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Geometry

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1. Foundations and notation

Geometry studies shape, size, position, and relationships in the plane and in space.

Core undefined terms:

  • Point: location with no size

  • Line: infinite straight path

  • Plane: flat two-dimensional surface

Common notation:

  • Segment: $\overline{AB}$

  • Ray: $\overrightarrow{AB}$

  • Line through $A$ and $B$: $\overleftrightarrow{AB}$

  • Length of segment $AB$: $AB$

  • Angle $ABC$: $\angle ABC$

Basic ideas

  • A line segment has length, a line does not.

  • An angle is formed by two rays with a common endpoint.

  • A polygon is a closed figure made of line segments.

  • A circle is the set of points a fixed distance from a center.

Postulates and theorems you use constantly

  • Through any two points there is exactly one line.

  • A segment can be measured and compared.

  • A circle can be drawn with any center and radius.

  • All right angles are congruent.

  • If two lines intersect, vertical angles are congruent.

2. Angles, lines, and parallelism

Angle types

TypeMeasure
Acute$0^\circ < \theta < 90^\circ$
Right$90^\circ$
Obtuse$90^\circ < \theta < 180^\circ$
Straight$180^\circ$
Reflex$180^\circ < \theta < 360^\circ$

Angle relationships

  • Complementary angles sum to $90^\circ$.

  • Supplementary angles sum to $180^\circ$.

  • Vertical angles are congruent.

  • Linear pairs are supplementary.

If a line cuts two parallel lines, then:

  • Corresponding angles are congruent.

  • Alternate interior angles are congruent.

  • Same-side interior angles are supplementary.

Slope intuition for parallel and perpendicular lines

In coordinate geometry:

  • Parallel lines have equal slope.

  • Perpendicular lines have slopes whose product is $-1$, when both slopes are defined.

If a line has slope $m$, a perpendicular line has slope

$$ m_\perp = -\frac{1}{m} $$

when $m \ne 0$.

3. Triangles

Triangles are central because many geometric problems reduce to triangle relationships.

Angle sum and exterior angles

For any triangle:

$$ A + B + C = 180^\circ $$

An exterior angle equals the sum of the two remote interior angles:

$$ m\angle ext = m\angle 1 + m\angle 2 $$

Special triangles

Isosceles triangle

  • Two sides are congruent.

  • Base angles are congruent.

  • The altitude from the vertex to the base bisects the base and the vertex angle.

Equilateral triangle

  • All sides are congruent.

  • All angles are $60^\circ$.

Right triangle

  • One angle is $90^\circ$.

  • The side opposite the right angle is the hypotenuse.

Triangle inequality

For side lengths $a$, $b$, and $c$:

$$ a+b>c,\quad a+c>b,\quad b+c>a $$

A triangle exists only if the sum of any two side lengths is greater than the third.

Medians, altitudes, angle bisectors, and perpendicular bisectors

  • A median connects a vertex to the midpoint of the opposite side.

  • An altitude is perpendicular to the opposite side or its extension.

  • An angle bisector divides an angle into two equal angles.

  • A perpendicular bisector is perpendicular to a segment and passes through its midpoint.

Key fact:

  • The perpendicular bisector of a segment contains all points equidistant from the segment's endpoints.

Interactive visual

Triangle angle and area

Adjust an included angle and the two side lengths to see how triangle area and the third side respond.

Area 0
Third side 0

4. Congruence and similarity

Congruence

Figures are congruent if they have the same shape and size. Congruent figures can be matched by rigid motions.

Triangle congruence criteria:

  • SSS: three corresponding sides congruent

  • SAS: two sides and the included angle congruent

  • ASA: two angles and the included side congruent

  • AAS: two angles and a non-included side congruent

  • HL: hypotenuse and one leg of right triangles congruent

Similarity

Figures are similar if they have the same shape and proportional side lengths.

