Honors Geometry Chapter 2 Review Test Study Guide

Essential Geometric Principles and Theorems

• Definition of Midpoint: A point on a line segment that divides the segment into two congruent segments.
• Definition of Angle Bisector: A ray that divides an angle into two congruent adjacent angles.
• Definition of Complementary Angles: Two angles whose measures sum to $90^\circ$.
• Definition of Supplementary Angles: Two angles whose measures sum to $180^\circ$.
• Right Angle Theorem: All right angles are congruent to one another.
• Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
• Vertical Angles Theorem: Vertical angles are always congruent.
• Congruent Complements Theorem: Complements of the same angle or of congruent angles are congruent.
• Congruent Supplements Theorem: Supplements of the same angle or of congruent angles are congruent.
• Addition Property of Congruence: If congruent segments or angles are added to congruent segments or angles, the resulting sums are congruent.
• Subtraction Property of Congruence: If congruent segments or angles are subtracted from congruent segments or angles, the resulting differences are congruent.
• Division Property of Congruence (Like Division Theorem): If two segments or angles are congruent, then their like divisions (such as halves) are congruent.
• Multiplication Property of Congruence (Like Multiple Theorem): If like divisions (such as halves) of segments or angles are congruent, then the whole segments or angles are congruent.
• Transitive Property of Congruence: If $A \cong B$ and $B \cong C$, then A \cong C$.\n- **Reflexive Property of Congruence**: Any segment or angle is congruent to itself (A \cong A).\n\n# Part I: Immediate Conclusions and Reasons\n\n- **Problem 1**\n - **Given**: \overline{AC} \cong \overline{DF}$;$B$is the midpoint of$\overline{AC}$;$E$is the midpoint of$\overline{DF}.\n\n  \n\n  \n\n - **Conclusion**: \overline{AB} \cong \overline{DE}$(or$\overline{AB} \cong \overline{BC} \cong \overline{DE} \cong \overline{EF})\n - **Reason**: If two segments are congruent, then their like divisions (halves) are congruent (Division Property of Congruence).\n\n- **Problem 2**\n - **Given**: \overline{AB} \cong \overline{EF}$;$B$is the midpoint of$\overline{AC}$;$E$is the midpoint of$\overline{DF}.\n - **Conclusion**: \overline{AC} \cong \overline{DF}\n - **Reason**: If like divisions (halves) of segments are congruent, then the whole segments are congruent (Multiplication Property of Congruence).\n\n- **Problem 3**\n - **Given**: Vertex B$with rays$\vec{BA}, \vec{BE}, \vec{BD}, \vec{BC}$, where$\vec{BE}$bisects$\angle ABD$and$\vec{BD}$bisects$\angle EBC$.

• Conclusion: $\angle ABE \cong \angle DBC$ (or $\angle ABE \cong \angle EBD \cong \angle DBC$)
• Reason: If angles are bisected and congruent divisions exist, then like divisions of congruent angles are congruent (Division Property of Congruence / Definition of Angle Bisector).

Part II: Identifying Reasons from Geometric Diagrams

• Diagram Setup: $\angle ABD$ and $\angle GED$ are right angles ($90^\circ$).

• Problem 4

  • Statement: If $\angle 1 \cong \angle 10$, why is $\angle 2 \cong \angle 9$?
  • Reason: Complements of congruent angles are congruent (Congruent Complements Theorem). Since $\angle 1$ and $\angle 2$ form right angle $\angle ABD$ (complementary), and $\angle 10$ and $\angle 9$ form right angle $\angle GED$ (complementary), if $\angle 1 \cong \angle 10$, then \angle 2 \cong \angle 9$.\n\n- **Problem 5**\n - **Statement**: If \angle 3 \cong \angle 8$and$\angle 4 \cong \angle 7$, why is$\angle ACD \cong \angle GFD?\n - **Reason**: Addition Property of Congruence. If congruent angles are added to congruent angles, the resulting sums are congruent.\n\n- **Problem 6**\n - **Statement**: Why are \angle 3$and$\angle 4 supplementary?\n - **Reason**: Linear Pair Postulate. If two angles form a linear pair (along line segment \overline{AD}), then they are supplementary.\n\n- **Problem 7**\n - **Statement**: If \overline{AD} \cong \overline{DG}$and$\overline{AC} \cong \overline{GF}$, why is$\overline{CD} \cong \overline{DF}?\n - **Reason**: Subtraction Property of Congruence. If congruent segments are subtracted from congruent segments, the remaining differences are congruent.\n\n- **Problem 8**\n - **Statement**: If C$is the midpoint of$\overline{AD}$,$F$is the midpoint of$\overline{DG}$, and$\overline{AD} \cong \overline{DG}$, why is$\overline{AC} \cong \overline{GF}?\n - **Reason**: Division Property of Congruence (Like Division Theorem). If two segments are congruent, then their halves (like divisions) are congruent.\n\n# Part III: Algebraic Problem Solving and Angle Relationships\n\n- **Problem 9**\n - **Problem**: One of two complementary angles has a measure that is six more than twice the other. Find the measure of the larger angle.\n - **Step-by-Step Solution**:\n 1. Let the measure of the smaller angle be x.\n 2. Then the measure of the larger angle is 2x + 6$.
  1. Since the angles are complementary, their sum is $90^\circ$:        $x + (2x + 6) = 90$
  2. Combine like terms:        $3x + 6 = 90$
  3. Subtract $6$ from both sides:        $3x = 84$
  4. Divide by $3$:        $x = 28^\circ$
  5. Calculate the larger angle:        $2(28^\circ) + 6^\circ = 56^\circ + 6^\circ = 62^\circ$
  • Final Answer: Measure of the larger angle = $62^\circ$.

