6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles

6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles

Geometry tests your ability to analyze geometric figures, compute angle measures, apply geometric theorems, and solve spatial relationships. in Mathematics Knowledge practice, geometry questions require a strong foundation in planar geometry properties, right triangle relationships, and circle definitions.

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Angle Relationships & Lines

An angle is formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees ($^\circ$).

Classification of Angles

• Acute Angle: $0^\circ < \theta < 90^\circ$
• Right Angle: $\theta = 90^\circ$ (indicated by a square symbol $\llcorner$)
• Obtuse Angle: $90^\circ < \theta < 180^\circ$
• Straight Angle: $\theta = 180^\circ$ (forms a straight line)

Key Angle Relationships

Parallel Lines Cut by a Transversal

When two parallel lines ($L_1 \parallel L_2$) are intersected by a third line (transversal $T$), eight angles are formed with unique relationships:

• Equal Angles:
  • Corresponding Angles: (e.g., Position 1 and Position 5)
  • Alternate Interior Angles: (e.g., Position 3 and Position 6)
  • Alternate Exterior Angles: (e.g., Position 1 and Position 8)
  • Vertical Angles: (e.g., Position 1 and Position 4)
• Supplementary Angles ($180^\circ$):
  • Consecutive Interior Angles (Same-side interior): $A + B = 180^\circ$
  • Consecutive Exterior Angles (Same-side exterior): $A + B = 180^\circ$

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Triangles & Triangle Theorems

A triangle is a three-sided polygon. The sum of the interior angles of any triangle is always $180^\circ$.

Classifications of Triangles

• By Sides:
  • Scalene: All three sides have different lengths.
  • Isosceles: At least two sides are equal in length; angles opposite equal sides are equal.
  • Equilateral: All three sides are equal; all three angles equal $60^\circ$.
• By Angles:
  • Acute: All three angles are $< 90^\circ$.
  • Right: One angle equals $90^\circ$.
  • Obtuse: One angle is $> 90^\circ$.

Core Triangle Theorems

1. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side ($a + b > c$).
2. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles ($m\angle \text{Ext} = m\angle A + m\angle B$).

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Right Triangles & Special Relationships

In any right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the $90^\circ$ angle):

$$\text{Pythagorean Theorem: } a^2 + b^2 = c^2$$

Common Pythagorean Triples

Recognizing common integer side length ratios saves unnecessary calculation time in practice:

• $3 - 4 - 5$ (and multiples: $6-8-10, 9-12-15, 12-16-20$)
• $5 - 12 - 13$ (and multiples: $10-24-26$)
• $8 - 15 - 17$
• $7 - 24 - 25$

Special Right Triangles

Practice questions may test two special right triangles derived from regular figures:

45°-45°-90° Triangle (Isosceles Right)     30°-60°-90° Triangle
         /|                                       /|
        / |                                      / |
  x√2  /  | x                             2x    /  | x√3
      /   |                                    /   |
     /____|                                   /____|
       x                                         x

1. $45^\circ - 45^\circ - 90^\circ$ Triangle (Isosceles Right Triangle):
   • Ratio of side lengths: $1 : 1 : \sqrt{2}$
   • Leg $= x$, Hypotenuse $= x\sqrt{2}$.
2. $30^\circ - 60^\circ - 90^\circ$ Triangle:
   • Ratio of side lengths: $1 : \sqrt{3} : 2$
   • Side opposite $30^\circ$ (Short Leg) $= x$
   • Side opposite $60^\circ$ (Long Leg) $= x\sqrt{3}$
   • Hypotenuse $= 2x$

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Quadrilaterals & Polygon Theorems

A polygon is a closed 2D shape with straight sides. An $n$-sided polygon has $n$ interior angles.

Polygon Formulas

• Sum of Interior Angles: $S = (n - 2) \times 180^\circ$
• Each Interior Angle (Regular Polygon): $I = \frac{(n - 2) \times 180^\circ}{n}$
• Sum of Exterior Angles: Always $360^\circ$ for any convex polygon.
• Each Exterior Angle (Regular Polygon): $E = \frac{360^\circ}{n}$

Quadrilateral Hierarchy

• Parallelogram: Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
• Rectangle: Parallelogram with four $90^\circ$ right angles; diagonals are equal.
• Rhombus: Parallelogram with four equal sides; diagonals are perpendicular bisectors.
• Square: Regular quadrilateral (both rectangle and rhombus); four equal sides and four $90^\circ$ angles.
• Trapezoid: Quadrilateral with exactly one pair of parallel bases.

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Circle Fundamentals

A circle is the set of all points in a plane equidistant from a fixed point (center).

Key Definitions

• Radius ($r$): Distance from center to any point on circle.
• Diameter ($d$): Distance across circle through center ($d = 2r$).
• Chord: Line segment connecting any two points on the circle.
• Tangent: Line intersecting circle at exactly one point (perpendicular to radius at point of tangency).

Essential Circle Formulas

• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$
• Arc Length: $L = \frac{\theta}{360^\circ} \times 2\pi r$
• Sector Area: $A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$