Parallel Lines & Transversals - Pre Algebra

1. Two parallel lines are intersected by a transversal. One pair of corresponding angles has measures $(5a + 10)°$ and $(7a - 30)°$. A pair of alternate exterior angles formed by the same transversal has measures $(3b + 20)°$ and $(b + 60)°$. What is the value of $a + b$?
   Answer: $40$ Corresponding angles are equal: $5a + 10 = 7a - 30$. Solving: $40 = 2a$, so $a = 20$. Alternate exterior angles are equal: $3b + 20 = b + 60$. Solving: $2b = 40$, so $b = 20$. Therefore $a + b = 20 + 20 = 40$. Choice A gives only the value of $a$ or $b$. Choice C results from incorrectly adding the angle measures instead of the variables. Choice D results from calculation errors like $a = b = 40$.
2. Two parallel lines are cut by two different transversals. The first transversal creates an angle of $72°$ with the first parallel line. The second transversal creates an angle with the second parallel line that is $18°$ more than half the corresponding angle formed by the first transversal. What is the measure of this angle on the second parallel line?
   Answer: $54°$ The first transversal creates a $72°$ angle with the first parallel line. The corresponding angle on the second parallel line is also $72°$ (corresponding angles are equal when parallel lines are cut by a transversal). The second transversal creates an angle that is $18°$ more than half of this corresponding angle: $\frac{72°}{2} + 18° = 36° + 18° = 54°$. Choice A gives only half of $72°$. Choice C gives the original $72°$ angle. Choice D results from incorrectly calculating $72° + 18° = 90°$.

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