Question

algebra $overrightarrow{bd}$ bisects $angle abc$. solve for $x$ and find $mangle abc$.

1. $mangle abd = 5x$, $mangle dbc = 3x + 10$
2. $mangle abc = 4x - 12$, $mangle abd = 24$
3. $mangle abd = 4x - 16$, $mangle cbd = 2x + 6$
4. $mangle abd = 3x + 20$, $mangle cbd = 6x - 16$

Explanation:

Problem 27

Step1: Use angle bisector property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), \(m\angle ABD = m\angle DBC\). So \(5x = 3x + 10\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(5x - 3x = 3x + 10 - 3x\) → \(2x = 10\). Divide by 2: \(x = 5\).

Step3: Find \(m\angle ABC\)

\(m\angle ABC = m\angle ABD + m\angle DBC = 5x + 3x + 10\). Substitute \(x = 5\): \(5(5) + 3(5) + 10 = 25 + 15 + 10 = 50\).

Step1: Use angle bisector property

\(\overrightarrow{BD}\) bisects \(\angle ABC\), so \(m\angle ABC = 2 \times m\angle ABD\). Thus, \(4x - 12 = 2 \times 24\).

Step2: Solve for \(x\)

Simplify right side: \(4x - 12 = 48\). Add 12: \(4x = 60\). Divide by 4: \(x = 15\).

Step3: Find \(m\angle ABC\)

Substitute \(x = 15\) into \(4x - 12\): \(4(15) - 12 = 60 - 12 = 48^\circ\).

Step1: Use angle bisector property

\(m\angle ABD = m\angle CBD\) (bisector), so \(4x - 16 = 2x + 6\).

Step2: Solve for \(x\)

Subtract \(2x\): \(2x - 16 = 6\). Add 16: \(2x = 22\). Divide by 2: \(x = 11\).

Step3: Find \(m\angle ABC\)

\(m\angle ABC = (4x - 16) + (2x + 6)\). Substitute \(x = 11\): \(4(11)-16 + 2(11)+6 = 44 - 16 + 22 + 6 = 56\).

Answer:

\(x = 5\), \(m\angle ABC = 50^\circ\)

Problem 28