Let’s analyze the given diagram and match the appropriate reasoning or theorem to each statement.

Statements:

1. If ∠A and ∠B are vertical angles and ∠A = 50°, then ∠B = 50°.

   • This follows from the Definition of Vertical Angles, which states that vertical angles are congruent.

2. If ∠A and ∠B are complementary angles and ∠A = 30, then ∠B = 60.

   • This uses the Definition of Complementary Angles, which says that complementary angles sum to 90°.

3. If AB = CD, then $\overline{AB} \cong \overline{CD}$.

   • This follows the Definition of Congruence, meaning equal lengths imply congruent segments.

4. $\overline{BC}$ bisects ∠ABD, so ∠ABC ≅ ∠CBD.

   • This involves the Definition of Angle Bisector, which states that a bisector divides an angle into two congruent angles.

5. AD = CB, so CB = AD.

   • This is an example of the Symmetric Property, which means if one quantity equals another, they can be reversed.

6. If ∠A is a right angle, it measures 90°.

   • This follows the Definition of a Right Angle, which is an angle that measures exactly 90°.

7. If Point B bisects $\overline{AC}$, then $\overline{AB} \cong \overline{BC}$.

   • This uses the Definition of Midpoint, as the midpoint divides a segment into two congruent parts.

8. AB = CD and AB = EF, so CD = EF.

   • This is the Transitive Property, which states that if one quantity equals two others, those two must be equal.

9. If ∠A and ∠B are supplementary angles and ∠A = 30°, then ∠B = 150°.

   • This follows from the Definition of Supplementary Angles, where the sum of two angles equals 180°.

10. B is the midpoint of $\overline{AC}$, so $\overline{AB} \cong \overline{BC}$.

    • This follows the Definition of Midpoint, same as number 7.

11. ∠ABC + ∠CDE = ∠NOP + ∠CDE.

    • This uses the Addition Property of Equality, where equal amounts can be added to both sides of an equation.

Matching Answers:

• Definition of Congruence: 3 (Statement 3)
• Definition of Midpoint: 7, 10 (Statements 7 and 10)
• Segment Addition Postulate: This does not apply directly to any of the given statements.
• Definition of Segment Bisector: 4 (Statement 4)
• Substitution: This does not apply directly to any of the given statements.
• Transitive Property: 8 (Statement 8)
• Reflexive Property: This does not apply directly to any of the given statements.
• Symmetric Property: 5 (Statement 5)
• Addition Property of Equality: 11 (Statement 11)

Do you want further details or have any questions about the reasoning? Here are 5 related questions to expand your understanding:

1. What is the difference between complementary and supplementary angles?
2. Can the Transitive Property be applied to both angles and segments?
3. How does the Definition of Midpoint differ from the Segment Bisector?
4. What is the Reflexive Property, and when would it be used?
5. How do vertical angles relate to the congruence of angles?

Tip: When studying properties of equality (like reflexive or transitive), try to visualize them with simple geometric diagrams to reinforce understanding!