1.In the figure, m∠ACD = (11x + 16)° and m∠BCE = (13x − 4)°. Find the measures of these
angles.
2.Two railroad tracks intersect, as shown. If the measure of ∠X is twice that of ∠W, find the
measures of ∠W, ∠X, ∠Y, and ∠Z.
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1.In the figure, m∠ACD = (11x + 16)° and m∠BCE = (13x − 4)°. Find the measures of these
angles.
2.Two railroad tracks intersect, as shown. If the measure of ∠X is twice that of ∠W, find the
measures of ∠W, ∠X, ∠Y, and ∠Z.
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The first problem involves vertical angles. These angles are formed when two lines intersect and congruent.
So, if m∠ACD and m∠BCE are vertical angles, then m∠ACD = m∠BCE. Now, we'll find first the value of x through substitution:
m∠ACD = m∠BCE
11x + 16 = 13x - 4
11x - 13x = -16 - 4
-2x = -20
-2x / -2 = -20 / -2
x = 10
Substituting the value of x to the equation:
11x + 16 = 13x - 4
11(10) + 16 = 13(10) - 4
110 + 16 = 130 - 4
126 = 126
Therefore, the measures of both angles are 126°.
In the second problem, it also involves vertical angles but with the concept of supplementary angles where two angles has a total measurement of 180°.
If m∠X and ∠W are supplementary, then ∠X + ∠W = 180. It is stated that ∠X is twice larger than ∠W. Therefore,
∠X + ∠W = 180
2(∠W) + ∠W = 180
3(∠W) = 180°
Now, we will find first the measurement of ∠W:
3(∠W) = 180
3(∠W) / 3 = 180 / 3
∠W = 60°
∠X is twice larger than ∠W. So,
∠X = 2(∠W)
∠X = 2(60)
∠X = 120°
∠Y is equal to ∠X, as well as, ∠Z to ∠W, because they are vertical angles. Therefore,
∠Y = ∠X
∠Y = 120°
∠Z = ∠W
∠Z = 60°
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