Course Review

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• Point: a location, no size (named $A$). Line: extends forever in two directions ($\overleftrightarrow{AB}$). Plane: a flat surface extending forever.
• Collinear points lie on one line; coplanar points lie in one plane.
• Segment $\overline{AB}$: two endpoints and all points between (has length $AB$). Ray $\overrightarrow{AB}$: one endpoint, goes forever one way.
• Congruent segments have equal length: $\overline{AB}\cong\overline{CD}$ means $AB = CD$.

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• Segment Addition Postulate: if $B$ is between $A$ and $C$, then $AB + BC = AC$.
• Midpoint $M$ of $\overline{AB}$: the point with $AM = MB$ (it bisects the segment).

On a coordinate plane, for $A(x_1,y_1)$ and $B(x_2,y_2)$:

Measured in degrees ($^\circ$): acute $<90^\circ$, right $=90^\circ$, obtuse between $90^\circ$ and $180^\circ$, straight $=180^\circ$. An angle bisector splits an angle into two equal halves.

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• Complementary: two angles that add to $90^\circ$.
• Supplementary: two angles that add to $180^\circ$.
• Linear pair: adjacent angles on a straight line --- they are supplementary ($=180^\circ$).
• Vertical angles: the opposite angles formed by two crossing lines --- they are congruent (equal).
• Angle Addition Postulate: if $D$ is inside $\angle ABC$, then $\angle ABD + \angle DBC = \angle ABC$.

When a transversal cuts two parallel lines, these pairs are formed:

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• Corresponding angles: congruent (equal).
• Alternate interior angles: congruent.
• Alternate exterior angles: congruent.
• Co-interior (same-side interior) angles: supplementary ($=180^\circ$).

The converse also works: if any of these relationships holds, the lines are parallel.

For slope $m$ (recall $m = \dfrac{y_2-y_1}{x_2-x_1}$, “rise over run”):

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• Parallel lines have equal slopes: $m_1 = m_2$.
• Perpendicular lines have opposite-reciprocal slopes: $m_1 \cdot m_2 = -1$, i.e. $m_2 = -\dfrac{1}{m_1}$.

Horizontal ($m=0$) and vertical (undefined slope) lines are perpendicular to each other.

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• Triangle Angle Sum: the three interior angles add to $180^\circ$.
• Exterior Angle Theorem: an exterior angle equals the sum of the two remote (non-adjacent) interior angles.
• Classify by sides: scalene (none equal), isosceles (two equal), equilateral (all equal). By angles: acute, right, obtuse.

Two triangles are congruent if any one of these matches: multicols2

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• SSS --- three sides
• SAS --- two sides + included angle
• ASA --- two angles + included side
• AAS --- two angles + a non-included side
• HL --- (right triangles) hypotenuse + a leg

multicols SSA and AAA do not prove congruence. After proving congruence, CPCTC lets you conclude the matching parts are equal.

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• Isosceles Triangle Theorem: the angles opposite the two equal sides (base angles) are equal --- and the converse holds.
• Triangle Inequality: the sum of any two sides is greater than the third: $a+b>c$. The third side lies between $|a-b|$ and $a+b$.

Similar figures ($\sim$) have equal corresponding angles and proportional corresponding sides. Prove triangles similar by:

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• AA --- two pairs of equal angles.
• SSS$\sim$ --- all three side ratios equal.
• SAS$\sim$ --- two side ratios equal with equal included angle.

The common ratio of the sides is the scale factor $k$.

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• Solve proportions by cross-multiplying: $\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc$.
• Triangle Midsegment: the segment joining midpoints of two sides is parallel to the third side and half its length.
• Side-Splitter: a line parallel to one side cuts the other two sides proportionally.

For scale factor $k$:   ratio of perimeters $= k$,   ratio of areas $= k^2$,   ratio of volumes $= k^3$.

In a right triangle with legs $a,b$ and hypotenuse $c$ (the side opposite the right angle):

Converse: if $a^2+b^2 = c^2$, the triangle is right. If $a^2+b^2 > c^2$ it is acute; if $a^2+b^2 < c^2$ it is obtuse. Common triples: $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$ (and their multiples).

(In $30$-$60$-$90$ the short leg is opposite $30^\circ$; the long leg is opposite $60^\circ$.)

For an acute angle $\theta$ in a right triangle:

To find a missing angle, use inverse trig: $\theta = \sin^{-1}\!\left(\tfrac{\text{opp}}{\text{hyp}}\right)$, and likewise $\cos^{-1}$, $\tan^{-1}$.

For a polygon with $n$ sides:

For a regular polygon: each interior angle $= \dfrac{(n-2)180^\circ}{n}$, each exterior angle $= \dfrac{360^\circ}{n}$.

In every parallelogram: opposite sides are parallel and congruent; opposite angles are congruent; consecutive angles are supplementary; and the diagonals bisect each other.

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• Rectangle: parallelogram with $4$ right angles; diagonals are congruent.
• Rhombus: parallelogram with $4$ equal sides; diagonals are perpendicular and bisect the angles.
• Square: both a rectangle and a rhombus (all properties of each).
• Trapezoid: exactly one pair of parallel sides; isosceles trapezoid has congruent legs, congruent base angles, and congruent diagonals.
• Kite: two pairs of adjacent congruent sides; diagonals are perpendicular.

With radius $r$ and diameter $d = 2r$:

For a central angle of $n^\circ$:   arc length $= \dfrac{n}{360}\cdot 2\pi r$,   sector area $= \dfrac{n}{360}\cdot \pi r^2$.

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• Central angle (vertex at center) $=$ its intercepted arc.
• Inscribed angle (vertex on circle) $= \tfrac{1}{2}$ its intercepted arc. An angle inscribed in a semicircle is $90^\circ$.
• Tangent $\perp$ radius at the point of tangency; two tangents from one outside point are congruent.

From a point, the products of the two chord/secant/tangent pieces are equal:

A circle with center $(h,k)$ and radius $r$:

The center uses the opposite sign of what appears; the right side is $r^2$, so take a square root for $r$.

multicols2

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• Rectangle: $A = bh$
• Parallelogram: $A = bh$
• Triangle: $A = \tfrac{1}{2}bh$
• Trapezoid: $A = \tfrac{1}{2}(b_1+b_2)h$
• Rhombus / kite: $A = \tfrac{1}{2}d_1 d_2$
• Circle: $A = \pi r^2$
• Regular polygon: $A = \tfrac{1}{2}(\text{apothem})(\text{perimeter})$

multicols Perimeter is the distance around; the circumference $C=2\pi r$ is a circle's perimeter.

Let $B$ = area of the base, $P$ = base perimeter, $h$ = height, $\ell$ = slant height.

These preserve size and shape (image $\cong$ pre-image):

Rotations about the origin (counterclockwise):

Dilation (center origin, scale factor $k$): $(x,y)\to(kx,\ ky)$. This is not rigid --- it changes size (lengths $\times k$, area $\times k^2$) but keeps shape, so figures are similar.

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• Line (reflection) symmetry: a fold line maps the figure onto itself.
• Rotational symmetry: a turn of less than $360^\circ$ about a center maps the figure onto itself (a regular $n$-gon has order $n$, smallest angle $\tfrac{360^\circ}{n}$).