Question

bd bisects ∠abc. find m∠abd, m∠cbd, and m∠abc.

1. a (6x + 14)° d b (3x + 29)° c
2. m∠abc=(2 - 16x)° a (2x + 81)° b d c

Explanation:

1. For problem 31:

• Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(m\angle ABD=m\angle CBD\).
• # Explanation:
• ## Step1: Set up the equation
• Because \(m\angle ABD = m\angle CBD\), we set \(6x + 14=3x + 29\).
• ## Step2: Solve for \(x\)
• Subtract \(3x\) from both sides: \(6x-3x + 14=3x-3x + 29\), which simplifies to \(3x+14 = 29\). Then subtract 14 from both sides: \(3x+14 - 14=29 - 14\), so \(3x=15\). Divide both - sides by 3: \(x = 5\).
• ## Step3: Find \(m\angle ABD\)
• Substitute \(x = 5\) into the expression for \(m\angle ABD\): \(m\angle ABD=(6x + 14)^{\circ}=(6\times5 + 14)^{\circ}=(30 + 14)^{\circ}=44^{\circ}\).
• ## Step4: Find \(m\angle CBD\)
• Since \(m\angle CBD=m\angle ABD\), \(m\angle CBD = 44^{\circ}\).
• ## Step5: Find \(m\angle ABC\)
• \(m\angle ABC=m\angle ABD + m\angle CBD\), so \(m\angle ABC=44^{\circ}+44^{\circ}=88^{\circ}\).
• # Answer:
• \(m\angle ABD = 44^{\circ}\), \(m\angle CBD = 44^{\circ}\), \(m\angle ABC = 88^{\circ}\)

1. For problem 32:

• Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(m\angle ABD=m\angle CBD=\frac{1}{2}m\angle ABC\). Also, \(m\angle ABD + m\angle CBD=m\angle ABC\), and \(m\angle ABD=(2x + 81)^{\circ}\), \(m\angle ABC=(2 - 16x)^{\circ}\).
• # Explanation:
• ## Step1: Set up the equation
• Because \(m\angle ABD=\frac{1}{2}m\angle ABC\), we have \(2(2x + 81)=2 - 16x\).
• ## Step2: Expand the left - hand side
• Using the distributive property, \(4x+162 = 2 - 16x\).
• ## Step3: Add \(16x\) to both sides
• \(4x+16x+162 = 2-16x + 16x\), which gives \(20x+162 = 2\).
• ## Step4: Subtract 162 from both sides
• \(20x+162 - 162=2 - 162\), so \(20x=-160\).
• ## Step5: Solve for \(x\)
• Divide both sides by 20: \(x=-8\).
• ## Step6: Find \(m\angle ABD\)
• Substitute \(x = - 8\) into \(m\angle ABD=(2x + 81)^{\circ}\), \(m\angle ABD=[2\times(-8)+81]^{\circ}=(-16 + 81)^{\circ}=65^{\circ}\).
• ## Step7: Find \(m\angle CBD\)
• Since \(m\angle CBD=m\angle ABD\), \(m\angle CBD = 65^{\circ}\).
• ## Step8: Find \(m\angle ABC\)
• \(m\angle ABC=(2 - 16x)^{\circ}=[2-16\times(-8)]^{\circ}=(2 + 128)^{\circ}=130^{\circ}\).
• # Answer:
• \(m\angle ABD = 65^{\circ}\), \(m\angle CBD = 65^{\circ}\), \(m\angle ABC = 130^{\circ}\)

Answer:

1. For problem 31:

• Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(m\angle ABD=m\angle CBD\).
• # Explanation:
• ## Step1: Set up the equation
• Because \(m\angle ABD = m\angle CBD\), we set \(6x + 14=3x + 29\).
• ## Step2: Solve for \(x\)
• Subtract \(3x\) from both sides: \(6x-3x + 14=3x-3x + 29\), which simplifies to \(3x+14 = 29\). Then subtract 14 from both sides: \(3x+14 - 14=29 - 14\), so \(3x=15\). Divide both - sides by 3: \(x = 5\).
• ## Step3: Find \(m\angle ABD\)
• Substitute \(x = 5\) into the expression for \(m\angle ABD\): \(m\angle ABD=(6x + 14)^{\circ}=(6\times5 + 14)^{\circ}=(30 + 14)^{\circ}=44^{\circ}\).
• ## Step4: Find \(m\angle CBD\)
• Since \(m\angle CBD=m\angle ABD\), \(m\angle CBD = 44^{\circ}\).
• ## Step5: Find \(m\angle ABC\)
• \(m\angle ABC=m\angle ABD + m\angle CBD\), so \(m\angle ABC=44^{\circ}+44^{\circ}=88^{\circ}\).
• # Answer:
• \(m\angle ABD = 44^{\circ}\), \(m\angle CBD = 44^{\circ}\), \(m\angle ABC = 88^{\circ}\)

1. For problem 32:

• Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(m\angle ABD=m\angle CBD=\frac{1}{2}m\angle ABC\). Also, \(m\angle ABD + m\angle CBD=m\angle ABC\), and \(m\angle ABD=(2x + 81)^{\circ}\), \(m\angle ABC=(2 - 16x)^{\circ}\).
• # Explanation:
• ## Step1: Set up the equation
• Because \(m\angle ABD=\frac{1}{2}m\angle ABC\), we have \(2(2x + 81)=2 - 16x\).
• ## Step2: Expand the left - hand side
• Using the distributive property, \(4x+162 = 2 - 16x\).
• ## Step3: Add \(16x\) to both sides
• \(4x+16x+162 = 2-16x + 16x\), which gives \(20x+162 = 2\).
• ## Step4: Subtract 162 from both sides
• \(20x+162 - 162=2 - 162\), so \(20x=-160\).
• ## Step5: Solve for \(x\)
• Divide both sides by 20: \(x=-8\).
• ## Step6: Find \(m\angle ABD\)
• Substitute \(x = - 8\) into \(m\angle ABD=(2x + 81)^{\circ}\), \(m\angle ABD=[2\times(-8)+81]^{\circ}=(-16 + 81)^{\circ}=65^{\circ}\).
• ## Step7: Find \(m\angle CBD\)
• Since \(m\angle CBD=m\angle ABD\), \(m\angle CBD = 65^{\circ}\).
• ## Step8: Find \(m\angle ABC\)
• \(m\angle ABC=(2 - 16x)^{\circ}=[2-16\times(-8)]^{\circ}=(2 + 128)^{\circ}=130^{\circ}\).
• # Answer:
• \(m\angle ABD = 65^{\circ}\), \(m\angle CBD = 65^{\circ}\), \(m\angle ABC = 130^{\circ}\)