Geometry Basics: Chapter Review

Vocabulary and notation

Line, segment, ray

$\overleftrightarrow{AB}$: no endpoints, forever both ways. $\overline{AB}$: two endpoints, the only one of the three with a length. $\overrightarrow{AB}$: one endpoint, $A$, named first.

Plane, collinear

A plane is an endless flat surface. Points on one straight line are collinear; two different lines meet at most once, so lines that do cross share exactly one point.

Angle $\angle ABC$

Two rays from a shared vertex, the middle letter naming it. Size is the turn between the arms, in degrees, however long they are drawn.

Adjacent angles, linear pair

Adjacent: sharing a vertex and a side, not overlapping. A linear pair is two adjacent angles whose outer sides form a straight line.

$\ell \parallel m$ and $\ell \perp m$

Parallel lines never meet. Perpendicular lines cross at a right angle, which makes all four angles there $90^\circ$.

Transversal

A line crossing two or more others. Its angle facts hold only when the crossed lines are parallel.

Perpendicular height $h$

Distance from a base to the opposite side or vertex, at a right angle. Never a slanted side; on an obtuse triangle it can land outside.

Circle parts

Radius $r$: center to edge, all equal. Chord: joins two edge points. Diameter $d$: the longest chord, through the center. Arc: part of the boundary. Circumference: all of it.

Prism, cylinder, face, net

Prism: two identical bases, unchanging cross-section. Cylinder: the same with a circular base. Face: one flat surface. Net: the solid unfolded flat.

Perimeter, area, volume, surface area

Distance around a flat shape; unit squares covering a region; unit cubes filling a solid; total area of every face. Units: linear, square, cubic, square.

Ordered pair $(x, y)$

Across first ($x$, right positive), then up ($y$), so $(3, 2) \neq (2, 3)$. The origin $(0, 0)$ is where the axes cross.

Formulas and theorems

• Angle types by measure

  $$\begin{gathered} \text{acute} < 90^\circ = \text{right} \\ 90^\circ < \text{obtuse} < 180^\circ = \text{straight} \\ 180^\circ < \text{reflex} < 360^\circ \end{gathered}$$

  Use when Degrees, full turn $360^\circ$. Any opening between $0^\circ$ and $180^\circ$ has a reflex partner completing $360^\circ$.

  e.g. The reflex angle beside $115^\circ$ is $245^\circ$.

• Complement and supplement of $x$

  $$\begin{gathered} \text{complement} = 90^\circ - x \\ \text{supplement} = 180^\circ - x \end{gathered}$$

  Text description

  A right angle split into x and its complement, beside a straight angle split into x and its supplement.

  Use when A complement needs $x < 90^\circ$, a supplement $x < 180^\circ$, or the result is zero or negative. The angles need not touch.

  e.g. $62^\circ$: complement $28^\circ$, supplement $118^\circ$.

• Angle sums: line and full turn

  $$\begin{gathered} \text{linear pair} = 180^\circ \\ \text{around a point} = 360^\circ \end{gathered}$$

  Use when The angles must share the vertex and fill the line or turn with no gaps or overlaps; a linear pair is therefore supplementary.

  e.g. $110^\circ$, $135^\circ$, $x$ fill a turn: $x = 115^\circ$.

• Vertical angles are equal

  $$\angle 1 = \angle 3, \qquad \angle 2 = \angle 4$$

  Use when Only the opposite pairs where two straight lines cross, sharing the vertex and no side; a neighbour is supplementary to it instead.

• Parallel lines cut by a transversal

  $$\begin{gathered} \text{corresponding: equal} \\ \text{alternate interior: equal} \\ \text{co-interior: sum } 180^\circ \end{gathered}$$

  Text description

  Two parallel lines cut by a transversal, with the corresponding pair of angles marked a and the alternate interior pair marked b.

  Use when All three need the crossed lines PARALLEL. Corresponding: matching positions. Alternate interior: opposite sides, between the lines. Co-interior: same side, between them.

  e.g. An interior $78^\circ$ gives alternate interior $78^\circ$ and co-interior $102^\circ$.

• Rectangle and square

  $$\begin{gathered} P = 2(l + w), \quad A = lw \\ P_{\text{square}} = 4s, \quad A_{\text{square}} = s^2 \end{gathered}$$

  Use when Four right angles; the area counts $w$ rows of $l$ unit squares, and any other polygon's perimeter is the sum of its sides. Put lengths in ONE unit first: squaring a length squares the factor, so $1 \text{ m}^2 = 10{,}000 \text{ cm}^2$.

• Area of a parallelogram

  $$A = b \times h$$

  Use when $h$ is the perpendicular distance between base $b$ and the side opposite, never the slanted side, which is longer.

• Area of a triangle

  $$A = \tfrac{1}{2} \, b \, h$$

  Use when Any side may be base $b$ if $h$ is the perpendicular height to THAT base, and $bh$ is the same whichever is chosen; on an obtuse triangle that height can fall outside the triangle.

• Area of a trapezoid

  $$A = \tfrac{1}{2}(a + b)\,h = \frac{a + b}{2} \times h$$

  Use when $a$ and $b$ must be the PARALLEL sides and $h$ perpendicular between them; slanted sides never enter. Average the two, then multiply.

