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Geometry Fundamentals: Lines, Angles, and Triangles Explained

Learn the basics of geometry - from lines and angles to triangles and polygons. Understand the types of angles, lines, and triangles, and discover essential formulas for calculating perimeters and areas of polygons and triangles.

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Geometry Fundamentals: Lines, Angles, and Triangles Explained

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  1. Chapter 2 Geometry

  2. Section 2.1 Lines and Angles

  3. A B B B A A A The Foundation Point: Usually named using a capital letter Line: (1-dimensional) Usually named using two points included on the line Plane: (2-dimensional) Extends indefinitely in all directions on it. Ray: (half-line) Has one endpoint and extends infinitely in one direction

  4. A B C Angles An angle is formed by two ___________ with a common _____________, called the ____________ of the angle. The amount of ________________ of the terminal ray is called the angle.

  5. Types of Angles Measure: Right Angle ________________ Straight Angle ________________ Acute Angle ________________ Obtuse Angle _________________

  6. Types of Lines Lines in the same plane that do not intersect are called ____________________ lines. Lines that intersect at right angles are called _______________________ lines. B A AB CD D C

  7. H E F G More on Angles ________________________ angles are two angles whose sum is 90°. ________________________ angles are two angles whose sum is 180°. ________________________ angles share a common vertex and a common side.

  8. When two lines intersect, the angles that are formed on opposite sides of the point of intersection are called ___________________ angles. _______________ angles are equal in measure (congruent). 2 3 1 4

  9. M N Q P If a line intersects two or more lines in a plane, it is called a _____________________________. • When two parallel lines are cut by a transversal, the • corresponding angles are equal (congruent) • alternate interior angles are equal (congruent) • alternate exterior angles are equal (congruent)

  10. 1 2 4 3 5 6 M N 7 8 Q P Pairs of corresponding angles: ________&_________; ________&_________; ________&_________; ________&_________ Pairs of alternate interior angles: ________&_________; ________&_________ Pairs of alternate exterior angles: ________&_________; ________&_________

  11. 1 2 4 3 5 6 M N 7 8 Q P Example If 4 = 41°, find the measures of the remaining angles. 1 =______ 2 = ______ 3 = ______ 5 =______ 6 = ______ 7 = ______ 8 =______

  12. A B W X C D Z Y E F • When more than two parallel lines are cut by two transversals, the segments of the transversals between the same two parallel lines are called corresponding segments. Ratios of corresponding segments of the transversal are equal.

  13. Section 2.2 Triangles

  14. Polygons • A polygon is a closed figure with straight sides. • Some common polygons: • 3 sides: __________________ • 4 sides: __________________ • 5 sides: __________________ • 6 sides: __________________ • 7 sides: __________________ • 8 sides: __________________ • 9 sides: __________________ • 10 sides: __________________ • 12 sides: __________________

  15. Polygons • Perimeter: _______________________________________________________________________________________________________________________________________ • Units used for perimeter: • Area: __________________________________________________________________________________________ Units used for area:

  16. Triangles • Triangles are polygons with three sides and three interior angles. • The sum of the measures of the three angles of a triangle is __________. • Triangles are often classified according to the types of sides or the types of angles they contain.

  17. Types of Triangles (classified by sides) • Scalene: • Isosceles: • Equilateral:

  18. hypotenuse leg leg Right Triangles Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides (legs). Example: If a = 5 and c = 11, find the exact value of b. c a b

  19. h b Area of a Triangle • The area of a triangle is one-half the product of a base and its height. The height (or altitude) of a triangle is the line segment drawn from a vertex perpendicular to the opposite side (base). Area of a triangle:

  20. Find the perimeter and area of the triangle shown below (all numbers are approximate). 6.5 cm 1.3 cm 2.2 cm 4.8 cm

  21. Find the perimeter and area (to 2 s.d.) of an isosceles triangle whose two equal sides measure 35 mm and height from the vertex angle to the base measures 28 mm.

  22. Another Formula for the Area of a Triangle:A Super Hero! • If we know the three lengths of the sides of a triangle, we can use Hero’s Formula (Alternate: Heron’s Formula) to calculate the area. Hero’s Formula

  23. Example • A triangular-shaped park is bounded by three streets. The lengths of the three sides of the park are found to be 358 ft, 437 ft, and 509 ft. What is the area of the park (to 3 sig dig)?

