Math Problem Statement
Solution
The problem states that BD bisects ∠ABC. We're given the measures of ∠ABD and ∠CBD as expressions involving , and we're tasked with solving for and finding the measure of ∠ABC.
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Since BD bisects ∠ABC, it means that: Using the expressions provided:
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To solve for :
- Subtract from both sides:
- Add 3 to both sides:
- Divide both sides by 2:
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Now that we have , substitute it back into the expressions for the angles:
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Since BD bisects ∠ABC, we can find:
Thus, and .
Would you like more details or have any questions?
Here are some related questions for further exploration:
- What happens if BD doesn't bisect ∠ABC? How would you solve it?
- If ∠ABD were twice ∠CBD, how would the equation change?
- How would you solve if one of the angle expressions involved a quadratic?
- Can bisectors be used to solve problems in three-dimensional geometry?
- How does the angle bisector theorem help in solving problems involving triangles?
Tip: In angle bisector problems, remember that the two bisected angles are always equal unless other conditions are specified.
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Math Problem Analysis
Mathematical Concepts
Algebra
Angle Bisectors
Equations
Formulas
m∠ABD = m∠CBD
Equation: 5x - 3 = 3x + 3
Theorems
Angle Bisector Theorem
Suitable Grade Level
Grades 8-10