• Circumference, and $d = 2r$

  $$d = 2r, \qquad C = \pi d = 2\pi r$$

  Use when Halve a diameter whenever a formula wants $r$. Circumference is a length, and it goes round a WHOLE circle: a part-circle boundary adds its straight edges.

• Pi

  $$\pi = \frac{C}{d} \approx 3.14$$

  Use when The same for every circle, all circles being scaled copies. Computing with $3.14$ or $\tfrac{22}{7}$ makes any circle answer approximate: write $\approx$.

• Area of a circle and a semicircle

  $$\begin{gathered} A = \pi r^2 \\ A_{\text{semi}} = \tfrac{1}{2}\pi r^2 \\ P_{\text{semi}} = \pi r + 2r \end{gathered}$$

  Use when The RADIUS, squared before multiplying by $\pi$; halve a given diameter first, or the answer is four times too large. A semicircle's area halves; its boundary does not, the cut exposing the straight diameter $2r$.

• Volume of a prism or a cylinder

  $$\begin{gathered} V = B \times h \\ V_{\text{box}} = lwh, \quad V_{\text{cube}} = s^3 \\ V_{\text{cylinder}} = \pi r^2 h \end{gathered}$$

  Use when $B$ is one base's area and $h$ the perpendicular distance to the identical opposite base, so the cross-section must never change; a cylinder takes the RADIUS. Cubic units; cones, pyramids, and spheres are not covered.

  e.g. Triangular end $7$ by $4$, length $6$: $B = 14$, so $V = 84 \text{ m}^3$.

• Surface area from a net

  $$\begin{gathered} S_{\text{box}} = 2(lw + lh + wh) \\ S_{\text{cube}} = 6s^2 \\ S_{\text{cylinder}} = 2\pi r^2 + 2\pi r h \end{gathered}$$

  Use when Six box faces in three opposite pairs, so each term doubles; the cylinder's $2\pi r^2$ is its two caps and $2\pi r h$ its side unrolled to width $2\pi r$. An open box or tube drops the faces it lacks. SQUARE units, not cubic.

• Quadrant sign patterns

  $$\begin{gathered} \text{I } (+,+) \qquad \text{II } (-,+) \\ \text{III } (-,-) \qquad \text{IV } (+,-) \end{gathered}$$

  Text description

  Coordinate axes with the four quadrants numbered counterclockwise from the upper right, each labelled with the signs of its coordinates.

  Use when Counterclockwise from the upper right. A coordinate of $0$ puts the point ON an axis and in NO quadrant, the origin included.

• Horizontal and vertical distance

  $$\begin{gathered} \text{horizontal } |x_2 - x_1| \\ \text{vertical } |y_2 - y_1| \end{gathered}$$

  Use when Only when the OTHER coordinate matches, putting the segment on a gridline; the absolute value keeps the length positive either way.

• Reflecting a point across an axis

  $$\begin{gathered} \text{over } x\text{-axis: } (x, y) \to (x, -y) \\ \text{over } y\text{-axis: } (x, y) \to (-x, y) \end{gathered}$$

  Text description

  A point three right and two up, reflected across the x-axis to the point the same distance below with its y-sign flipped.

  Use when One sign flips: crossing the $x$-axis flips the $y$-sign, crossing the $y$-axis flips the $x$-sign. A point already on the axis crossed stays put.

Problem types, step by step

Exam traps

• Trap Reporting a measurement in the wrong kind of unit, or converting $1 \text{ m}^2$ to $100 \text{ cm}^2$.

  Fix Count the factors of length the formula multiplies: one linear, two square, three cubic. Squaring a length squares the conversion factor, so it is $10{,}000 \text{ cm}^2$.

• Trap Putting the diameter where the radius belongs, so $d = 12$ cm gives $A \approx 3.14 \times 12^2 = 452.16$.

  Fix Halve first: $r = 6$ and $A \approx 113.04 \text{ cm}^2$. Squaring turns the factor of $2$ into $4$, so the slip is always four times too big.

• Trap Answering a distance-around question with $\pi r^2$, or a space-inside question with $2\pi r$.

  Fix Circumference carries $r$ once, area carries $r$ squared. At $r = 10$: about $62.8$ around, $314$ inside.

• Trap Multiplying by the slanted side of a parallelogram, triangle, or trapezoid.

  Fix Only the perpendicular height counts: base $9$, slant $6$, height $5$ gives $45 \text{ cm}^2$, not $54$.

• Trap Dropping the $\tfrac{1}{2}$ from the triangle formula, or using $a + b$ instead of $\tfrac{a+b}{2}$ for a trapezoid.

  Fix Either doubles the area: each shape is half of a parallelogram made from two copies of it.

• Trap Calling a semicircle's boundary half the circumference.

  Fix The cut exposes a straight diameter, so $P = \pi r + 2r$: at $r = 4$, about $20.56$ cm, not $12.56$.

• Trap Charging a closed can only its unrolled side, $2\pi r h$.

  Fix Add both caps: $S = 2\pi r^2 + 2\pi r h$. The side alone is a tube open at each end.

• Trap Subtracting coordinates to get the distance between two diagonal points.

  Fix It works only when the points SHARE a coordinate; $(1, 2)$ and $(4, 6)$ are neither $3$ nor $4$ apart.