Math Problem Statement

Given that
BA and
BD are opposite rays, m∠CBA=12n+4, and m∠DBC=14n−6, find m∠ABC.

Solution

Given that BA\overrightarrow{BA} and BD\overrightarrow{BD} are opposite rays, this means that the angles CBA\angle CBA and DBC\angle DBC are adjacent angles on a straight line. The sum of the measures of these angles must therefore be 180°, because they form a linear pair.

Given:

  • mCBA=12n+4m\angle CBA = 12n + 4
  • mDBC=14n6m\angle DBC = 14n - 6

Since CBA\angle CBA and DBC\angle DBC form a straight line, we can set up the following equation:

(12n+4)+(14n6)=180(12n + 4) + (14n - 6) = 180

Now, let's solve for nn:

  1. Combine like terms:

26n2=18026n - 2 = 180

  1. Add 2 to both sides:

26n=18226n = 182

  1. Divide both sides by 26:

n=7n = 7

Now that we know n=7n = 7, we can substitute it back into the expressions for mCBAm\angle CBA and mDBCm\angle DBC:

  • mCBA=12(7)+4=84+4=88m\angle CBA = 12(7) + 4 = 84 + 4 = 88^\circ
  • mDBC=14(7)6=986=92m\angle DBC = 14(7) - 6 = 98 - 6 = 92^\circ

Thus, the measure of ABC\angle ABC, which is the entire straight line, is:

mABC=mCBA+mDBC=88+92=180m\angle ABC = m\angle CBA + m\angle DBC = 88^\circ + 92^\circ = 180^\circ

This confirms that the angles form a linear pair, and mABCm\angle ABC is 180180^\circ.

Would you like more details or have any questions? Here are 5 related questions to expand on this topic:

  1. What are opposite rays, and how do they relate to straight angles?
  2. How do linear pairs of angles help in solving angle problems?
  3. What is the difference between adjacent and supplementary angles?
  4. How would the problem change if the angles were complementary instead of forming a straight line?
  5. Can you explain the relationship between linear pairs and vertical angles?

Tip: Always remember that opposite rays create a straight angle, meaning the sum of the angles must equal 180°.

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Math Problem Analysis

Mathematical Concepts

Angles
Linear pairs
Opposite rays

Formulas

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Theorems

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Suitable Grade Level

Grades 9-10