Midterm Prep. Modern College Geometry

2/2/2022 3-minute read

Notes consists of logic, Euclid’s 5th, Congruence and Length Theorem, Congruence and Angle Theorem, angles and parallel lines, and triangle congruence theorems.

Logic

Conditional Statement: \(P\rightarrow Q\)

Negation: \(\exists P\Rightarrow \neg Q\)

Inverse: \(\neg P\Rightarrow \neg Q\)

Contrapositve: \(\neg Q \Rightarrow \neg P\)

Converse: \(Q\Rightarrow P\)

Remember:

  • Inverse applies to the conditional statement.

  • Converse applies to the contrapositve.

  • Conditional statement and contrapositive are logically equivalent. Hence why we can prove things by way of contradiction.

  • Inverse and converse are logically equivalent

  • Negation is the only one where there could exist.

Example

Conditional Statement: If it is cloudy then it is raining.

Negation: It could be cloudy and not raining.

Inverse: If it is not cloudy then it is not raining.

Contrapositive: If it is not raining then it is not cloudy.

Converse. If it is rainy, then it is cloudy.

Euclids 5th

If straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This relates to parallel lines, and helps us make statements about lines transverse to parallel lines.

Example

When proving supplementary angles add to \(180^\circ\) we were able to use Euclids 5th element to say that the supplementary interior angles added up to \(\geq 180^\circ\) because the lines are parallel (and don’t intersect).

Congruence and Length Theorem

  • \(\overline{AB}=\overline{CD}\Leftrightarrow AB=CD\)

Proofs

  1. “bc … are congruent then there is an isometry f that superimposes AB onto CD. Isometries preserve length, so AB and CD have the same length.”

  2. “Suppose a and b have the same length.”, then explain what isometries would be needed to move AB to CD. “Translation, rotation, and reflection are all isometries, so we’ve shown AB and CD are congruent.”

Congruence and Angle Theorem

  • \(\angle ABC\cong \angle DEF \Leftrightarrow m\angle ABC = m\angle DEF\)

Proofs

  1. “bc … are congruent then there is an isometry f that superimposes angle ABC on angle DEF. Isometries preserve angle measure, so angles ABC and DEF must have had the same measure.”

  2. “Suppose angles ABC and DEF have the same measure.”, then explain isometries needed to move angle ABC to angle DEF. “Translation, rotation, and reflection are all isometries, so we’ve shown angles ABC and DEF are congruent.”

Angles and Parallel Lines

  • Vertical Angles are Congruent (Proved Quiz 1)

  • Alternative Interior Angles are Congruent

  • Supplementary Angles add to 180

  • Corresponding Angles are Congruent

Triangle Congruence Theorems

  • AAS, SSS, SAS, ASA

Remember

SSS is the most obvious and intuitive (it also wouldn’t be AAA).

ASA and SAS have one A or one S sandwhiched in the middle.

AAS not ASS (bc you can’t say that in school).