• Problem 10

  • Problem: The measure of the supplement of an angle is five times that of the angle's complement. Find the measure of half of the complement of the angle.
  • Step-by-Step Solution:
  1. Let the measure of the angle be $x$.
  2. The supplement of the angle is $180 - x$.
  3. The complement of the angle is $90 - x$.
  4. Set up the equation according to the problem statement:        $180 - x = 5(90 - x)$
  5. Expand the right side:        $180 - x = 450 - 5x$
  6. Add $5x$ to both sides:        $180 + 4x = 450$
  7. Subtract $180$ from both sides:        $4x = 270$
  8. Divide by $4$:        $x = 67.5^\circ$
  9. Find the measure of the complement:        $\text{Complement} = 90^\circ - 67.5^\circ = 22.5^\circ$
  10. Find half of the complement:         $\frac{22.5^\circ}{2} = 11.25^\circ$
  • Final Answer: Half of the complement = $11.25^\circ$ (or $11\frac{1}{4}^\circ$).

• Problem 11

• Given: $\vec{OP}$ bisects $\angle MOE$, $m\angle MOP = 10 - 3x$, and $m\angle POE = x^2 - 6x$.
• Find: $m\angle MOE$
• Step-by-Step Solution:
  1. Since $\vec{OP}$ bisects $\angle MOE$, the two divided angles are equal in measure:        $m\angle MOP = m\angle POE$
  2. Equate the given expressions:        $10 - 3x = x^2 - 6x$
  3. Rearrange into standard quadratic form:        $x^2 - 3x - 10 = 0$
  4. Factor the quadratic equation:        $(x - 5)(x + 2) = 0$
  5. Solve for potential values of $x$:        $x = 5 \quad \text{or} \quad x = -2$
  6. Test $x = 5$:        $m\angle MOP = 10 - 3(5) = 10 - 15 = -5^\circ$        Since an angle measure cannot be negative, $x = 5$ is an extraneous solution.
  7. Test $x = -2$:        $m\angle MOP = 10 - 3(-2) = 10 + 6 = 16^\circ$$m\angle POE = (-2)^2 - 6(-2) = 4 + 12 = 16^\circ$        Both measures yield positive, valid angle values of $16^\circ$.
  8. Calculate total angle measure $m\angle MOE$:        $m\angle MOE = m\angle MOP + m\angle POE = 16^\circ + 16^\circ = 32^\circ$
• Final Answer: $m\angle MOE = 32^\circ$.

Part IV: Formal Two-Column Proofs

• Problem 12
  • Given: $\vec{BE}$ bisects $\angle ABC$, $\angle 1 \text{ comp } \angle 2$, $\angle 3 \text{ comp } \angle 4$, and $\angle 3 \cong \angle 4$.
  • Prove: $\angle 1 \cong \angle 3$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\vec{BE}$ bisects $\angle ABC$ | 1. Given |   | 2. $\angle 2 \cong \angle 4$ | 2. Definition of Angle Bisector |   | 3. $\angle 3 \cong \angle 4$ | 3. Given |   | 4. $\angle 2 \cong \angle 3$ | 4. Transitive Property of Congruence |   | 5. $\angle 1 \text{ is complementary to } \angle 2$ | 5. Given |   | 6. $\angle 3 \text{ is complementary to } \angle 4$ | 6. Given |   | 7. $\angle 1 \cong \angle 3$ | 7. Congruent Complements Theorem (Complements of congruent angles are congruent) |

