Question

in the diagram, $overrightarrow{bd}$ bisects $angle abc$. find $mangle abc.
7.
8.

Explanation:

1. For problem 7:

• Explanation:
• Step 1: Use the angle - bisector property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(\angle ABD=\angle DBC\). So, we set up the equation \(8x - 16=4x + 20\).

• Step 2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(8x-4x - 16=4x-4x + 20\), which simplifies to \(4x-16 = 20\). Then add 16 to both sides: \(4x-16 + 16=20 + 16\), giving \(4x=36\). Divide both sides by 4: \(x = 9\).

• Step 3: Find \(m\angle ABC\)

\(m\angle ABC=m\angle ABD + m\angle DBC\). Since \(m\angle ABD=8x - 16\) and \(m\angle DBC=4x + 20\), and \(x = 9\), \(m\angle ABD=8\times9-16=72 - 16 = 56^{\circ}\), \(m\angle DBC=4\times9 + 20=36+20 = 56^{\circ}\). So, \(m\angle ABC=56^{\circ}+56^{\circ}=112^{\circ}\).

1. For problem 8:

• Explanation:
• Step 1: Use the angle - bisector property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(\angle ABD=\angle DBC\). So, we set up the equation \(5x-11=4x + 1\).

• Step 2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(5x-4x-11=4x-4x + 1\), which simplifies to \(x-11 = 1\). Then add 11 to both sides: \(x-11 + 11=1 + 11\), giving \(x = 12\).

• Step 3: Find \(m\angle ABC\)

\(m\angle ABC=m\angle ABD + m\angle DBC\). Since \(m\angle ABD=5x - 11\) and \(m\angle DBC=4x + 1\), and \(x = 12\), \(m\angle ABD=5\times12-11=60 - 11 = 49^{\circ}\), \(m\angle DBC=4\times12 + 1=48 + 1 = 49^{\circ}\). So, \(m\angle ABC=49^{\circ}+49^{\circ}=98^{\circ}\).

Answer:

1. \(m\angle ABC = 112^{\circ}\)
2. \(m\angle ABC = 98^{\circ}\)