Foundations and Reasoning: Chapter Review

Vocabulary and notation

Point, line, and plane

Undefined starting terms. A point is pictured as a location without size, a line as a straight path extending both ways, and a plane as an unlimited flat surface. Drawn dots, strokes, and plane outlines have physical limits the objects do not.

Line, ray, segment, and plane notation

$\overleftrightarrow{AB}$ names the whole line; $\overrightarrow{AB}$ starts at $A$ and continues through $B$; $\overline{AB}$ has endpoints $A,B$. Use distinct named points. Reversing ray letters changes its endpoint. Opposite rays share an endpoint and form a line. Write $\ell\parallel m$ for parallel lines and $\ell\perp m$ for perpendicular lines. Name a plane with a script letter such as $\mathcal P$ or three noncollinear points.

Collinear and coplanar

Collinear points lie on one line; coplanar points lie in one plane. A noncollinear triple does not lie on one line.

Objects, measures, and congruence

$AB$ is the length of $\overline{AB}$; $m\angle AOB$ is the selected angle's measure. Congruent segments have equal lengths; congruent angles have equal measures. Write $\overline{AB}\cong\overline{CD}$ for objects and $AB=CD$ for lengths.

Betweenness, midpoint, and segment bisector

Betweenness requires three distinct collinear points in the stated order. A midpoint lies on its segment and makes equal halves. A segment bisector is a line, ray, segment, or plane intersecting the segment only at its midpoint; perpendicularity is not required.

Angle, vertex, and selected opening

An angle is formed by two rays with a common endpoint, its vertex. The middle letter names the vertex. Three-letter notation selects an opening of at most a half turn unless a different opening is specified; name a reflex opening explicitly.

Circle and disk

A circle is the set of points in a plane at one fixed positive distance from its center. A disk includes that boundary and its interior, including the center.

Radius, chord, diameter, and arc

Radius names the fixed distance or a center-to-circle segment. A chord joins two circle points; a diameter is a chord through the center. An arc is a portion of a circle.

Construction tools

A compass draws a circle or arc with a chosen positive radius; the compass used here retains its opening when moved. An unmarked straightedge draws through two distinct points without reading a measurement scale.

Definition, postulate, theorem, observation

A definition fixes meaning. A postulate, or axiom, is accepted as a foundation. A theorem has a proof from available facts. A measurement or moving figure supplies an experimental observation, not an exact general proof.

Inductive and deductive reasoning

Inductive reasoning uses examples to suggest a conjecture. Deductive reasoning uses accepted facts and valid steps to justify a conclusion. Repeated successful tests alone do not prove a universal conjecture.

Conditional and negation

In $p\to q$, $p$ is the hypothesis and $q$ the conclusion: if $p$, then $q$. Negation $\neg p$ says that the entire statement $p$ is false.

Counterexample

An allowed case with a true hypothesis and a false conclusion refutes a universal conditional. An object outside the hypothesis does not refute it.

Inclusive square and rectangle definitions

A rectangle is a quadrilateral with four right angles. A square is a rectangle with four congruent sides, so every square is a rectangle.

Equilateral triangle

A triangle with three equal side lengths.

Givens, goal, and proof formats

Givens are supplied facts; the goal is the claim to establish. Paragraph proofs connect reasons in sentences; two-column proofs pair statements with reasons; flow proofs show dependencies with arrows. All require a complete argument.

Formulas and theorems

• Accepted incidence foundations

  Two distinct points determine exactly one line. Three noncollinear points determine exactly one plane. A line with two distinct points in a plane lies entirely in that plane. Two distinct intersecting planes share exactly one line.

  Use when Keep distinctness and noncollinearity. Every line contains at least two distinct points; every plane contains at least three noncollinear points; space contains noncoplanar points.

• Line relationships

  Parallel lines are distinct, coplanar, and disjoint. Skew lines are noncoplanar. Perpendicular lines meet at a right angle. Coincident lines are the same line; distinct lines share at most one point (proved from the unique-line foundation).

  Text description

  A rectangular prism has front face A B F E and back face D C G H. The complete front bottom and front top lines are parallel. The front bottom and front right vertical lines are perpendicular. The front bottom line and the top right line extending toward the back are skew: they do not meet and do not share a plane.

  Use when Use complete lines, including extensions. In the labeled rectangular prism, lines AB and EF are parallel, AB and BF are perpendicular, and AB and FG are skew.

• Plane relationships

  Distinct planes either intersect in one line or are parallel with no common points. Coincident planes are the same plane and share every point.

  Use when Ordinary three-dimensional Euclidean space. The intersection-line claim excludes coincident planes.

• Number-line distance and midpoint coordinate

  $$\begin{gathered} AB=|b-a|\\ m=\frac{a+b}{2} \end{gathered}$$

  Use when $a,b$ are endpoint coordinates on one number line with a fixed unit; $m$ is the midpoint coordinate. Distance is an accepted foundation; the midpoint formula follows from equal halves. Length is nonnegative; a coordinate may be negative.

  e.g. Endpoints $-21$ and $16$ give length $37$ and midpoint coordinate $-2.5$.

• Accepted segment addition

  $$AB+BC=AC$$

  Use when $B$ is between $A$ and $C$; the distinct points are collinear. All lengths use the same unit.

• Midpoint halves

  $$AM=MB=\frac{AB}{2}$$

  Use when $M$ is the midpoint of the nonzero segment $\overline{AB}$. Equal distances alone do not supply its location on the segment.