• Problem 13
  • Given: $\angle WXT \cong \angle YXZ$
  • Prove: $\angle WXZ \cong \angle TXY$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\angle WXT \cong \angle YXZ$ | 1. Given |   | 2. $\angle TXZ \cong \angle TXZ$ | 2. Reflexive Property of Congruence |   | 3. $\angle WXT + \angle TXZ \cong \angle YXZ + \angle TXZ$ | 3. Addition Property of Congruence |   | 4. $\angle WXZ \cong \angle TXY$ | 4. Angle Addition Postulate |

• Problem 14
  • Given: $\overline{FG} \cong \overline{JH}$, $N$ is the midpoint of $\overline{FG}$, $O$ is the midpoint of $\overline{JH}$.
  • Prove: $\overline{NG} \cong \overline{OH}$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\overline{FG} \cong \overline{JH}$ | 1. Given |   | 2. $N$ is the midpoint of $\overline{FG}$, $O$ is the midpoint of $\overline{JH}$ | 2. Given |   | 3. $\overline{NG} \cong \overline{OH}$ | 3. Division Property of Congruence (If two segments are congruent, their like divisions/halves are congruent) |

• Problem 15
  • Given: $\angle 1 \cong \angle 3$
  • Conclusion / Prove: $\angle 1 \cong \angle 2$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\angle 1 \cong \angle 3$ | 1. Given |   | 2. $\angle 2 \cong \angle 3$ | 2. Vertical Angles Theorem |   | 3. $\angle 1 \cong \angle 2$ | 3. Transitive Property of Congruence |

• Problem 16
  • Given: $\angle EGF \cong \angle EFG$, $\angle EGH \cong \angle EFJ$
  • Conclusion / Prove: $\angle HGF \cong \angle JFG$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\angle EGF \cong \angle EFG$ | 1. Given |   | 2. $\angle EGH \cong \angle EFJ$ | 2. Given |   | 3. $\angle HGF \cong \angle JFG$ | 3. Subtraction Property of Congruence (If congruent angles are subtracted from congruent angles, the differences are congruent) |

• Problem 17
  • Given: $\vec{VY}$ bisects $\angle TVZ$, $\vec{ZY}$ bisects $\angle TZV$, $\angle TVZ \cong \angle TZV$
  • Conclusion / Prove: $\angle 3 \cong \angle 1$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\vec{VY}$ bisects $\angle TVZ$, $\vec{ZY}$ bisects $\angle TZV$ | 1. Given |   | 2. $\angle TVZ \cong \angle TZV$ | 2. Given |   | 3. $\angle 2 \cong \angle 3$ | 3. Division Property of Congruence (Halves of congruent angles are congruent) |   | 4. $\angle 1 \cong \angle 2$ | 4. Vertical Angles Theorem |   | 5. $\angle 3 \cong \angle 1$ | 5. Transitive Property of Congruence |

• Problem 18
  • Given: $\angle 1 \text{ is comp. to } \angle 3$, $\angle 4 \text{ is comp. to } \angle 2$
  • Conclusion / Prove: $\angle 1 \cong \angle 4$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\angle 1 \text{ is complementary to } \angle 3$ | 1. Given |   | 2. $\angle 4 \text{ is complementary to } \angle 2$ | 2. Given |   | 3. $\angle 2 \cong \angle 3$ | 3. Vertical Angles Theorem |   | 4. $\angle 1 \cong \angle 4$ | 4. Congruent Complements Theorem (Complements of congruent angles are congruent) |

• Problem 19
  • Given: $\angle NOP \cong \angle SRP$, $\angle NOP \text{ is comp. to } \angle POR$, $\angle SRP \text{ is comp. to } \angle PRO$
  • Prove: $\angle POR \cong \angle PRO$

  | Statements | Reasons |   | :--- | :--- |   | 1. $\angle NOP \cong \angle SRP$ | 1. Given |   | 2. $\angle NOP \text{ is complementary to } \angle POR$ | 2. Given |   | 3. $\angle SRP \text{ is complementary to } \angle PRO$ | 3. Given |   | 4. $\angle POR \cong \angle PRO$ | 4. Congruent Complements Theorem (Complements of congruent angles are congruent) |