Segment and Angle Addition Postulate Notes and Worksheets ... - Free Printable

Problem Set: Angle Addition Postulate Practice

We will solve each problem step by step using the Angle Addition Postulate, which states that if a point lies in the interior of an angle, then the measure of the whole angle is the sum of the measures of the two smaller angles.

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#### Problem 1:
If \( m\angle RUS = 32^\circ \) and \( m\angle SUT = 31^\circ \), what is the \( m\angle TUR \)?

Solution:
- According to the Angle Addition Postulate:
\[
m\angle TUR = m\angle RUS + m\angle SUT
\]
- Substitute the given values:
\[
m\angle TUR = 32^\circ + 31^\circ = 63^\circ
\]

Answer:
\[
\boxed{63^\circ}
\]

---

#### Problem 2:
The \( m\angle GJI = 103^\circ \). The \( m\angle GJH = 2x + 6 \) and \( m\angle IJH = 3x + 7 \). What is the value of \( x \)?

Solution:
- According to the Angle Addition Postulate:
\[
m\angle GJI = m\angle GJH + m\angle IJH
\]
- Substitute the given values:
\[
103^\circ = (2x + 6) + (3x + 7)
\]
- Simplify the equation:
\[
103 = 2x + 6 + 3x + 7
\]
\[
103 = 5x + 13
\]
- Solve for \( x \):
\[
103 - 13 = 5x
\]
\[
90 = 5x
\]
\[
x = \frac{90}{5} = 18
\]

Answer:
\[
\boxed{18}
\]

---

#### Problem 3:
Find the \( m\angle CBD \) if \( m\angle ABC = 87^\circ \).

Solution:
- From the diagram, \( \angle ABC \) is composed of \( \angle ABD \) and \( \angle DBC \):
\[
m\angle ABC = m\angle ABD + m\angle DBC
\]
- Let \( m\angle ABD = 3x \) and \( m\angle DBC = 5x - 9 \). Substitute these into the equation:
\[
87 = 3x + (5x - 9)
\]
- Simplify the equation:
\[
87 = 3x + 5x - 9
\]
\[
87 = 8x - 9
\]
- Solve for \( x \):
\[
87 + 9 = 8x
\]
\[
96 = 8x
\]
\[
x = \frac{96}{8} = 12
\]
- Now find \( m\angle CBD \):
\[
m\angle CBD = 5x - 9 = 5(12) - 9 = 60 - 9 = 51^\circ
\]

Answer:
\[
\boxed{51^\circ}
\]

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#### Problem 4:
The \( m\angle SQR = 52^\circ \). The \( m\angle PQS = 3x \) and \( m\angle RQP = 8x - 8 \). What is the value of \( x \)?

Solution:
- According to the Angle Addition Postulate:
\[
m\angle SQR = m\angle PQS + m\angle RQP
\]
- Substitute the given values:
\[
52 = 3x + (8x - 8)
\]
- Simplify the equation:
\[
52 = 3x + 8x - 8
\]
\[
52 = 11x - 8
\]
- Solve for \( x \):
\[
52 + 8 = 11x
\]
\[
60 = 11x
\]
\[
x = \frac{60}{11}
\]

Answer:
\[
\boxed{\frac{60}{11}}
\]

---

#### Problem 5:
The \( m\angle DEG = 46^\circ \). The \( m\angle DEF = 24x + 10 \) and \( m\angle GEF = 10x - 8 \). What is the value of \( x \)?

