Answer:
a) angle DEF = 75°
b) DF = 21/4 cm = 5 1/4 cm
c) BC = 20/3 cm = 6 2/3 cm
Step-by-step explanation:
I sometimes find it helpful to write down corresponding angles and sides:
Angles:
- A and D
- B and E (75°)
- C and F
Sides:
- AB (4 cm) and DE (3 cm)
- BC (??) and EF (5 cm)
- CA (7 cm) and FD (??)
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The triangles are similar, so corresponding angles are congruent, and corresponding sides are proportional. This lets us write proportions for the missing side lengths:
DF/CA = DE/AB
DF = CA(DE/AB) = (7 cm)(3/4) = 21/4 cm
and
BC/EF = AB/DE
BC = EF(AB/DE) = (5 cm)(4/3) = 20/3 cm
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Additional comment
We have written the proportions so the unknown value is in the numerator of the fraction. That lets us solve it in a single step: multiply the equation by the denominator. This means the proportion in part (b) has the smaller length in the numerator, while the proportion in part (c) has the larger length in the numerator.
