Two-Column Geometric Proofs Study Guide

Fundamental Definitions and Theorems in Geometric Proofs

• Definition of Angle Bisector: A ray, segment, or line that divides an angle into two congruent angles of equal measure. If $\overline{KM}$ bisects $\angle JKL$, then $m\angle JKM = m\angle MKL$.

• Angle Addition Postulate: If a point lies in the interior of an angle, the measures of the two adjacent angles sum to the measure of the overall angle. For $\angle JKL$ divided by ray $KM$, $m\angle JKM + m\angle MKL = m\angle JKL$.

• Definition of Perpendicular Lines: Two lines, segments, or rays that intersect to form right angles. If $\overline{BD} \perp \overline{BC}$, then $\angle DBC$ is a right angle.

• Definition of Right Angle: An angle with a measure equal to exactly $90^\circ$.

• Definition of Congruent Angles: Angles that possess identical angle measures. If $\angle ABD \cong \angle DBE$, then $m\angle ABD = m\angle DBE$

• Definition of Complementary Angles: Two angles whose measures add up to $90^\circ$.

• Linear Pair Theorem: If two angles form a linear pair, then they are supplementary (their measures sum to $180^\circ$).

• Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then those two angles are congruent to each other.

• Vertical Angles Theorem: Vertical angles formed by two intersecting lines are congruent.

Proof 1: Bisected Angle Measure Derivation

• Given: $\overline{KM}$ bisects $\angle JKL$

• Prove: $m\angle MKL = \frac{1}{2} m\angle JKL$

• Two-Column Proof Steps:

  • Statement 1: $\overline{KM}$ bisects $\angle JKL$
  • Reason 1: Given
  • Statement 2: $m\angle JKM = m\angle MKL$
  • Reason 2: Definition of Angle Bisector
  • Statement 3: $m\angle JKM + m\angle MKL = m\angle JKL$
  • Reason 3: Angle Addition Postulate
  • Statement 4: $m\angle MKL + m\angle MKL = m\angle JKL$
  • Reason 4: Substitution Property of Equality
  • Statement 5: $2 \cdot m\angle MKL = m\angle JKL$
  • Reason 5: Combine Like Terms
  • Statement 6: $m\angle MKL = \frac{1}{2} m\angle JKL$
  • Reason 6: Division Property of Equality

Proof 2: Perpendicularity and Complementary Angles

• Given: $\overline{BD} \perp \overline{BC}$; $\angle ABD \cong \angle DBE$

• Prove: $\angle ABD$ and $\angle EBC$ are complementary

• Two-Column Proof Steps:

  • Statement 1: $\overline{BD} \perp \overline{BC}$
  • Reason 1: Given
  • Statement 2: $\angle DBC$ is a right angle
  • Reason 2: Definition of Perpendicular Lines
  • Statement 3: $m\angle DBC = 90^\circ$
  • Reason 3: Definition of Right Angle
  • Statement 4: $m\angle DBE + m\angle EBC = m\angle DBC$
  • Reason 4: Angle Addition Postulate
  • Statement 5: $m\angle DBE + m\angle EBC = 90^\circ$
  • Reason 5: Substitution Property of Equality
  • Statement 6: $\angle ABD \cong \angle DBE$
  • Reason 6: Given
  • Statement 7: $m\angle ABD = m\angle DBE$
  • Reason 7: Definition of Congruent Angles
  • Statement 8: $m\angle ABD + m\angle EBC = 90^\circ$
  • Reason 8: Substitution Property of Equality
  • Statement 9: $\angle ABD$ and $\angle EBC$ are complementary
  • Reason 9: Definition of Complementary Angles

Proof 3: Linear Pairs and Congruent Supplements

• Given: $\angle 1$ and $\angle 4$ form a linear pair; $\angle 1$ and $\angle 2$ are supplementary

• Prove: $\angle 3 \cong \angle 4$

• Two-Column Proof Steps:

  • Statement 1: $\angle 1$ and $\angle 4$ form a linear pair
  • Reason 1: Given
  • Statement 2: $\angle 1$ and $\angle 4$ are supplementary
  • Reason 2: Linear Pair (Supplement) Theorem
  • Statement 3: $\angle 1$ and $\angle 2$ are supplementary
  • Reason 3: Given
  • Statement 4: $\angle 2 \cong \angle 4$
  • Reason 4: Congruent Supplements Theorem
  • Statement 5: $\angle 2 \cong \angle 3$
  • Reason 5: Vertical Angles Theorem
  • Statement 6: $\angle 3 \cong \angle 4$
  • Reason 6: Transitive Property of Congruence