01.06 Introduction to Proofs

Module 01 — Geometry Foundations

The big idea A proof is a chain of statements in which every step names the definition, postulate, theorem, or equality property that makes it valid.

By the end you can

  • Recognize common angle relationships.
  • Use midpoint and angle-bisector definitions to set equal measures.
  • Choose a valid justification for each algebra or geometry step.

A proof starts with what you know

The first lines usually restate the Given. After that, every statement needs a named reason: a definition, a postulate, a theorem, or a Property of Equality.

“Proven” is not a justification. Neither is “it looks equal.” State the rule that connects the new line to something already known.

A proof starts with what you know (2)

Read a proof one arrow at a time: What changed from the previous line? If the same amount was added, subtracted, multiplied, or divided on both sides, name that Property of Equality. If terms only swapped order, use the commutative property.

Angle relationships

  • Vertical: opposite angles at a crossing; nonadjacent and congruent.
  • Adjacent: share a side without overlapping interiors.
  • Linear pair: adjacent and supplementary, forming a straight line.
  • Supplementary: measures add to 180°; they need not touch.
  • Complementary: measures add to 90°; they may be adjacent or separate.

Angle relationships (2)

Start with placement, then use the relationship. Across the X means vertical. Side by side on a straight line means a linear pair. Two parts filling a right angle are complementary.

Lookalike: adjacent only says the angles share a side. It does not automatically make them congruent, complementary, or supplementary.

Equal pieces create equations

A midpoint divides a segment into two equal pieces. An angle bisector divides an angle into two congruent angles. Those definitions authorize the equation that sets the two expressions equal.

After solving for the variable, substitute it back into the requested measure. If the problem asks for the whole bisected angle, add the two halves or double one half.

Equal pieces create equations (2)

The definition of congruence works both ways: congruent figures have equal measures, and figures with equal measures are congruent. That bridge often appears at the end of a proof.