Points and Lines
- Point: An exact location in space, often named by a capital letter (e.g., A).
- Line: A straight path extending in both directions without end; has infinite length but no thickness. Named by two points (e.g., line AB) or a lowercase letter.
- Line Segment: A part of a line between two endpoints. Example: Segment AB.
- Ray: A part of a line that starts at one endpoint and extends infinitely in one direction. Example: Ray AB starts at A and passes through B.
Measuring Angles & Types of Angles
- Angle: Formed by two rays (the sides) with a common endpoint (the vertex).
- Measuring Angles: Angles are measured in degrees (°).
- Types of Angles:
- Acute: Less than 90°.
- Right: Exactly 90°.
- Obtuse: Between 90° and 180°.
- Straight: Exactly 180°.
- Reflex: Between 180° and 360°.
- Adjacent, Complementary, Supplementary Angles
Triangles
- Classified by Sides:
- Equilateral: 3 equal sides
- Isosceles: 2 equal sides
- Scalene: No equal sides
- Classified by Angles:
- Acute Triangle: All angles < 90°
- Right Triangle: One angle is 90°
- Obtuse Triangle: One angle > 90°
- Triangle Sum Theorem: The sum of the interior angles is 180°.
Quadrilaterals & Polygons
- Quadrilaterals: 4-sided polygons.
- Parallelogram
- Rectangle
- Square
- Rhombus
- Trapezoid
- Polygons: Closed shapes with 3 or more straight sides (triangle, pentagon, hexagon, etc.).
- Regular Polygon: All sides and angles are equal.
- Irregular Polygon: Sides and/or angles are not all equal.
Circles
- Circle: Set of points equidistant from a center point.
- Radius: Distance from center to any point on the circle.
- Diameter: A chord passing through the center (twice the radius).
- Circumference: The perimeter of the circle.
- Chord: A segment with endpoints on the circle.
- Arc: A part of the circumference.
- Sector: A region bounded by two radii and an arc.
Solid Figures
- Cuboid (Rectangular Prism)
- Cube
- Sphere
- Cylinder
- Cone
- Pyramid
Surface Area
- Cuboid: 2(lw + lh + wh)
- Cube: 6a2
- Cylinder: 2πr(h + r)
- Sphere: 4πr2
- Cone: πr(l + r), where l is the slant height
Graphing Figures
- Plotting points on the Cartesian plane (x, y).
- Graphing lines: y = mx + b (slope-intercept form).
- Graphing polygons by connecting plotted vertices.
- Graphing a circle: (x - h)2 + (y - k)2 = r2
Chapter: Key Concepts Explained
- Congruence: Figures that have the same shape and size.
- Similarity: Figures that have the same shape but not necessarily the same size (corresponding angles equal, sides proportional).
- Perimeter: The distance around a figure.
- Area: The measure of the surface inside a figure.
- Volume: The amount of space inside a solid figure.
- Midpoint: The point that divides a segment into two equal parts.
- Bisect: To divide into two equal parts (can be an angle or a segment).
- Parallel lines: Lines in a plane that never meet.
- Perpendicular lines: Lines that meet at a right angle (90°).
- Coordinate Geometry: Using algebra and coordinates to study geometric properties.
Chapter: Examples
Example 1:
Classify the triangle with sides 5 cm, 5 cm, and 8 cm.
Solution: Since two sides are equal, it is isosceles.
Classify the triangle with sides 5 cm, 5 cm, and 8 cm.
Solution: Since two sides are equal, it is isosceles.
Example 2:
Find the area of a rectangle with length 10 cm and width 4 cm.
Solution: Area = length × width = 10 × 4 = 40 cm2.
Find the area of a rectangle with length 10 cm and width 4 cm.
Solution: Area = length × width = 10 × 4 = 40 cm2.
Example 3:
Name the angle type if it measures 120°.
Solution: It is an obtuse angle.
Name the angle type if it measures 120°.
Solution: It is an obtuse angle.
Example 4:
Find the circumference of a circle with radius 7 cm (π = 3.14).
Solution: Circumference = 2πr = 2 × 3.14 × 7 = 43.96 cm.
Find the circumference of a circle with radius 7 cm (π = 3.14).
Solution: Circumference = 2πr = 2 × 3.14 × 7 = 43.96 cm.
Example 5:
What is the midpoint of segment AB if A(2, 3) and B(6, 11)?
Solution: Midpoint = ((2+6)/2, (3+11)/2) = (4, 7)
What is the midpoint of segment AB if A(2, 3) and B(6, 11)?
Solution: Midpoint = ((2+6)/2, (3+11)/2) = (4, 7)
Pre-Test: Geometry (40 Questions)
Chapter: Questions & Answers
Questions and answers will appear here after you complete the pre-test.
Each incorrect pre-test answer will be explained below.
Each incorrect pre-test answer will be explained below.