  24. Congruent Triangles • Two or more triangles are congruent if the measures of each of the ___________ and the measures of each of the ____________ are the same. A P 84 84 5.8 cm 5.8 cm 4.6 cm 4.6 cm 41 55 55 41 C B Q R 7.0 cm 7.0 cm

  25. B D C A E Similar Triangles • Similar triangles have congruent ________________ but do not necessarily have congruent _______________. Properties of Similar Triangles Corresponding angles of similar triangles are equal. Corresponding sides of similar triangles are proportional. ABC  ADE Corresponding angles: Corresponding sides:

  26. Example • A woman is standing next to a building on level ground. The woman, who is 5’6” tall, casts a shadow that is 3.0 feet long. At the same time, a building casts a shadow that is 92 feet long. How tall is the building? Round ans to 2 sig digits.

  27. Section 2.3 Quadrilaterals

  28. A quadrilateral is a polygon with ______ sides and _____ interior angles. The sum of the measures of the interior angles of a quadrilateral is ___________. A ________________ of a polygon is a line segment joining any two non-adjacent vertices.

  29. Special Types of Quadrilaterals • Parallelogram • Rhombus • Rectangle • Square • Trapezoid

  30. Determine if each statement is true sometimes, always, or never. A rectangle is a parallelogram. _______________ A rhombus is a square. _______________ A square is a rhombus. _______________ A trapezoid is a parallelogram. _______________ A square is a rectangle. _______________ A parallelogram is a rectangle. _______________ A square is a trapezoid. _______________

  31. Perimeter • The perimeter of a quadrilateral is the sum of the lengths of the four sides. (The distance around the figure.) • Some Special Formulas: • Perimeter of a Rectangle: P = ________________ • Perimeter of a Square: P =________________

  32. Area Formulas • Area of a Parallelogram: A = ________________ • Area of a Rectangle: A =________________ • Area of a Square: A = ________________ • Area of a Trapezoid: A = ________________

  33. Examples • 1) Text p. 62 # 30

  34. Examples • 2) Text p. 62 # 32

  35. Examples • 3) Text p. 63 # 38

  36. Section 2.4 Circles

  37. Some Circle Vocab • ________________: The set of all points equidistant from a fixed point, called the __________. • ________________: A line segment with endpoints on the circle that passes through its center. • ________________: A line segment from the center to a point on the circle. • ________________: A line segment with endpoints on the circle.

  38. o More Circle Vocab • ________________: A line that intersects (touches) the circle at EXACTLY ONE POINT. • ________________: A line that passes through two points of the circle. The radius is perpendicular to the tangent at the point of tangency.

  39. N o P M An example Let O be the center of the circle. MN is tangent to the circle. If MNO = 14°, find MOP.

  40. Circumference •  (pi) is an irrational number that is approximately equal to ______ •  is the ratio of the circumference (perimeter) of a circle to its diameter. Circumference of a Circle:

  41. Area Area of a Circle: Example Find the area of a circle with diameter 18.5 cm. Give answer to three significant digits.

  42. An example • A circular walkway, with a uniform width of 3.0 ft, is to be installed around a traffic circle that has a diameter of 25 ft. Find the area of the walkway to two significant digits.

  43. 15.8 cm 21.2 cm Another example • What is the area of the largest circle that can be cut from a rectangular plate 21.2 cm by 15.8 cm? How much waste is there? Give answers to 3 sig digits.

  44. One More Example • Neglecting waste, how much would it cost to lay down a hardwood floor on the indoor rink if flooring costs $22.50 per square meter? Round to nearest dollar. Note: Ends are semicircular. 14 m 20 m

  45. A B o Arcs & Angles • A _________ __________ is an angle formed by two radii (its vertex is the center of the circle.) • An _________ is a part of the outside of the circle (a curved segment) The measure of an arc is the same as the measure of the central angle that forms it. (Measured in degrees.)

  46. A B o A B Regions of a Circle • A ___________________ is a region bounded by two radii and the arc they intercept. • A ____________ is a region bounded by a chord and its arc.

  47. A Intercepted arc B C Inscribed angle More Angles • An ____________ ___________ is formed by two chords of the circle and has its vertex on the circle. The measure of an inscribed angle is half of its intercepted arc.

  48. An Example • If the measure of an inscribed angle is 45°, find the length in cm (to 2 sig dig) of its intercepted arc, given that the radius of the circle is 8.0 cm.

  49. Section 2.6 Solid Geometric Figures

  50. Volume & Surface Area • Volume is the measure of space occupied by a solid geometric figure. • Units used for volume: • Surface Area is the total areaof all of the faces of a solid geometric figure. • Units used for surface area:

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