HESTEN ACADEMIC REFERENCE ARCHIVE CALL NO: QA453.H47 2026 • DEWEY: 516.3 • VOL. X

Volume X • Euclidean Geometry & Trigonometry Codex

Tenth Grade Mathematics Reference Codex (High School Geometry)

The Complete Scholastic Guide to Deductive Proofs, Triangle Congruence & Similarity, SOH-CAH-TOA, Special Triangles, Circle Theorems, Arc & Sector Mensuration & Synoptic Concordance

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An exhaustive reference codex compiled for secondary geometry scholars, honors students, and collegiate preparatory researchers. Codifying Euclidean deductive proof structures, rigorous axiomatic congruence and similarity criteria, right triangle trigonometry, coordinate conic circles, differential arc measures, and computational geometric heuristics.

§ 10.1

Formal Logic, Deductive Proofs & CPCTC

LOGICAL CONDITIONAL

Conditional & Variations

$$\begin{aligned} \text{Conditional: } &p \to q \\ \text{Converse: } &q \to p \\ \text{Inverse: } &\sim p \to \sim q \\ \text{Contrapositive: } &\sim q \to \sim p \ (\equiv p \to q) \end{aligned}$$

A conditional statement and its contrapositive are logically equivalent!

EUCLIDEAN PRINCIPLE

The CPCTC Axiom

$$\Delta ABC \cong \Delta DEF \implies \begin{cases} \overline{AB}\cong\overline{DE}, \overline{BC}\cong\overline{EF}, \overline{AC}\cong\overline{DF} \\ \angle A\cong\angle D, \angle B\cong\angle E, \angle C\cong\angle F \end{cases}$$

Corresponding Parts of Congruent Triangles are Congruent. Proves segment/angle equality after establishing triangle congruence.

§ 10.2

Triangle Congruence Postulates & Theorems ($\cong$)

Congruence Criterion	Required Geometric Evidence	Scholastic Invariant Note
SSS (Side-Side-Side)	All 3 pairs of corresponding sides are congruent ($\cong$)	Rigidly locks all 3 interior angles
SAS (Side-Angle-Side)	2 sides and the included angle between them	Angle MUST lie strictly between the two sides
ASA (Angle-Side-Angle)	2 angles and the included side between them	Side connects the two angle vertices
AAS (Angle-Angle-Side)	2 consecutive angles and a non-included side	Equivalent to ASA by Triangle Angle Sum ($180^\circ$)
HL (Hypotenuse-Leg)	Right triangle: Congruent hypotenuse and 1 leg	Valid strictly in right triangles ($90^\circ$)
AAA & SSA	INVALID FOR CONGRUENCE! AAA proves similarity only. SSA produces the ambiguous two-triangle case.

§ 10.3

Triangle Similarity ($\sim$) & Geometric Mean Theorems

THEOREM 10.3.1 Topic: Geometric Mean Altitude & Leg Theorems

In right triangle $\Delta ABC$ with altitude $h$ drawn to hypotenuse $c$, partitioning $c$ into segments $p$ and $q$: $$h^2 = p \cdot q \implies h = \sqrt{p \cdot q} \quad (\text{Altitude is Geometric Mean of Segments})$$ $$a^2 = p \cdot c \implies a = \sqrt{p \cdot c} \quad \text{and} \quad b^2 = q \cdot c \implies b = \sqrt{q \cdot c}$$

§ 10.4

Right Triangle Trigonometry: SOH-CAH-TOA

PRIMARY RATIO I

Sine Ratio ($\sin\theta$)

$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad (\text{SOH})$$

Reciprocal: Cosecant $\csc\theta = \frac{\text{Hyp}}{\text{Opp}} = \frac{1}{\sin\theta}$.

PRIMARY RATIO II

Cosine Ratio ($\cos\theta$)

$$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad (\text{CAH})$$

Reciprocal: Secant $\sec\theta = \frac{\text{Hyp}}{\text{Adj}} = \frac{1}{\cos\theta}$.

PRIMARY RATIO III

Tangent Ratio ($\tan\theta$)

$$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta} \quad (\text{TOA})$$

Reciprocal: Cotangent $\cot\theta = \frac{\text{Adj}}{\text{Opp}} = \frac{1}{\tan\theta}$.

§ 10.5

Special Right Triangles ($45^\circ$-$45^\circ$-$90^\circ$ & $30^\circ$-$60^\circ$-$90^\circ$)

Special Triangle	Side Length Ratio	$\sin\theta$ Exact	$\cos\theta$ Exact	$\tan\theta$ Exact
$45^\circ$-$45^\circ$-$90^\circ$ (Isosceles Right)	$x : x : x\sqrt{2}$	$\sin(45^\circ) = \frac{\sqrt{2}}{2}$	$\cos(45^\circ) = \frac{\sqrt{2}}{2}$	$\tan(45^\circ) = 1$
$30^\circ$-$60^\circ$-$90^\circ$ ($30^\circ$ Angle)	$x : x\sqrt{3} : 2x$	$\sin(30^\circ) = \frac{1}{2}$	$\cos(30^\circ) = \frac{\sqrt{3}}{2}$	$\tan(30^\circ) = \frac{\sqrt{3}}{3}$
$30^\circ$-$60^\circ$-$90^\circ$ ($60^\circ$ Angle)	$x : x\sqrt{3} : 2x$	$\sin(60^\circ) = \frac{\sqrt{3}}{2}$	$\cos(60^\circ) = \frac{1}{2}$	$\tan(60^\circ) = \sqrt{3}$

§ 10.6

Circle Geometry: Chords, Tangents & Inscribed Angles

INSCRIBED ANGLE

Inscribed Angle Theorem

$$\angle_{\text{inscribed}} = \frac{1}{2} \cdot \text{Arc Measure}$$

An angle inscribed in a semicircle is strictly a right angle ($90^\circ$).

