Geometry Angle Addition Worksheet - Free Printable

Problem Analysis and Solution

The worksheet involves several problems related to angle bisectors, solving for unknown variables, and using the Angle Addition Postulate. Let's solve each problem step by step.

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#### Problem 12:
- Given: \( \overrightarrow{PT} \) is the angle bisector of \( \angle RPS \).
- Diagram: \( \angle RPS = 27^\circ \).

Since \( \overrightarrow{PT} \) is the angle bisector, it divides \( \angle RPS \) into two equal angles:
\[
\angle RPT = \angle TPS = \frac{\angle RPS}{2} = \frac{27^\circ}{2} = 13.5^\circ
\]

Answer for Problem 12:
\[
\boxed{13.5^\circ}
\]

---

#### Problem 13:
- Given: \( \angle RPS = 48^\circ \).
- Diagram: \( \overrightarrow{PT} \) is the angle bisector of \( \angle RPS \).

Using the same logic as in Problem 12:
\[
\angle RPT = \angle TPS = \frac{\angle RPS}{2} = \frac{48^\circ}{2} = 24^\circ
\]

Answer for Problem 13:
\[
\boxed{24^\circ}
\]

---

#### Problem 14:
- Given: \( \angle RPS = 79^\circ \).
- Diagram: \( \overrightarrow{PT} \) is the angle bisector of \( \angle RPS \).

Again, using the angle bisector property:
\[
\angle RPT = \angle TPS = \frac{\angle RPS}{2} = \frac{79^\circ}{2} = 39.5^\circ
\]

Answer for Problem 14:
\[
\boxed{39.5^\circ}
\]

---

#### Problem 15:
- Given: \( \overrightarrow{BT} \) bisects \( \angle ABC \).
- Diagram: \( \angle ABT = (5x - 17)^\circ \) and \( \angle TBC = (3x + 14)^\circ \).

Since \( \overrightarrow{BT} \) is the angle bisector, the two angles it creates are equal:
\[
5x - 17 = 3x + 14
\]

Solve for \( x \):
\[
5x - 3x = 14 + 17
\]
\[
2x = 31
\]
\[
x = \frac{31}{2} = 15.5
\]

Answer for Problem 15:
\[
\boxed{15.5}
\]

---

#### Problem 16:
- Given: \( \overrightarrow{BT} \) bisects \( \angle ABC \).
- Diagram: \( \angle ABT = (3x - 14)^\circ \) and \( \angle TBC = (2x + 36)^\circ \).

Using the angle bisector property:
\[
3x - 14 = 2x + 36
\]

Solve for \( x \):
\[
3x - 2x = 36 + 14
\]
\[
x = 50
\]

Answer for Problem 16:
\[
\boxed{50}
\]

---

#### Problem 17:
- Given: \( \overrightarrow{BT} \) bisects \( \angle ABC \).
- Diagram: \( \angle ABT = (3x - 24)^\circ \) and \( \angle TBC = (3x - 48)^\circ \).

Using the angle bisector property:
\[
3x - 24 = 3x - 48
\]

This equation simplifies to:
\[
-24 = -48
\]

This is a contradiction, indicating an error in the problem setup or diagram. However, if we assume the problem is solvable, we might need additional information. For now, let's assume the problem is correctly stated and solve it as given.

Answer for Problem 17:
\[
\boxed{\text{No solution (contradiction)}}
\]

---

#### Problem 7:
- Given: \( \overrightarrow{BD} \) bisects \( \angle ABC \).
- Diagram: \( \angle ABD = (2x - 13)^\circ \) and \( \angle DBC = (1 - x + 16)^\circ \).

Simplify \( \angle DBC \):
\[
\angle DBC = 1 - x + 16 = 17 - x
\]

Using the angle bisector property:
\[
2x - 13 = 17 - x
\]

Solve for \( x \):
\[
2x + x = 17 + 13
\]
\[
3x = 30
\]
\[
x = 10
\]

Answer for Problem 7:
\[
\boxed{10}
\]

---

#### Problem 8:
- Given: \( \overrightarrow{BD} \) bisects \( \angle ABC \).
- Diagram: \( \angle ABD = (5x + 10)^\circ \) and \( \angle DBC = 6x^\circ \).

Using the angle bisector property:
\[
5x + 10 = 6x
\]

Solve for \( x \):
\[
10 = 6x - 5x
\]
\[
x = 10
\]

Answer for Problem 8:
\[
\boxed{10}
\]

---

#### Problem 4:
- Given: Use the Angle Addition Postulate to find \( m\angle FDC \).
- Diagram: \( \angle CDF = 21^\circ \) and \( \angle FDA = 23^\circ \).

Using the Angle Addition Postulate:
\[
m\angle FDC = m\angle CDF + m\angle FDA = 21^\circ + 23^\circ = 44^\circ
\]

Answer for Problem 4:
\[
\boxed{44^\circ}
\]

---

#### Problem 5:
- Given: Use the Angle Addition Postulate to find \( m\angle CDE \).
- Diagram: \( \angle EDF = 52^\circ \) and \( \angle FDA = 48^\circ \).

Using the Angle Addition Postulate:
\[
m\angle CDE = m\angle EDF + m\angle FDA = 52^\circ + 48^\circ = 100^\circ
\]

Answer for Problem 5:
\[
\boxed{100^\circ}
\]

---

#### Problem 6:
- Given: Use the Angle Addition Postulate to find \( m\angle XYZ \).
- Diagram: \( \angle WXY = 25^\circ \) and \( \angle WXZ = 32^\circ \).

Using the Angle Addition Postulate:
\[
m\angle XYZ = m\angle WXZ - m\angle WXY = 32^\circ - 25^\circ = 7^\circ
\]

Answer for Problem 6:
\[
\boxed{7^\circ}
\]

---

Final Answers:

1. Problem 12: \( \boxed{13.5^\circ} \)
2. Problem 13: \( \boxed{24^\circ} \)
3. Problem 14: \( \boxed{39.5^\circ} \)
4. Problem 15: \( \boxed{15.5} \)
5. Problem 16: \( \boxed{50} \)
6. Problem 17: \( \boxed{\text{No solution (contradiction)}} \)
7. Problem 7: \( \boxed{10} \)
8. Problem 8: \( \boxed{10} \)
9. Problem 4: \( \boxed{44^\circ} \)
10. Problem 5: \( \boxed{100^\circ} \)
11. Problem 6: \( \boxed{7^\circ} \)