Part a
Angle ABC = angle CDA (given by the angle markers)
Angle BAC = angle DCA (alternate interior angles)
Segment AC = segment AC (reflexive property)
Through AAS (angle angle side) we can prove the two triangles are congruent. We have a pair of congruent angles, and we have a pair of congruent sides that are not between the previously mentioned angles.
If two triangles are congruent, they are always similar as well (scale factor = 1).
The same cannot be said the other way around. Not all similar triangles are congruent.
Answer: Congruent
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Part b
Angle FGH = angle JIH (both shown to be 50 degrees)
Angle FHG = angle JHI (vertical angles)
We have enough information to prove the triangles to be similar triangles. This is through the AA (angle angle) similarity rule. Since FG and JI are different lengths, this means the triangles are not congruent.
Answer: Similar but not congruent
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Part c
For each right triangle shown, divide the longer leg over the shorter leg
larger triangle: (long leg)/(short leg) = 6/3 = 2
smaller triangle: (long leg)/(short leg) = 3/2 = 1.5
The two results are different, so the sides are not in proportion to one another, therefore the triangles are not similar.
Any triangles that are not similar will also never be congruent.
Answer: Not similar
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Part d
Use the pythagorean theorem to find that PQ = 5 and KL = 12
We have two triangles with corresponding sides that are the same length
So we use the SSS (side side side) triangle congruence theorem to prove the triangles congruent. The triangles are also similar triangles (scale factor = 1)
Answer: Congruent
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Summary of the answers:
a. Congruent
b. Similar but not congruent
c. Not similar
d. Congruent
