Question

finding missing measures using the angle addition postulate
date
draw and label four - point diagram with the given information. solve using the angle addition postulate.
assume that bd is between ab and bc.
(1) if m∠abc = 46° and m∠cbd = 34°, find m∠abd.
(2) if m∠cbd = 67° and m∠abd = 124°, find m∠abc.
(3) if m∠abc = 163° and m∠abd = 60°, find m∠cbd.
(4) if m∠abc=(3x + 22)°, m∠abd = 84°, and m∠cbd=(x + 24)°, find the measurements of ∠abc and ∠cbd.
(5) if m∠cbd=(3x + 23)°, m∠abc = 123°, and m∠abd=(x + 14)°, find the measurements of ∠cbd and ∠abd.
(6) if m∠cbd=(2x)°, m∠abc = 134°, and m∠abd=(4x + 14)°, find the measurements of ∠cbd and ∠abd.
(7) if m∠abc=(4x + 2)°, m∠abd=(x + 9)°, and m∠cbd=(-2x + 18)°, find the measurements of ∠abc and ∠abd.
(8) if m∠abc=(2x + 4)°, m∠cbd = 24°, and m∠abd=(x + 12)°, find the measurements of ∠abc and ∠abd.

Explanation:

Step1: Recall angle - addition postulate

The angle - addition postulate states that if point \(D\) lies in the interior of \(\angle ABC\), then \(m\angle ABD+m\angle DBC = m\angle ABC\).

Step2: Solve problem (1)

Given \(m\angle ABC = 46^{\circ}\) and \(m\angle CBD=34^{\circ}\), then \(m\angle ABD=m\angle ABC + m\angle CBD\). So \(m\angle ABD=46 + 34=80^{\circ}\).

Step3: Solve problem (2)

Given \(m\angle CBD = 67^{\circ}\) and \(m\angle ABD = 119^{\circ}\), then \(m\angle ABC=m\angle ABD - m\angle CBD\). So \(m\angle ABC=119 - 67 = 52^{\circ}\).

Step4: Solve problem (3)

Given \(m\angle ABC = 163^{\circ}\) and \(m\angle ABD = 60^{\circ}\), then \(m\angle CBD=m\angle ABC - m\angle ABD\). So \(m\angle CBD=163 - 60=103^{\circ}\).

Step5: Solve problem (4)

Since \(m\angle ABC=(3x + 22)^{\circ}\), \(m\angle ABD = 84^{\circ}\), and \(m\angle CBD=(x + 24)^{\circ}\), by the angle - addition postulate \(m\angle ABC=m\angle ABD + m\angle CBD\). So \((3x + 22)=84+(x + 24)\).
First, expand the right - hand side: \(3x+22=x + 108\).
Subtract \(x\) from both sides: \(3x - x+22=x - x + 108\), \(2x+22 = 108\).
Subtract 22 from both sides: \(2x=108 - 22=86\).
Divide both sides by 2: \(x = 43\).
Then \(m\angle ABC=3\times43+22=129 + 22=151^{\circ}\) and \(m\angle CBD=43 + 24 = 67^{\circ}\).

Step6: Solve problem (5)

Since \(m\angle CBD=(3x + 23)^{\circ}\), \(m\angle ABC = 123^{\circ}\), and \(m\angle ABD=(x + 14)^{\circ}\), and \(m\angle ABC=m\angle ABD + m\angle CBD\), then \(123=(x + 14)+(3x + 23)\).
Expand the right - hand side: \(123=4x+37\).
Subtract 37 from both sides: \(4x=123 - 37 = 86\).
Divide both sides by 4: \(x=\frac{86}{4}=21.5\).
\(m\angle CBD=3\times21.5+23=64.5 + 23=87.5^{\circ}\) and \(m\angle ABD=21.5 + 14=35.5^{\circ}\).

Step7: Solve problem (6)

Since \(m\angle CBD=(2x)^{\circ}\), \(m\angle ABC = 134^{\circ}\), and \(m\angle ABD=(4x + 14)^{\circ}\), and \(m\angle ABC=m\angle ABD + m\angle CBD\), then \(134=(4x + 14)+2x\).
Combine like terms: \(134=6x + 14\).
Subtract 14 from both sides: \(6x=134 - 14 = 120\).
Divide both sides by 6: \(x = 20\).
\(m\angle CBD=2\times20 = 40^{\circ}\) and \(m\angle ABD=4\times20+14=80 + 14=94^{\circ}\).

Step8: Solve problem (7)

Since \(m\angle ABC=(4x + 2)^{\circ}\), \(m\angle ABD=(x + 9)^{\circ}\), and \(m\angle CBD=(-2x + 18)^{\circ}\), and \(m\angle ABC=m\angle ABD + m\angle CBD\), then \((4x + 2)=(x + 9)+(-2x + 18)\).
Expand the right - hand side: \(4x+2=-x + 27\).
Add \(x\) to both sides: \(4x+x+2=-x+x + 27\), \(5x+2 = 27\).
Subtract 2 from both sides: \(5x=27 - 2=25\).
Divide both sides by 5: \(x = 5\).
\(m\angle ABC=4\times5+2=20 + 2=22^{\circ}\) and \(m\angle ABD=5 + 9=14^{\circ}\).

Step9: Solve problem (8)

Since \(m\angle ABC=(2x + 4)^{\circ}\), \(m\angle CBD = 24^{\circ}\), and \(m\angle ABD=(x + 12)^{\circ}\), and \(m\angle ABC=m\angle ABD + m\angle CBD\), then \((2x + 4)=(x + 12)+24\).
Expand the right - hand side: \(2x+4=x + 36\).
Subtract \(x\) from both sides: \(2x - x+4=x - x + 36\), \(x+4 = 36\).
Subtract 4 from both sides: \(x = 32\).
\(m\angle ABC=2\times32+4=64 + 4=68^{\circ}\) and \(m\angle ABD=32 + 12=44^{\circ}\).

Answer:

1. \(m\angle ABD = 80^{\circ}\)
2. \(m\angle ABC = 52^{\circ}\)
3. \(m\angle CBD = 103^{\circ}\)
4. \(m\angle ABC = 151^{\circ}\), \(m\angle CBD = 67^{\circ}\)
5. \(m\angle CBD = 87.5^{\circ}\), \(m\angle ABD = 35.5^{\circ}\)
6. \(m\angle CBD = 40^{\circ}\), \(m\angle ABD = 94^{\circ}\)
7. \(m\angle ABC = 22^{\circ}\), \(m\angle ABD = 14^{\circ}\)
8. \(m\angle ABC = 68^{\circ}\), \(m\angle ABD = 44^{\circ}\)