Math Problem Statement

BD bisects ∠ABC. Find m∠ABD, m∠CBD, and m∠ABC given that m∠ABD = (3x + 6)° and m∠CBD = (7x - 18)°.

Solution

To solve for the angles ABD\angle ABD, CBD\angle CBD, and ABC\angle ABC, we can use the information provided in the diagram and the fact that BDBD bisects ABC\angle ABC.

Given:

  • BDBD bisects ABC\angle ABC, meaning that mABD=mCBDm\angle ABD = m\angle CBD.
  • mABD=(3x+6)m\angle ABD = (3x + 6)^\circ
  • mCBD=(7x18)m\angle CBD = (7x - 18)^\circ

Since BDBD bisects ABC\angle ABC, we can set the two expressions for ABD\angle ABD and CBD\angle CBD equal to each other:

3x+6=7x183x + 6 = 7x - 18

Step 1: Solve for xx

3x+6=7x183x + 6 = 7x - 18 Subtract 3x3x from both sides: 6=4x186 = 4x - 18 Add 18 to both sides: 24=4x24 = 4x Divide both sides by 4: x=6x = 6

Step 2: Find mABDm\angle ABD and mCBDm\angle CBD

Substitute x=6x = 6 into the expressions for ABD\angle ABD and CBD\angle CBD:

mABD=3(6)+6=18+6=24m\angle ABD = 3(6) + 6 = 18 + 6 = 24^\circ mCBD=7(6)18=4218=24m\angle CBD = 7(6) - 18 = 42 - 18 = 24^\circ

Step 3: Find mABCm\angle ABC

Since BDBD bisects ABC\angle ABC, the total angle ABC\angle ABC is the sum of ABD\angle ABD and CBD\angle CBD:

mABC=mABD+mCBD=24+24=48m\angle ABC = m\angle ABD + m\angle CBD = 24^\circ + 24^\circ = 48^\circ

Final Answers:

mABD=24m\angle ABD = 24^\circ mCBD=24m\angle CBD = 24^\circ mABC=48m\angle ABC = 48^\circ

Would you like further details on this, or do you have any questions?

Here are some related questions you might find helpful:

  1. How would the solution change if BDBD did not bisect ABC\angle ABC?
  2. What would happen if the expression for ABD\angle ABD involved quadratic terms?
  3. How can you verify that the angles in a bisected angle are equal?
  4. Can you use trigonometric methods to find the angles in different ways?
  5. How would you approach the problem if the diagram was scaled or rotated?

Tip: Always verify whether the bisector splits the angle into equal parts to apply symmetry and simplify the problem.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Bisectors
Linear Equations

Formulas

m∠ABD = m∠CBD because BD bisects ∠ABC

Theorems

Angle Bisector Theorem

Suitable Grade Level

Grades 8-10