Math Problem Statement

BD bisects ∠ABC. Solve for x and find m∠ABC. m∠ABD = 5x - 3, m∠CBD = 3x + 3

Solution

The problem states that BD bisects ∠ABC. We're given the measures of ∠ABD and ∠CBD as expressions involving xx, and we're tasked with solving for xx and finding the measure of ∠ABC.

  1. Since BD bisects ∠ABC, it means that: mABD=mCBDm\angle ABD = m\angle CBD Using the expressions provided: 5x3=3x+35x - 3 = 3x + 3

  2. To solve for xx:

    • Subtract 3x3x from both sides: 5x3x3=32x3=35x - 3x - 3 = 3 \quad \Rightarrow \quad 2x - 3 = 3
    • Add 3 to both sides: 2x=62x = 6
    • Divide both sides by 2: x=3x = 3
  3. Now that we have x=3x = 3, substitute it back into the expressions for the angles:

    • mABD=5(3)3=153=12m\angle ABD = 5(3) - 3 = 15 - 3 = 12^\circ
    • mCBD=3(3)+3=9+3=12m\angle CBD = 3(3) + 3 = 9 + 3 = 12^\circ
  4. Since BD bisects ∠ABC, we can find: mABC=mABD+mCBD=12+12=24m\angle ABC = m\angle ABD + m\angle CBD = 12^\circ + 12^\circ = 24^\circ

Thus, x=3x = 3 and mABC=24m\angle ABC = 24^\circ.


Would you like more details or have any questions?

Here are some related questions for further exploration:

  1. What happens if BD doesn't bisect ∠ABC? How would you solve it?
  2. If ∠ABD were twice ∠CBD, how would the equation change?
  3. How would you solve if one of the angle expressions involved a quadratic?
  4. Can bisectors be used to solve problems in three-dimensional geometry?
  5. 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