Chapter 10: Introducing Geometry

10.1Basic Ideas of Geometry

  • Geometry in nature
  • Honey combs
  • Snow flakes
  • Fibonacci sequence
  • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
  • Sunflowers
  • Ratio of counterclockwise spirals to clockwise spirals is often 55:34 or 34:21
  • pine cone
  • Ratio = 13:8 or 8:5
  • Golden ratio
  • Approximately 1.618
  • Ratio of successive Fibonacci numbers
  • Starfish
  • Snail shell
  • Geometry in human endeavors
  • Egyptian pyramids
  • Pentagon in Washington, D.C.
  • Defining basic ideas
  • Points, lines, planes, and space
  • Segments, rays, angles
  • Special angles and perpendicular lines
  • Circles and polygons
  • Triangles
  • Quadrilaterals

10.2 Solving Problems in Geometry

  • A traversable network is also considered to be a simple path
  • Network Traversability Theorem
  • All even vertices = traversable type 1 (start from any vertex)
  • Exactly 2 odd vertices = traversable type 2 (start at one odd vertex end at the other odd vertex)
  • >2 odd vertices = NOT traversable
  • Concurrency Relationships in Triangles Theorem
  • Centroid = intersection of all three triangle medians
  • Balance point
  • Center of gravity
  • Two thirds the distance from each vertex to the opposite side
  • Orthocenter = intersection of all three triangle heights
  • Circumcenter = intersection of all three triangle perpendicular bisectors
  • Center of the circle containing the triangle vertices or
  • Center of the circle that circumscribes said triangle
  • The triangle would be inscribed in the circle
  • Incenter = intersection of all three triangle angle bisectors
  • Center of a circle tangent to all three sides of the triangle
  • Center of the circle inscribed in the triangle
  • Euler’s line
  • contains 3 of the four points of concurrency
  • Centroid, Orthocenter, and Circumcenter form Euler’s line
  • Leonard Euler (1707-1783) Pretty famous guy!
  • Tangrams

10.3 More About Angles

  • Angles in Intersecting Lines
  • transversal – a line cutting through two or more distinct lines
  • alternate interior angles – congruent angles formed on opposite sides of a transversal between the two lines intersected
  • alternate exterior angles – congruent angles formed on opposite sides of a transversal outside the two lines intersected
  • corresponding angles – congruent angles formed on the same side of a transversal where one angle is between the two lines including one line and the other angle is outside including the other line of the two lines intersected
  • same-side interior angles – same-side interior angles are supplementary angles
  • same-side exterior angles – same-side exterior angles are supplementary angles
  • vertical angles – congruent angles formed by the intersection of any two distinct lines such that opposite pairs of angles are congruent
  • Angles in Polygons
  • sum of the interior angles of any polygon – the sum of the measures of the interior angles of an n-gon is (n – 2) 180
  • sum of the exterior angles of any polygon – the sum of the exterior angles of any polygon is 360
  • Interior angle measures for a regular polygon – the measure of each interior angle of a regular n-gon is
  • exterior angle measures for a regular polygon – the measure of an exterior angle of a regular n-gon is
  • central angle measure for a regular polygon – the measure of the central angle of a regular n-gon is
  • Angles in Circles
  • arc – portion of a circle cut off by a pair of rays
  • relating arc measure to angle measure –
  • mP = m(arc s)
  • angle inside the circle
  • angle vertex on circle
  • mP = [m(arc s) – m(arc r)]
  • angle outside the circle
  • mP = [m(arc s) + m(arc r)]
  • angle inside the circle
  • angle vertex NOT on the circle

10.4 More About Triangles

  • Congruent Triangles
  • Definition of congruent triangles – Two triangles are congruent if and only if, for some correspondence between the two triangles, each pair of corresponding sides are congruent and each pair of corresponding angles are congruent
  • Triangle congruence postulates
  • SSS – if all of the corresponding pairs of sides of a triangle are congruent, then the two triangles are congruent
  • SAS – If two sides and the included angle of the corresponding pairs of sides and angles of a triangle are congruent, then the two triangles are congruent
  • ASA – If two angles and the shared side of the corresponding pairs of angles and sides of a triangle are congruent, then the two triangles are congruent
  • AAS – If two angles and a non-shared side of the corresponding pairs of angles and sides of a triangle are congruent, then the two triangles are congruent
  • For Right Triangles ONLY –
  • HA – If the hypotenuse and one angle of the corresponding pairs of angles and sides of a right triangle are congruent, then the two right triangles are congruent
  • HL – If the hypotenuse and one leg of the corresponding pairs of sides of a right triangle are congruent, then the two right triangles are congruent
  • The Pythagorean Theorem
  • a and b are legs of a right triangle
  • c is ALWAYS the hypotenuse of the right triangle
  • Pythagorean triples
  • Special Right Triangles
  • 45, 45, 90
  • c = OR
  • c =
  • 30, 60, 90
  • c = 2a where a is the shorter leg
  • b =

10.5 More About Quadrilaterals

  • Properties of Quadrilaterals
  • parallelogram – quadrilateral with two pairs of parallel sides
  • opposite sides are parallel
  • opposite sides are congruent
  • one pair of opposite sides are parallel and congruent
  • opposite angles are congruent
  • consecutive angles are supplementary
  • diagonals bisect each other
  • rectangle – quadrilateral with four right angles
  • a parallelogram is a rectangle if and only if
  • it has at least one right angle
  • its diagonals are congruent
  • rhombus – quadrilateral with four congruent sides
  • a parallelogram is a rhombus if and only if
  • it has four congruent sides
  • its diagonals bisect the angles
  • its diagonals are perpendicular bisectors of each other
  • square – quadrilateral with four right angles and four congruent sides
  • a square is a parallelogram if and only if
  • it is a rectangle with four congruent sides
  • it is a rhombus with a right angle
  • its diagonals are congruent and perpendicular bisectors of each other
  • its diagonals are congruent and bisect the angles

Chapter Summary – p. 589

Key Terms, Concepts, and Generalizations – p. 591

Chapter Review – p. 592

  • Work on problems 1-22 in your groups
  • Questions?