Angles ABC and DBE are vertical angles where m∠ABC = (5x + 5)° and m∠DBE = (6x − 3)°.
Part A: Solve for x. Show every step of your work. (2 points)
Part B: Are the vertical angles also complementary angles? Explain.
Part A:
Since angles ABC and DBE are vertical angles, they are congruent to each other. We can set their measures equal to each other and solve for x.
m∠ABC = m∠DBE
5x + 5 = 6x - 3
To isolate the variable x, we can subtract 5x from both sides of the equation:
5x - 5x + 5 = 6x - 5x - 3
5 = x - 3
Next, we can add 3 to both sides of the equation to isolate x:
5 + 3 = x - 3 + 3
8 = x
Therefore, x = 8.
Part B:
Complementary angles are pairs of angles that add up to 90 degrees. To determine if angles ABC and DBE are complementary, we need to find their measures and check if their sum is 90 degrees.
m∠ABC = (5x + 5)°
Substituting x = 8:
m∠ABC = (5(8) + 5)°
m∠ABC = 40 + 5
m∠ABC = 45°
m∠DBE = (6x - 3)°
Substituting x = 8:
m∠DBE = (6(8) - 3)°
m∠DBE = 48 - 3
m∠DBE = 45°
The measures of angles ABC and DBE are both equal to 45 degrees. Therefore, the vertical angles ABC and DBE are congruent, but they are not complementary angles because their sum is not 90 degrees.