12tuulis

=Symmetry in Polygons=


 * Objectives:**
 * Define polygon.
 * Define and use refletional symmetry and rotational symmetry.
 * Define regular polygon, center of a regular polygon, central angle of a regular polygon, and axis of symmetry.


 * Vocabulary:**

__Polygon__- A polygon is a plane figure formed by three or more segments. Each segment intersects exactly two other segments, one at each endpoint, and no two segments with a common endpoint are collinear. __Sides of a polygon__- The segments of a polygon are called the sides of the polygon. __Vertices of the polygon__- The common endpoints of a polygon are called the vertices of the polygon. __Equiangular polygon__- A polygon in which all angle are congruent. __Equilateral polygon__- A polugon in which all sides are congruent. __Regular polygon__- A polygon that is both equiangular and equilateral. __Center of a regular polygon__- The point that is equidistant from all vertices of the polygon. __Central angle of a regular polygon__- An angle whose vertex is the center of the polygon and whose sides pass through two consecutive vertices. __Reflectional symmetry__- A figure has reflectional symmetry if and only if its reflected image across a line coinsides exactly with the preimage. __Axis of symmetry__- The line that the image is reflected across. __Triangles classified by number of congruent sides__- Three congruent sides: equilateral. At least two conggruent sides: isosceles. No congruent sides: scalene. __Rotational symmetry__- A figure has rotational symmetry if and only if it has at least one rotation image, not counting rotation images of 0° or multiples 360°, that coincides with the original image.


 * Key Concepts:**

__Polygons classified by number of sides__-
 * Triangle- 3 || Nonagon- 9 ||
 * Quadrilateral- 4 || Decagon- 10 ||
 * Pentagon- 5 || 11-gon- 11 ||
 * Hexagon- 6 || Dodecagon- 12 ||
 * Heptagon- 7 || 13-gon- 13 ||
 * Octagon- 8 || n-gon- n ||


 * **Type of triangle** || **Number of axes of symmetry** || **Number of congruent angles** ||
 * equilateral || 3 || 3 ||
 * isosceles || 1 || 2 ||
 * scalene || 0 || 0 ||

__Triangle symmetry conjecture__- An axis of symmetry in a triangle is the perpendicular bisector of the side it intersects, and it passes through the vertex of the angle opposite that side of the triangle. __The central angle of a regular polygon__- The measure of a central angle of a regular polygon with n sides is given by the following formula; 360÷n.

=Properties of Quadrilaterals=


 * Objectives:**
 * Define quadrilateral, parallelogram, rhombus, rectangle, square, and trapezoid.
 * Identify the properties of quadrilaterals and the relationships among the properties.


 * Vocabulary:**

__Quadrilateral__- Any four- sided polygon. __Trapezoid__- A quadrilateral with one and only one pair of parallel sides. __Parallelogram__- A quadrilateral with two pairs of parallel sides. __Rhombus__- A quadrilateral with four congruent sides. __Rectangle__- A quadrilateral with four right angles. __Square__- A quadrilateral with four congruent sides and four right angles.


 * Key Concepts:**

__Conjectures: properties of parallelograms__- 1. Opposite sides of a parallelogram are congruent. 2. Opposite angles of a parallelogram are congruent. 3. Diagonals of a parallelogram bisect eachother. 4. Consecutive angles of a parallelogram are supplementary. __Conjectures: properties of rhombuses__- 1. A rhombus is a parallelogram. 2. The diagonals of a rhombus are perpendicular. __Conjectures: properties of rectangles__- 1. A rectangle is a parallelogram. 2. The diagonals of a rectangle congruent. __Conjectures: properties of squares__- 1. A square is a rectangle. 2. The diagonals of a square congruent and the perpendicular bisectors of eachother.

=Parallel Lines and Transversals=


 * Objectives:**
 * Define transversal, alternate interior angles, alternate exterior angles, same- side interior angles, and corresponding angles.
 * Make conjectures and prove theorems by using postulates and properties of parallel lines ad transversals.