Triangle similarity criteria:

  • AA: two angles congruent

  • SAS: proportional sides with included angle congruent

  • SSS: all corresponding sides proportional

If triangles are similar with scale factor $k$, then:

  • Corresponding side lengths scale by $k$

  • Perimeters scale by $k$

  • Areas scale by $k^2$

Common uses

  • Indirect measurement

  • Proportional reasoning in maps and scale drawings

  • Solving unknown lengths in nested shapes

Example pattern:

If $\triangle ABC \sim \triangle DEF$, then

$$ \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} $$

and corresponding angles are equal.

5. Right triangles and trigonometry

Pythagorean theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$$ a^2+b^2=c^2 $$

Common triples:

  • $3,4,5$

  • $5,12,13$

  • $8,15,17$

  • $7,24,25$

Distance and midpoint

For points $(x_1,y_1)$ and $(x_2,y_2)$:

$$ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} $$

Midpoint:

$$ M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) $$

Basic trigonometric ratios

In a right triangle:

$$ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} $$
$$ \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} $$
$$ \tan\theta=\frac{\text{opposite}}{\text{adjacent}} $$

Use trig when the problem involves an angle and a side length that is not directly reachable with similarity or the Pythagorean theorem.

6. Quadrilaterals and polygons

Quadrilateral families

ShapeDefining properties
ParallelogramOpposite sides parallel and congruent; opposite angles congruent
RectangleA parallelogram with four right angles
RhombusA parallelogram with four congruent sides
SquareBoth a rectangle and a rhombus
TrapezoidAt least one pair of parallel sides
KiteTwo pairs of adjacent congruent sides

Important parallelogram facts:

  • Diagonals bisect each other.

  • Opposite sides are parallel.

  • Opposite angles are equal.

Polygon angle sums

For an $n$-gon, the sum of interior angles is:

$$ (n-2)180^\circ $$

Each interior angle of a regular $n$-gon is:

$$ \frac{(n-2)180^\circ}{n} $$

Each exterior angle of a regular $n$-gon is:

$$ \frac{360^\circ}{n} $$

The exterior angles of any polygon sum to $360^\circ$.

7. Circles

Core definitions

  • Center: fixed point in the middle

  • Radius: segment from center to circle

  • Diameter: chord through the center, equal to $2r$

  • Chord: segment connecting two points on the circle

  • Secant: line intersecting the circle at two points

  • Tangent: line touching the circle at exactly one point

Circle relationships

  • A radius is perpendicular to a tangent at the point of tangency.

  • Congruent circles have equal radii.

  • Equal chords in the same circle subtend equal arcs and equal central angles.

Angle measures in circles

  • Central angle measure equals its intercepted arc.

  • Inscribed angle measure is half its intercepted arc.

If an inscribed angle intercepts arc $\widehat{AB}$, then

$$ m\angle ACB=\frac{1}{2}m\widehat{AB} $$

Arc and sector length

Arc length:

$$ s=\frac{\theta}{360^\circ}\cdot 2\pi r $$

Sector area:

$$ A=\frac{\theta}{360^\circ}\cdot \pi r^2 $$

Power of a point

Useful circle facts:

  • Tangent segments from the same external point are congruent.

  • Intersecting chords satisfy

$$ a\cdot b=c\cdot d $$

  • For two secants from one external point:

$$ (\text{external})(\text{whole})=(\text{external})(\text{whole}) $$

  • For a tangent and a secant:

$$ (\text{tangent})^2=(\text{external})(\text{whole}) $$

8. Coordinate geometry

Coordinate geometry lets you translate geometric questions into algebra.

Line formulas

Slope between two points:

$$ m=\frac{y_2-y_1}{x_2-x_1} $$

Slope-intercept form:

$$ y=mx+b $$

Point-slope form:

$$ y-y_1=m(x-x_1) $$

Equation of a circle

Circle centered at $(h,k)$ with radius $r$:

$$ (x-h)^2+(y-k)^2=r^2 $$

Common coordinate proofs

  • Show two segments are congruent by comparing distances.

  • Show a midpoint or bisector by using midpoint formula or equal slopes.

  • Show perpendicularity using negative reciprocal slopes.

  • Show parallelism using equal slopes.