• Angle types and turns

  Acute: $0^\circ<m<90^\circ$. Right: $m=90^\circ$. Obtuse: $90^\circ<m<180^\circ$. Straight: $m=180^\circ$. Reflex: $180^\circ<m<360^\circ$. Full turn: $360^\circ$.

  Use when $m$ is the selected opening's measure. One degree is one three-hundred-sixtieth of a full turn. A zero turn and a full turn end in the same direction but record different turns.

• Accepted angle addition

  $$m\angle AOB+m\angle BOC=m\angle AOC$$

  Use when Ray OB lies inside the selected opening AOC, and the selected parts fill it with no overlapping interiors. Specify a reflex whole explicitly.

• Angle bisector equalities

  $$\begin{gathered} m\angle AOB=m\angle BOC\\ 2m\angle AOB=m\angle AOC \end{gathered}$$

  Use when Ray OB bisects the selected opening AOC. It halves the turn, regardless of drawn ray lengths.

• Circle and disk distance tests

  Circle membership means $OP=r$; disk membership means $OP\le r$. A point with $OP>r$ is outside the disk.

  Use when Points lie in the circle's plane; $O$ is its center and $r>0$. The center belongs to the disk, not the circle.

• Diameter and radius

  $$d=2r$$

  Use when $d$ is a diameter's length, not an arbitrary chord's length; $r$ is the positive radius.

• Equal-radius construction results

  If Q lies on the circle centered at P with retained compass opening AB, then $PQ=AB$. An intersection C of circles centered at A and B with common radius AB gives $AC=AB$ and $BC=AB$.

  Use when Work in one plane with distinct $A,B$ and retained positive compass settings. Accept the unique center-ray intersection and the two intersections for each specified equilateral or angle-copy circle pair; the latter needs a source opening strictly between $0^\circ$ and $180^\circ$. Equal radii justify the copied and equilateral lengths. The angle-copy proof comes in 4.5.

• Four conditional forms

  $$\begin{gathered} \text{Conditional: }p\to q\\ \text{Converse: }q\to p\\ \text{Inverse: }\neg p\to\neg q\\ \text{Contrapositive: }\neg q\to\neg p \end{gathered}$$

  Use when Keep the meanings of $p,q$ and the allowed objects fixed when changing order or negating.

• Logical equivalence and biconditionals

  A conditional is logically equivalent to its contrapositive; its converse is logically equivalent to its inverse. The biconditional $p\leftrightarrow q$ requires both $p\to q$ and $q\to p$.

  Use when A true original does not establish its converse or inverse. Check every geometric condition in each direction.

• Detachment and syllogism

  Detachment: $p\to q$ and a matching fact $p$ give $q$. Syllogism: $p\to q$ and $q\to r$ give $p\to r$.

  Use when Use valid conditionals in their stated directions. A conditional chain alone does not supply a starting fact about a particular object.

• Equality and congruence properties

  Reflexive: $a=a$. Symmetric: $a=b$ gives $b=a$. Transitive: $a=b$ and $b=c$ give $a=c$. Substitute equals and apply equal operations to both sides. Addition is commutative.

  Use when Here $a,b,c$ are lengths or measures. Divide only by a nonzero number. Segment and angle congruence inherit reflexive, symmetric, and transitive properties from equal measures.

• Two proved geometric results

  For distinct collinear points in order A,B,C,D, $AB=CD$ gives $AC=BD$; conversely, $AC=BD$ gives $AB=CD$ by subtracting $BC$. Concentric circles with unequal positive radii have no common point.

  Use when Keep the point order for both directions of the first result. For the second, concentric means the same center; equal radii give coincident circles when the circles also lie in the same plane.

Problem types, step by step

Exam traps

• Trap Dropping distinctness or noncollinearity

  Fix Use two distinct points for a unique line and three noncollinear points for a unique plane.

• Trap Calling every disjoint pair of lines parallel

  Fix Parallel lines must be coplanar as well as distinct and disjoint. Skew lines are noncoplanar.

• Trap Using equality of parts without their location

  Fix Midpoint requires a point on the segment; segment addition requires betweenness. Equal distances alone do not supply either.

• Trap Treating a bisector as automatically perpendicular

  Fix Bisection means equal halves. A square-corner marking establishes perpendicularity, not equal halves.

• Trap Reading an endpoint instead of measuring a gap

  Fix Subtract ruler readings, count intervals, and convert to one unit. Match reported precision to the measuring tool.

• Trap Reading the wrong protractor scale or opening

  Fix Start at the zero on the baseline ray. Specify the selected opening before adding, halving, or finding its reflex partner.

• Trap Treating a circle as its filled interior

  Fix Circle points have exactly the radius distance; a disk also includes smaller distances. Not every chord is a diameter.

• Trap Changing the wrong compass setting

  Fix Retain the radius when transferring it. In an angle copy, change to the source chord only for the chord-transfer step.

• Trap Reversing a conditional from its conclusion

  Fix Knowing $q$ does not establish $p$ from $p\to q$. The converse needs its own justification; examples alone are insufficient.

• Trap Promoting a sketch or measurement to a proof

  Fix Use definitions, accepted foundations, prior proved results, and valid algebra. Do not mark the goal as a given.

• Trap Stopping after assuming the opposite

  Fix A contradiction proof must identify an impossibility conflicting with known facts, then reject the temporary assumption and conclude the goal.