Chapter 1: Making Connections with Proof

Rules of Logic and Valid Conclusions

• Deductive reasoning requires the use of established rules of logic to form valid conclusions. Every conclusion must be supported by a reason written in if-then (conditional) form.

• Example 1: Angle Congruence based on Measure

  • Given: $\angle A = 90^{\circ}$ and $\angle B = 90^{\circ}$.
  • Conclusion: $\angle A \cong \angle B$.
  • Reason: If two angles have the same measure, then they are congruent.

• Example 2: Midpoints and Segment Congruence

  • Given: $M$ is the midpoint of $\overline{XY}$.
  • Conclusion: $\overline{XM} \cong \overline{MY}$.
  • Reason: If a point is a midpoint of a segment, then it splits it into two congruent segments.

• Example 3: Trisection of Angles

  • Given: $\angle ABD \cong \angle DBE \cong \angle EBC$.
  • Conclusion: $\vec{BD}$ and $\vec{BE}$ trisect $\angle ABC$.
  • Reason: If three congruent angles are formed by two rays, then the rays trisect the angle.

Error Analysis in Geometric Proofs

• Proofs are considered invalid when the reasons used to support a statement are factually incorrect or presented in the wrong logical order (converse error).

• Invalid Logic in Angle Bisectors

  • Given: $\vec{BD}$ bisects $\angle ABC$.
  • Statement: $\angle ABD \cong \angle DBC$.
  • Invalid Reason Provided: If a ray divides an angle into 2 congruent angles, then it bisects the angle.
  • Error Explanation: The reason is in the wrong order. It provides the definition of a bisector as the conclusion rather than the justification for the congruent angles.
  • Correct Reason: If a ray bisects an angle, then it makes two congruent angles.

• Invalid Logic in Congruence Definitions

  • Given: $\angle A = 30^{\circ}$ and $\angle B = 30^{\circ}$.
  • Statement: $\angle A \cong \angle B$.
  • Invalid Reason Provided: If two angles are congruent, then they have the same measure.
  • Error Explanation: This is technically the converse of the logic needed. The goal is to prove congruence based on measure, not measure based on congruence.
  • Correct Reason: If two angles have the same measure, then the two angles are congruent.

Assumptions and Angle Relationships

• Diagram Assumptions

  • Often, diagrams may visually suggest that specific conditions exist, such as right angles or segment congruence.
  • It is invalid to assume two angles are right angles (e.g., $\angle 1$ and $\angle 2$) simply because they appear that way in a drawing.
  • If the given information only specifies a "Diagram as shown" without explicit right angle markers or statements, concluding that $\angle 1 \cong \angle 2$ based on the Right Angle Congruence Theorem is invalid because you cannot assume they are right angles.

• Right Angle Congruence Proof

  • Given: $\angle CAT$ is a right angle; $\angle DOG$ is a right angle.
  • Statement: $\angle CAT \cong \angle DOG$.
  • Reason: If both angles are right angles, then they are congruent.

Segment Bisectors and Compound Angle Proofs

• Segment Bisectors

  • Given: $\overline{XZ}$ bisects $\overline{AB}$.
  • Statement: $\overline{AY} \cong \overline{YB}$.
  • Reason: If a ray/line/segment bisects a segment, then there are two congruent segments.

• Multi-Step Proof for Angle Congruence

  • To prove that two angles are congruent when they are composed of multiple smaller angles, addition and substitution are used.
  • Example Proof:
    1. $\angle ABD = 50^{\circ}$ (Given)
    2. $\angle DBC = 40^{\circ}$ (Given)
    3. $\angle ABC = 90^{\circ}$ (Addition)
    4. $\angle XYZ$ is a right angle (Given)
    5. $\angle XYZ = 90^{\circ}$ (If an angle is a right angle, then it measures $90^{\circ}$)
    6. $\angle ABC \cong \angle XYZ$ (If two angles have the same measure, then they are congruent.)