Geometry Lines and Angles Vocabulary Point Line segment
Geometry
Lines and Angles
Vocabulary • • Point Line segment Congruent line segments Midpoint Plane Congruent angles Vertex
Rules for Lines and Angles A P C E • • G D F B A straight line has an angle of 180 o on either side Opposite angles of intersecting lines are equal Perpendicular lines: the angle created between two perpendicular lines is 90 o Parallel lines: “F” and “Z”
Lines and Angles 19.
Lines and Angles 2 0
Polygons – sides and angles n (number of sides) Name of polygon Sum of interior angles (n 2)*180 o 3 Triangle (1)*180 o=180 o 4 Quadrilateral (2)*180 o=360 o 5 Pentagon (3)*180 o=540 o 6 Hexagon (4)*180 o=720 o 7 Heptagon (5)*180 o=900 o 8 Octagon (6)*180 o=1080 o 9 Nonagon (7)*180 o=1260 o 10 decagon (8)*180 o=1440 o
Polygons, Quadrilaterals & Three-dimensional Figures • Vocabulary – Sides – Vertices – Polygon – Regular polygon • Quadrilaterals -rectangle -square -parallelogram (two sets of parallel sides) -rhombus (think of a kite) -trapezoid (one set of parallel sides)
Perimeter, Area, Surface Area, Volume • Perimeter = sum of side lengths (“peri” = around “meter” = measure) • Area = the size of the region enclosed by a two dimensional shape • Surface Area = the area on the outside of a three dimensional shape • Volume = the size of the region enclosed by a three dimensional shape
Formulas for Perimeter & Area • b b
Formulas for Perimeter & Area • b a a b
Formulas for Perimeter & Area • b h a a b
Formulas for Perimeter & Area Trapezoids
Formulas for Surface Area
Formulas for Surface Area and Volume
Formulas for Perimeter & Area •
Triangles •
Equilateral Triangles • • Equilateral triangles have three sides of equal length and three angles of equal measure (i. e. 60 o). if you are given a triangle with three equal sides or three equal angles you know it is equilateral https: //en. wikipedia. org/wiki/Equilateral_triangle
Isosceles Triangles • • Isosceles triangles have two sides of equal length and two (opposite) angles of equal measure. if you are given a triangle with two equal sides or two equal angles you know it is an isosceles triangle https: //en. wikipedia. org/wiki/Isosceles_triangle
Right Triangles • • • Right triangles have one 90 o angle The side opposite the 90 o angle is called the hypotenuse the relationship between the side lengths is a 2 + b 2 = c 2 (c is the length of the hypotenuse and a and b are the other side lengths) It is helpful to be able to recognize some Pythagorean triples: (3, 4, 5) and (5, 12, 13) if you are given a right triangle with two side lengths you can compute third https: //en. wikipedia. org/wiki/Right_triangle
Triangle Inequality • the length of any side of a triangle is less than the sum of and greater than the difference of the other two sides
Congruent and Similar Triangles* • • congruent triangles are the same shape and same size (i. e. their side lengths and angles are the same) similar triangles are the same shape but not the same size; their measurements are related to each other by a scale factor
Special Triangles • The main point to keep in mind with these triangles is that you can solve problems involving them with less information than you would for a non-special triangle.
Isosceles right triangle •
30 o-60 o-90 o triangle •
Special Triangles Can you recognize these triangles?
Circles
Radius and Diameter •
Formulas for Perimeter & Area •
Circle Formulas •
Other Circle Vocabulary • Chord (a line segment joining any two points on the circle) • Arc (any portion of the outline of the circle) • Central angle (the measure of the angle corresponding to an arc) • Sector (think of a pie slice) • Tangent • Inscribed (drawn inside of) • Circumscribed (drawn around) • Concentric
Helpful relationships •
Helpful relationships • If a triangle inscribed inside a circle has one side that is the diameter of the circle, it is a right triangle • If a square is inscribed inside a circle, its diagonal is the diameter of the circle • If a circle is inscribed inside a square, its diameter is the side length of the square
Geometry Hints • Try to understand why these formulas work rather than memorizing. • The base (of a triangle, parallelogram, etc) is arbitrary and can be chosen for convenience. • The height of a parallelogram or trapezoid will often need to be computed using the Pythagorean theorem or special triangles. • For other shapes, it may be easier to compute the perimeter and area by subdividing into simpler shapes. • To compute the surface area or volume of a cylinder, sphere or cone you may need to calculate the radius based on the circumference or diameter.
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