Summary of Geometry I Concepts
eometry I focuses on the study of shapes, sizes, properties of space, and their relationships.
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Below is a summary of the key topics covered in Geometry I.
1. Basics of Geometry
● oints, Lines, and Planes: Fundamental elements ofgeometry.
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● Line Segments and Rays: Parts of a line with definedendpoints.
● Collinear and Coplanar Points: Points on the sameline or plane.
● Angles:
○ Acute (<90°), Right (90°), Obtuse (>90°), and Straight (180°).
○ Complementary Angles(sum to 90°) andSupplementaryAngles(sum to
180°).
○ Vertical Angles(opposite angles formed by two intersectinglines, always
congruent).
○ Adjacent Angles(share a common side).
2. Logic and Proofs
● onditional Statements: "If-then" statements.
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● Converse, Inverse, and Contrapositive: Variationsof conditional statements.
● Biconditional Statements: "If and only if" statements.
● Laws of Logic:
○ Law of Detachment: If p→qp \to q is true and pp istrue, then qq is true.
○ Law of Syllogism: If p→qp \to q and q→rq \to r aretrue, then p→rp \to r is true.
Types of Proofs:
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○ Two-Column Proofs(statements and justifications).
○ Paragraph Proofs(written in logical steps).
○ Flowchart Proofs(diagram-based reasoning).
3. Parallel and Perpendicular Lines
● Properties of Parallel Lines Cut by a Transversal:
○ Corresponding Angles (congruent).
○ Alternate Interior Angles (congruent).
○ Alternate Exterior Angles (congruent).
○ Consecutive Interior Angles (supplementary).
● Slopes of Lines:
○ Parallel lines have thesame slope.
○ Perpendicular lines haveopposite reciprocal slopes.
4. Triangles
● Classifications by Sides:
○ Scalene(no equal sides).
○ Isosceles(two equal sides).
○ Equilateral(three equal sides).
● Classifications by Angles:
○ Acute(all angles <90°).
○ Right(one angle = 90°).
○ Obtuse(one angle >90°).
● Triangle Angle Sum Theorem: The sum of interior anglesis180°.
● Pythagorean Theorem: a2+b2=c2a^2 + b^2 = c^2 (forright triangles).
● Congruence Theorems:
○ SSS (Side-Side-Side)
○ SAS (Side-Angle-Side)
○ ASA (Angle-Side-Angle)
○ AAS (Angle-Angle-Side)
○ HL (Hypotenuse-Leg for right triangles)
● Triangle Inequality Theorem: The sum of any two sidesmust be greater than the third
side.
5. Quadrilaterals and Polygons
● Types of Quadrilaterals:
○ Parallelogram: Opposite sides parallel and equal.
○ Rectangle: Parallelogram with four right angles.
○ Rhombus: Parallelogram with four equal sides.
○ Square: Rectangle with four equal sides.
○ Trapezoid: One pair of parallel sides.
● Polygon Interior Angles Theorem: (n−2)×180°(n-2) \times180° (sum of interior angles
for an nn-sided polygon).
● Exterior Angle Theorem: The sum of the exterior anglesof any polygon is360°.
6. Circles
● Parts of a Circle:
○ Radius: Distance from center to any point on the circle.
○ Diameter: Longest chord, twice the radius.
○ Chord: A segment joining two points on a circle.
○ Tangent: A line touching the circle at one point.
○ Secant: A line passing through the circle at two points.
● Arc Measures:
○ Minor Arc (<180°).
○ Major Arc (>180°).
○ Semicircle (180°).
● Circumference: C=2πrC = 2\pi r or C=πdC = \pi d.
● Area of a Circle: A=πr2A = \pi r^2.
● Central Angles and Inscribed Angles: Inscribed anglesarehalfthe measure of their
intercepted arc.
7. Transformations
● Types of Transformations:
○ Translation(sliding).
○ Reflection(flipping over a line).
○ Rotation(turning around a point).
○ Dilation(resizing by a scale factor).
● Rigid Motions: Transformations that preservedistanceand angle measures
(translations, reflections, rotations).
● Similarity Transformations: Dilations createsimilarfigures with proportional sides.
8. Perimeter, Area, and Volume
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● erimeter: The sum of the sides of a shape.
● Area Formulas:
○ Rectangle: A=lwA = lw
○ Triangle: A=12bhA = \frac{1}{2} bh
○ Trapezoid: A=12(b1+b2)hA = \frac{1}{2} (b_1 + b_2) h
○ Circle: A=πr2A = \pi r^2
● Surface Area and Volume of 3D Shapes:
○ Prisms:
■ SA: Sum of all faces.
■ V: V=BhV = Bh (Base area × height).
○ Cylinders:
■ SA: 2πr2+2πrh2\pi r^2 + 2\pi rh
■ V: πr2h\pi r^2 h
○ Pyramids and Cones:
■ V: 13Bh\frac{1}{3} Bh
○ Spheres:
■ SA: 4πr24\pi r^2
■ V: 43πr3\frac{4}{3} \pi r^3
9. Coordinate Geometry
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● istance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2- x_1)^2 + (y_2 - y_1)^2}
● Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1+ x_2}{2}, \frac{y_1 + y_2}{2}
\right)
● Equation of a Circle: (x−h)2+(y−k)2=r2(x - h)^2 +(y - k)^2 = r^2
10. Trigonometry Basics
● Right Triangle Trigonometry:
○ Sine: sinθ=oppositehypotenuse\sin \theta =
\frac{\text{opposite}}{\text{hypotenuse}}
○ Cosine: cosθ=adjacenthypotenuse\cos \theta =
\frac{\text{adjacent}}{\text{hypotenuse}}
○ Tangent: tanθ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
● Using Trig Ratios to Find Missing Sides and Angles.
his summary covers the fundamental concepts inGeometryI, providing a solid foundation for
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advanced studies inGeometry II, Trigonometry, andPre-Calculus. Let me know if you need
further explanations or examples! 🚀
