2 · 7 it is <span>2 · 3^2 · 5.
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Two numbers have prime factorizations of 2^2 • 3 • 5 and 2 • 3^2 • 5 (note 2 squared & 3 squared).
Now, to choose the GCF, you choose, for each base factor in either number, the least exponent-ed one; so the GCF needs a factor 2, a factor 3, and a factor 5. Thus the GCF is 30 (their product). [i.e,2 squared is not a common factor]
<span>To create the LCM, you choose, for each base factor in either number, the greatest exponented one. Thus, LCM needs a factor 2 squared, 3 squared, and 5, giving LCM = 4(9)(5) = 180.</span><span />
Problem 1) The triangles are similar because of the AA (angle angle) Similarity Theorem. The first A is the pair of congruent 39 degree angles. The second pair is unmarked, but look at where the triangles meet. They form a pair of vertical angles which are congruent. So we have two pairs of congruent angles allowing us to use the AA Similarity Theorem.
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Problem 2) We can use the SAS (Side Angle Side) Similarity Theorem to prove that these two triangles are similar. The angles are congruent. They are both 29 degrees. So that checks off the "A" portion of SAS. Then notice how the bottom sides are 32 and 64 for the small and large triangle respectively. They form the ratio 32/64 = 1/2, ie the smaller triangle's side is 1/2 as long as the longer counter part. Similarly, 8/16 = 1/2 as well. The ratio is constant at 1/2. This allows us to use the other "S" portions of SAS.
