5.3 Geometric Principles, Coordinate Geometry, and Trigonometry Fundamentals

5.3 Geometric Principles, Coordinate Geometry, and Trigonometry Fundamentals

Core Principle: Geometry and coordinate trigonometry questions on the CAT-ASVAB test spatial logic and rapid formula execution. You will not write formal geometric proofs; rather, you must extract geometric properties from diagrams, establish algebraic equations for unknown dimensions, and compute slopes, lengths, areas, and volumes comfortably inside the roughly 124 seconds per item that the Mathematics Knowledge subtest allows.

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1. Angle Classifications, Transversals & Polygon Angle Theorems

Angle Classifications:

• Acute Angle: $0^\circ < \theta < 90^\circ$
• Right Angle: $\theta = 90^\circ$ (indicated by a square vertex marker)
• Obtuse Angle: $90^\circ < \theta < 180^\circ$
• Straight Angle: $\theta = 180^\circ$ (a collinear straight line)
• Complementary Angles: Two angle measures summing to $90^\circ$ ($37^\circ + 53^\circ = 90^\circ$)
• Supplementary Angles: Two angle measures summing to $180^\circ$ ($115^\circ + 65^\circ = 180^\circ$)
• Vertical Angles: Opposite non-adjacent angles formed by intersecting lines; vertical angles are always equal ($m\angle 1 = m\angle 3$).

                  Parallel Lines L1 || L2 cut by Transversal T
                                 T
                                 |
                            1  / | \  2
                          ----+--+----+---- Line 1 (L1)
                            3  \ | /  4
                                 |
                            5  / | \  6
                          ----+--+----+---- Line 2 (L2)
                            7  \ | /  8
                                 |

Transversal Theorems ($L_1 \parallel L_2$):

• All Acute Angles are EQUAL: $\angle 1 = \angle 4 = \angle 5 = \angle 8$
• All Obtuse Angles are EQUAL: $\angle 2 = \angle 3 = \angle 6 = \angle 7$
• Any Acute Angle + Any Obtuse Angle = $180^\circ$ (Supplementary)
• Alternate Interior Angles (Equal): $(\angle 3, \angle 6)$ and $(\angle 4, \angle 5)$
• Alternate Exterior Angles (Equal): $(\angle 1, \angle 8)$ and $(\angle 2, \angle 7)$
• Corresponding Angles (Equal): $(\angle 1, \angle 5)$, $(\angle 2, \angle 6)$, $(\angle 3, \angle 7)$, $(\angle 4, \angle 8)$
• Consecutive Interior Angles (Sum = $180^\circ$): $\angle 3 + \angle 5 = 180^\circ$ and $\angle 4 + \angle 6 = 180^\circ$

Polygons: Interior & Exterior Angle Formulas

• Sum of Interior Angles of an $n$-sided Polygon: $S = (n - 2) \times 180^\circ$ (Triangle: $n=3 \rightarrow 180^\circ$; Quadrilateral: $n=4 \rightarrow 360^\circ$; Pentagon: $n=5 \rightarrow 540^\circ$; Hexagon: $n=6 \rightarrow 720^\circ$; Octagon: $n=8 \rightarrow 1,080^\circ$)
• Interior Angle of a Regular $n$-gon: $I = \frac{(n - 2) \times 180^\circ}{n}$
• Sum of Exterior Angles: Always $360^\circ$ for any convex polygon regardless of side count.

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2. Triangles, Pythagorean Theorem & Special Right Triangles

Core Triangle Theorems

1. Interior Angle Sum: The interior angles of any triangle sum to $180^\circ$ ($\angle A + \angle B + \angle C = 180^\circ$).
2. Exterior Angle Theorem: The measure of an exterior angle equals the sum of the two non-adjacent (remote) interior angles: $d = a + b$.
3. Triangle Inequality Theorem: The sum of the lengths of any two sides must exceed the third side: $a + b > c, ; a + c > b, ; b + c > a$.
4. Area of a Triangle: $A = \frac{1}{2}bh$.

High-Yield Pythagorean Triples

For right triangles with legs $a, b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.

