Answer:
- x = 24
- y = 36
- vertical angles = 120°
- top angle = 60°
Explanation:
You want to find x, y, and the angle measures in the given figure.
Angle relationships
You solve a problem like this by expressing the known angle relationships in the figure using the expressions for the angle measures.
You know vertical angles are congruent, so you can use that fact to write the equation ...
5x° = (4y -x)°
You also know that a linear pair of angles has a sum of 180°. That relationship lets you write another equation (or two).
(y+x)° +(4y-x)° = 180°
Solution
The first equation can be rewritten as ...
6x = 4y . . . . . divide by °, add x
x = 2/3y . . . . divide by 6
The second equation can be rewritten as ...
5y = 180 . . . . . collect terms, divide by °
y = 36
Using this with the equation for x gives ...
x = 2/3·36 = 24
Angles
Now, you can find the angles:
5x = 120° . . . . . . . . left and right angles
180° -5x = 60° . . . . . top and bottom angles
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Additional comment
You could also write the equation (5x) +(y +x) = 180. Having the other two equations makes this one "dependent," so it does not contribute any new information.
That is, vertical angles are congruent because they are both supplementary to the same angle. You can use the congruence relation or the supplementary relation. Each tells you essentially the same thing.
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