Download Chapter 10: Introducing Geometry

Chapter 10: Introducing Geometry
10.1 Basic Ideas of Geometry
• Geometry in nature
o Honey combs
o Snow flakes
o 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
o Egyptian pyramids
o Pentagon in Washington, D.C.
• Defining basic ideas
o Points, lines, planes, and space
o Segments, rays, angles
o Special angles and perpendicular lines
o Circles and polygons
o Triangles
o Quadrilaterals
10.2 Solving Problems in Geometry
• A traversable network is also considered to be a simple path
• Network Traversability Theorem
o All even vertices = traversable type 1 (start from any vertex)
o Exactly 2 odd vertices = traversable type 2 (start at one odd vertex end at
the other odd vertex)
o >2 odd vertices = NOT traversable
• Concurrency Relationships in Triangles Theorem
o Centroid = intersection of all three triangle medians
§ Balance point
§ Center of gravity
§ Two thirds the distance from each vertex to the opposite side
o Orthocenter = intersection of all three triangle heights
o 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
o 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
o contains 3 of the four points of concurrency
o Centroid, Orthocenter, and Circumcenter form Euler’s line
o Leonard Euler (1707-1783) Pretty famous guy!
• Tangrams
10.3 More About Angles
• Angles in Intersecting Lines
o transversal – a line cutting through two or more distinct lines
o alternate interior angles – congruent angles formed on opposite sides of
a transversal between the two lines intersected
o alternate exterior angles – congruent angles formed on opposite sides of
a transversal outside the two lines intersected
o 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
o same-side interior angles – same-side interior angles are supplementary
angles
o same-side exterior angles – same-side exterior angles are
supplementary angles
o vertical angles – congruent angles formed by the intersection of any two
distinct lines such that opposite pairs of angles are congruent
• Angles in Polygons
o 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°
o sum of the exterior angles of any polygon – the sum of the exterior angles
of any polygon is 360°
o Interior angle measures for a regular polygon – the measure of each
(n − 2 )180°
interior angle of a regular n-gon is
n
o exterior angle measures for a regular polygon – the measure of an exterior
360°
angle of a regular n-gon is
n
o central angle measure for a regular polygon – the measure of the central
360°
angle of a regular n-gon is
n
• Angles in Circles
o arc – portion of a circle cut off by a pair of rays
o relating arc measure to angle measure –
1
§ m∠P =
m(arc s)
2
• angle inside the circle
• angle vertex on circle
1
§ m∠P =
[m(arc s) – m(arc r)]
2
• angle outside the circle
1
§ m∠P =
[m(arc s) + m(arc r)]
2
• angle inside the circle
• angle vertex NOT on the circle
10.4 More About Triangles
• Congruent Triangles
o 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
o 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 cong ruent
§ 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
o a2 + b2 = c 2
o a and b are legs of a right triangle
o c is ALWAYS the hypotenuse of the right triangle
o Pythagorean triples
§ Special Right Triangles
• 45°, 45°, 90°
o c= a 2
OR
o c= b 2
• 30°, 60°, 90°
o c = 2a
where a is the shorter leg
o b= a 3
10.5 More About Quadrilaterals
• Properties of Quadrilaterals
o 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
o 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
o 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
o 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?