INTERSECTING CHORDS

Chord Segment Products

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

Products of intersecting chord segments inside a circle are equal.

TANGENT THEOREM

Radius-Tangent Perpendicularity

$$\text{Radius } \perp \text{ Tangent Line } (90^\circ)$$

Two tangents drawn from the same external point are congruent.

§ 10.7

Arc Length, Sector Area & Coordinate Circle Equation

ARC LENGTH

Arc Length ($s$)

$$s = \frac{\theta}{360^\circ}(2\pi r) = r\theta \quad (\theta \text{ rad})$$

Curved length along the circumference subtended by central angle $\theta$.

SECTOR AREA

Sector Area ($A_{\text{sector}}$)

$$A = \frac{\theta}{360^\circ}(\pi r^2) = \frac{1}{2}r^2\theta$$

Fractional area of circle enclosed by two radii and the arc.

CONIC SECTION

Standard Equation of a Circle

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

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

§ 10.8

Coordinate Geometry Proofs & Quadrilateral Verification

SLOPE CRITERIA

Perpendicular Lines

$$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1}$$

Slopes are opposite reciprocals. (Parallel: $m_1 = m_2$).

MIDPOINT FORMULA

Segment Bisection

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

Used to prove diagonals of a parallelogram bisect each other!

§ 10.9

3D Solids, Cavalieri's Principle & Density

PRINCIPLE 10.9.1 Topic: Cavalieri's Cross-Sectional Invariant

If two solids have the same height $h$ and the same cross-sectional area at every level parallel to their bases, then they have equal volumes! $$\text{Pyramid / Cone Volume: } V = \frac{1}{3} B h \quad \vert \quad \text{Sphere Volume: } V = \frac{4}{3}\pi r^3$$ $$\text{Density: } \text{Density} = \frac{\text{Mass}}{\text{Volume}} \quad \vert \quad \text{Population Density} = \frac{\text{Population}}{\text{Area}}$$

§ 10.10

Scholia & Computational Geometry Speed Hacks

HACK I

Trigonometric Left-Hand Trick

$$\sin\theta = \frac{\sqrt{\text{Fingers Below}}}{2} \quad \vert \quad \cos\theta = \frac{\sqrt{\text{Fingers Above}}}{2}$$

Assign fingers to $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$. Fold finger to read exact radical fractions instantly!

HACK II

30-60-90 Short-Leg Anchor

$$\text{Short Leg } x \implies \text{Hypotenuse } = 2x, \ \text{Long Leg } = x\sqrt{3}$$

Always isolate the short leg across from $30^\circ$ first! Everything scales from that single number.

HACK III

Thales' Semicircle Right Angle Shortcut

$$\text{Inscribed angle intercepting diameter } \implies 90^\circ \text{ ALWAYS}$$

Any triangle with one side as the circle diameter is automatically a right triangle.

HACK IV

Area Scale Factor Squared Rule

$$\text{Linear Scale } k \implies \text{Area Scale } = k^2, \ \text{Volume Scale } = k^3$$

If dimensions double ($k=2$), area quadruples ($\times 4$) and volume octuples ($\times 8$)!

§ 10.11

The Grand Synoptic Tables & Complete Grade 10 / Geometry Student Cheat Sheet

Grade 10 / Geometry Master Reference Concordance

Authorized curriculum reference concordance • High School Geometry Complete

Congruence & Similarity

Congruent ($\cong$): SSS, SAS, ASA, AAS, HL
Similar ($\sim$): AA~, SAS~, SSS~
CPCTC: Corresponding parts congruent
Altitude: $h = \sqrt{pq}$
Leg: $a = \sqrt{pc}, b = \sqrt{qc}$

Trigonometry (SOH-CAH-TOA)

$\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$ • $\csc\theta = \frac{\text{Hyp}}{\text{Opp}}$
$\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$ • $\sec\theta = \frac{\text{Hyp}}{\text{Adj}}$
$\tan\theta = \frac{\text{Opp}}{\text{Adj}}$ • $\cot\theta = \frac{\text{Adj}}{\text{Opp}}$
Pythagorean ID: $\sin^2\theta + \cos^2\theta = 1$
$\tan\theta = \sin\theta / \cos\theta$

Special Triangles

45-45-90: $x : x : x\sqrt{2}$
$\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$
30-60-90: $x : x\sqrt{3} : 2x$
$\sin(30^\circ) = \frac{1}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}$
$\sin(60^\circ) = \frac{\sqrt{3}}{2}, \cos(60^\circ) = \frac{1}{2}$

Circle Geometry

Inscribed Angle: $\frac{1}{2} \times \text{Arc}$
Chords: $a \cdot b = c \cdot d$
Radius $\perp$ Tangent ($90^\circ$)
Circle Eq: $(x-h)^2 + (y-k)^2 = r^2$
Thales: Semicircle angle $= 90^\circ$

Arcs & Sectors

Arc Length: $s = r\theta = \frac{\theta}{360}(2\pi r)$
Sector Area: $A = \frac{1}{2}r^2\theta = \frac{\theta}{360}(\pi r^2)$
Radians to Deg: $\times \frac{180^\circ}{\pi}$
Deg to Radians: $\times \frac{\pi}{180^\circ}$
$2\pi \text{ rad} = 360^\circ$

3D Solids & Slopes

Perpendicular: $m_1 \cdot m_2 = -1$
Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Pyramid/Cone: $V = \frac{1}{3}Bh$
Sphere: $V = \frac{4}{3}\pi r^3, SA = 4\pi r^2$
Area Scale: $k^2$ • Volume: $k^3$