9. Transformations and symmetry

Rigid motions

Rigid motions preserve distance and angle measure:

  • Translation

  • Rotation

  • Reflection

Because rigid motions preserve size and shape, they prove congruence.

Dilations

A dilation changes size but preserves shape.

If the scale factor is $k$:

  • Lengths multiply by $k$

  • Areas multiply by $k^2$

  • Volumes multiply by $k^3$

Symmetry

  • Line symmetry: a figure matches itself across a line

  • Rotational symmetry: a figure matches itself after a rotation less than $360^\circ$

Symmetry is often a shortcut for finding equal lengths, angles, or missing parts.

10. Area, perimeter, surface area, and volume

Perimeter and area

FigureFormula
Rectangle$A=lw$
Square$A=s^2$
Triangle$A=\frac12 bh$
Parallelogram$A=bh$
Trapezoid$A=\frac12(b_1+b_2)h$
Circle$A=\pi r^2$

Common volumes

SolidFormula
Rectangular prism$V=lwh$
Prism$V=Bh$
Cylinder$V=\pi r^2h$
Pyramid$V=\frac13Bh$
Cone$V=\frac13\pi r^2h$
Sphere$V=\frac43\pi r^3$

Surface area mindset

Surface area is the total area of all outer faces. A reliable method is:

  1. Unfold or visualize the net.

  2. Compute each face separately.

  3. Add all exposed areas.

  4. Check for shared faces or overlaps.

11. Proof strategies and problem-solving workflow

Common proof tools

  • Definitions: use the meaning of congruent, midpoint, bisector, tangent, and so on.

  • Algebra: substitute known equalities and solve equations.

  • CPCTC: corresponding parts of congruent triangles are congruent.

  • Similar triangles: use proportional sides and equal angles.

  • Circle theorems: use radii, tangents, arcs, and inscribed angles.

  • Coordinate methods: use slopes, distances, and equations.

Clean problem-solving workflow

  1. Sketch the figure and label all givens.

  2. Identify the topic: triangles, circles, similarity, coordinates, or transformations.

  3. Mark equal angles, equal lengths, parallel lines, or right angles.

  4. Choose the shortest path:

    • Pythagorean theorem for right triangles

    • Similarity for proportions

    • Angle chasing for unknown angles

    • Coordinate formulas for algebraic setups

  5. Write equations from the marked relationships.

  6. Solve carefully and verify the result fits the geometry.

Common pitfalls

  • Mixing up diameter and radius

  • Using the Pythagorean theorem on a non-right triangle

  • Forgetting that similar figures have area scale factor $k^2$, not $k$

  • Assuming lines are parallel or perpendicular without evidence

  • Using degrees and radians inconsistently in circle work

12. Formula sheet

Angle and triangle formulas

$$ A+B+C=180^\circ $$
$$ a^2+b^2=c^2 $$
$$ (\text{triangle exterior angle})=(\text{two remote interior angles}) $$

Similarity and scaling

$$ \text{scale factor}=\frac{\text{new length}}{\text{original length}} $$
$$ \text{area scale factor}=k^2 $$
$$ \text{volume scale factor}=k^3 $$

Coordinate geometry

$$ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} $$
$$ M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) $$
$$ m=\frac{y_2-y_1}{x_2-x_1} $$

Circle formulas

$$ C=2\pi r $$
$$ A=\pi r^2 $$
$$ s=\frac{\theta}{360^\circ}\cdot 2\pi r $$
$$ A_{sector}=\frac{\theta}{360^\circ}\cdot \pi r^2 $$

Surface area and volume

$$ V_{prism}=Bh $$
$$ V_{cylinder}=\pi r^2h $$
$$ V_{pyramid}=\frac13Bh $$
$$ V_{cone}=\frac13\pi r^2h $$
$$ V_{sphere}=\frac43\pi r^3 $$

Quick self-check

When you finish a geometry problem, verify:

  • Units are correct

  • The answer matches the figure type

  • The value is reasonable compared with the diagram

  • Any angle measure lies between $0^\circ$ and $180^\circ$ when expected

  • Any length is nonnegative

Sources