x = 14°, y = 127°
Step-by-step explanation:
Definitions:
Parallel lines: Two lines are considered as parallel if, and only if, they are coplanar and do not intersect each other.
Linear pair: Two adjacent angles whose opposite rays comprise of their noncommon side. These angles are also supplements; hence, the sum of their measures equal 180°.
Corresponding angles: Two angles with corresponding or relative positions (same side along the transversal that are non-adjacent).
Corresponding Angles Theorem: if there are two lines cut by a transversal, then their corresponding angles are congruent.
Solution:
Given that m ║n, and that we are asked to find the value of x and y:
Solve for x:
As defined in the previous section, a linear pair are two adjacent angles that are supplements. Therefore, ∠(4x - 6)° and ∠(8x + 18)° are linear pair.
We can establish the following equation to solve for the value of x (by definition of linear pair ):
⇒ ∠(4x - 6)° + ∠(8x + 18)° = 180°
4x° - 6° + 8x° + 18° = 180°
Combine like terms and constants:
12x° + 12° = 180°
Subtract 12° from both sides:
12x° + 12° - 12° = 180° - 12°
12x° = 168°
Divide both sides by 12 to solve for x:
x = 14°
Therefore, the value of x = 14°.
Solve for y:
As defined in the previous section of this post, corresponding angles are located in relative positions in between the intersection of the parallel lines and transversal. Moreover, according to the Corresponding Angles Theorem, corresponding angles are congruent.
⇒ Corresponding angles: ∠(8x + 18)° ≅ ∠(y + 3)°
Since these two angles are congruent, we can set up the following equality statement: ∠(8x + 18)° = ∠(y + 3)°.
Substitute the value of x = 14° into the equality statement to solve for y:
∠(8x + 18)° = ∠(y + 3)°
8(14)° + 18° = y° + 3°
112° + 18° = y° + 3°
130° = y° + 3°
Subtract 3° from both sides of the equation:
130° - 3° = y° + 3° - 3°
127° = y
Double-check Solution:
Linear pair: ∠(4x - 6)° + ∠(8x + 18)° = 180°
4(14)° - 6° + 8(14)° + 18° = 180°
56° - 6° + 112° + 18° = 180°
180° = 180° (True statement).
Corresponding Angles: ∠(8x + 18)° = ∠(y + 3)°
8(14)° + 18° = 127° + 3°
112° + 18° = 130°
130° = 130° (True statement).
Therefore, x = 14° and y = 127°.
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Keywords:
Linear pair
Corresponding Angles
Supplements
Parallel lines
Transversal
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