Solution:
- According to the Angle Addition Postulate:
\[
m\angle DEG = m\angle DEF + m\angle GEF
\]
- Substitute the given values:
\[
46 = (24x + 10) + (10x - 8)
\]
- Simplify the equation:
\[
46 = 24x + 10 + 10x - 8
\]
\[
46 = 34x + 2
\]
- Solve for \( x \):
\[
46 - 2 = 34x
\]
\[
44 = 34x
\]
\[
x = \frac{44}{34} = \frac{22}{17}
\]

Answer:
\[
\boxed{\frac{22}{17}}
\]

---

#### Problem 6:
The \( m\angle LMN = 108^\circ \). Find the measure of the two smaller angles.

Solution:
- From the diagram, \( \angle LMN \) is composed of \( \angle LMH \) and \( \angle NMH \):
\[
m\angle LMN = m\angle LMH + m\angle NMH
\]
- Let \( m\angle LMH = 5x - 2 \) and \( m\angle NMH = x \). Substitute these into the equation:
\[
108 = (5x - 2) + x
\]
- Simplify the equation:
\[
108 = 5x - 2 + x
\]
\[
108 = 6x - 2
\]
- Solve for \( x \):
\[
108 + 2 = 6x
\]
\[
110 = 6x
\]
\[
x = \frac{110}{6} = \frac{55}{3}
\]
- Now find the measures of the two smaller angles:
\[
m\angle NMH = x = \frac{55}{3}
\]
\[
m\angle LMH = 5x - 2 = 5\left(\frac{55}{3}\right) - 2 = \frac{275}{3} - 2 = \frac{275}{3} - \frac{6}{3} = \frac{269}{3}
\]

Answer:
\[
\boxed{\frac{55}{3}, \frac{269}{3}}
\]

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#### Problem 7:
The \( m\angle ADB = 39^\circ \), \( m\angle ADC = 8x + 5 \), and \( m\angle BDC = 3x - 4 \). Find the degree measure of \( \angle ADC \).

Solution:
- From the diagram, \( \angle ADC \) is composed of \( \angle ADB \) and \( \angle BDC \):
\[
m\angle ADC = m\angle ADB + m\angle BDC
\]
- Substitute the given values:
\[
8x + 5 = 39 + (3x - 4)
\]
- Simplify the equation:
\[
8x + 5 = 39 + 3x - 4
\]
\[
8x + 5 = 35 + 3x
\]
- Solve for \( x \):
\[
8x - 3x = 35 - 5
\]
\[
5x = 30
\]
\[
x = 6
\]
- Now find \( m\angle ADC \):
\[
m\angle ADC = 8x + 5 = 8(6) + 5 = 48 + 5 = 53^\circ
\]

Answer:
\[
\boxed{53^\circ}
\]

---

#### Problem 8:
If \( m\angle RSU = 10x - 9 \), find the degree measure of all 3 angles.

Solution:
- From the diagram, \( \angle RSU \) is composed of \( \angle RST \) and \( \angle TSU \):
\[
m\angle RSU = m\angle RST + m\angle TSU
\]
- Let \( m\angle RST = 2x \) and \( m\angle TSU = 2x + 75 \). Substitute these into the equation:
\[
10x - 9 = 2x + (2x + 75)
\]
- Simplify the equation:
\[
10x - 9 = 2x + 2x + 75
\]
\[
10x - 9 = 4x + 75
\]
- Solve for \( x \):
\[
10x - 4x = 75 + 9
\]
\[
6x = 84
\]
\[
x = 14
\]
- Now find the measures of all three angles:
\[
m\angle RST = 2x = 2(14) = 28^\circ
\]
\[
m\angle TSU = 2x + 75 = 2(14) + 75 = 28 + 75 = 103^\circ
\]
\[
m\angle RSU = 10x - 9 = 10(14) - 9 = 140 - 9 = 131^\circ
\]

Answer:
\[
\boxed{28^\circ, 103^\circ, 131^\circ}
\]

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Final Answers:

1. \(\boxed{63^\circ}\)
2. \(\boxed{18}\)
3. \(\boxed{51^\circ}\)
4. \(\boxed{\frac{60}{11}}\)
5. \(\boxed{\frac{22}{17}}\)
6. \(\boxed{\frac{55}{3}, \frac{269}{3}}\)
7. \(\boxed{53^\circ}\)
8. \(\boxed{28^\circ, 103^\circ, 131^\circ}\)