+-----------------------------------------------------------------------------------------+
|                              HIGH-YIELD PYTHAGOREAN TRIPLES                             |
+-----------------------+-----------------------------------------------------------------+
| 1.  (3 - 4 - 5)       | Scaled Multiples: (6-8-10), (9-12-15), (12-16-20), (15-20-25)   |
| 2.  (5 - 12 - 13)     | Scaled Multiples: (10-24-26), (15-36-39)                        |
| 3.  (8 - 15 - 17)     | Scaled Multiples: (16-30-34)                                    |
| 4.  (7 - 24 - 25)     | Scaled Multiples: (14-48-50)                                    |
| 5.  (20 - 21 - 29)    | Primitive Triple                                                |
+-----------------------+-----------------------------------------------------------------+

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

        45°-45°-90° (Isosceles Right)             30°-60°-90° (Half-Equilateral)
                    /|                                      /|
                   / |                                     / |
             x√2  /  | x                             2x   /  | x√3 (Long Leg, opp 60°)
                 /   |                                   /   |
                / 45°|                                  / 60°|
               +-----+                                 +-----+ 
                  x                                       x (Short Leg, opp 30°)

• $45^\circ-45^\circ-90^\circ$ Triangle (Side Ratios: $1 : 1 : \sqrt{2}$):
  • $\text{Leg}_1 = \text{Leg}_2 = x$
  • $\text{Hypotenuse} = x\sqrt{2}$
  • $\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{\text{Hypotenuse}\sqrt{2}}{2}$
• $30^\circ-60^\circ-90^\circ$ Triangle (Side Ratios: $1 : \sqrt{3} : 2$):
  • Short Leg ($x$): Opposite the $30^\circ$ angle
  • Long Leg ($x\sqrt{3}$): Opposite the $60^\circ$ angle ($x\sqrt{3} = \text{Short Leg} \times \sqrt{3}$)
  • Hypotenuse ($2x$): Opposite the $90^\circ$ angle ($2x = 2 \times \text{Short Leg}$)

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3. Trigonometry Fundamentals (SOH CAH TOA)

On the CAT-ASVAB, trigonometry items evaluate basic trigonometric definitions in right triangles:

$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Exact Benchmark Trigonometric Values Table

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4. Circle Mensuration & 3D Geometric Solids

Circle Formulas

• Diameter ($d$): $d = 2r$
• Circumference ($C$): $C = 2\pi r = \pi d$
• Area ($A$): $A = \pi r^2$
• Tangent Property: A tangent line touches a circle at exactly one point and is perpendicular ($90^\circ$) to the radius drawn to the point of tangency.
• Arc Length ($L$): $L = \left(\frac{\theta}{360^\circ}\right) 2\pi r$
• Sector Area ($K$): $K = \left(\frac{\theta}{360^\circ}\right) \pi r^2$

Three-Dimensional Mensuration

• Rectangular Prism: $\text{Volume} = l \cdot w \cdot h, \quad \text{Surface Area} = 2(lw + lh + wh)$
• Right Circular Cylinder: $\text{Volume} = \pi r^2 h, \quad \text{Total Surface Area} = 2\pi r^2 + 2\pi rh$
• Cube: $\text{Volume} = s^3, \quad \text{Surface Area} = 6s^2$

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5. Coordinate Geometry on the Cartesian Plane

                    y-axis
                      |
         Quadrant II  |  Quadrant I
         (-x, +y)     |  (+x, +y)
       ---------------+-------------- x-axis
         Quadrant III |  Quadrant IV
         (-x, -y)     |  (+x, -y)
                      |

Coordinate Formulas

1. Slope of a Line ($m$): Rate of steepness ($\frac{\text{rise}}{\text{run}}$): $m = \frac{y_2 - y_1}{x_2 - x_1}$
   • Horizontal Line: $y = c \implies m = 0$
   • Vertical Line: $x = c \implies m \text{ is undefined}$
2. Slope-Intercept Form: $y = mx + b$ ($m = \text{slope}, b = y\text{-intercept}$)
3. Point-Slope Form: $y - y_1 = m(x - x_1)$
4. Parallel vs. Perpendicular Slopes:
   • Parallel Lines: Identical slopes ($m_1 = m_2$).
   • Perpendicular Lines: Negative reciprocal slopes ($m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$).
5. Midpoint Formula: Exact center point between $(x_1, y_1)$ and $(x_2, y_2)$: $M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$
6. Distance Formula: Straight-line distance derived from Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$