 * Vocabulary:**

__Transversal__- A line, ray, or segment, that intersects two or more coplaner lines, rays, or segments, each at a different point. __Alternate interior angles__- Two nonadjacent interior angles that lie on opposite sides of a transversal. __Alternate exterior angles__- Two nonadjacent exterior angles that lie on opposite sides of a transversal. __same- side interior angles__- Interior angles that lie on the same side of a transversal. __Corresponding angles__- Two nonadjacent angles, one interior and one exterior, that lie on the same side of a transversal. __Corresponding angles postulate__- If two lines cut by a transversal are parallel, then corresponding angles are congruent. __Alternate interior angles theorem__- If two lines cut by a transversal are parallel, then alternate interior angles are congruent. __Alternate exterior angles theorem__- If two lines cut by a transversal are parallel, then alternate exterior angles are congruent. __Same- side interior angles theorem__- If two lines cut by a transversal are parallel, then same- side interior angles are supplementary.


 * Key Concepts:**

__Conjectures: for two lines cut by a transversal__- 1. Alternate interior angles are 2. Alternate exterior angles are 3. Same- side interior angles are 4. Corrsponding angles are = = = = =Proving That Lines Are Parallel=


 * Objectives:**
 * Identify and use the converse of the corresponding angles postulate.
 * Prove that lines are parallel by using theorems and postulates.


 * Vocabulary:**

__Original statement__- __Converse__- __Theorem: converse of the corresponding angles postulate__- __Converse of the same- side interior angles theorem__- __Converse of the alternate interior angles theorem__- __Converse of the alternate exterior angles theorem__- __Theorems: coplaner lines__- 1. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. 2. If two lines are parallel to the same line, then the two lines are parallel to each other.

=The Triangle Sum Theorem*=


 * Objectives:**
 * Identify and use the parallel postulate and the triangle sum theorem.


 * Vocabulary:**

__The parallel postulate__- Given a line and a point not on the line, there is one and only one line that contains the given point and is parallel to the given line. __The triangle sum theorem__- The sum of the measures of the angles of a triangle is 180º. __Exterior angle theorem__- The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles. = = = = =Angles in Polygons*=


 * Objectives:**
 * Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.


 * Vocabulary:**

__Convex polygon__- A polygon in which no part of a line segment connecting any two points on the polygon is outside the polygon. __Concave polygon__- A polygon that is not convex, or a polygon in which a line segment connecting any two points in the polygon is on the outside of the polygon. __Sum of the interior angles of a polygon__- The sum of the interior angles of a polygon with n sides is (n-2) • 180°. __The measure of an interior angle of a regular polygon__- The measure of an interior angle of a regular polygon with n sides is __(n-2) • 180 °__ ..........................2

__Theorem: sum of the exterior angles of a polygon__- The sum of the exterior angles of a polygon is 360°.


 * Key Concepts:**

2 || (n-2) • 180° || 360° ||
 * **Polygon** || **Number of** **sides** || **Number of triangular regions** || **Sum of interior and exterior angles** || **Measure of one interior angle** || **Sum of interior angles** || **Sum of exterior angles** ||
 * triangle || 3 || 1 || 540° || 60 || 180° || 360° ||
 * quadrilateral || 4 || 2 || 720° || 90° || 360° || 360° ||
 * pentagon || 5 || 3 || 900° || 108° || 540° || 360° ||
 * hexagon || 6 || 4 || 1080° || 120° || 720° || 360° ||
 * n- gon || n || n-2 || n • 180 ° || __(n-2) • 180 °__

=Midsegments of Triangles and Trapezoids*=


 * Objectives:**
 * Define midsegment of a triangle and midsegment of a trapezoid.
 * Develop and use formulas based on the properties of the triangle and trapezoid midsegments.


 * Vocabulary:**

__Definition: midsegment of a triangle__- A midsegment of a triangle is a segmen whose endpoints are the midpoints of two sides. __Definition: midsegment of a trapezoid__- A midsegment of a trapezoid is a segment whose endpoints are the midpoints of the nonparallel sides.


 * Key Concepts:**

__Triangle midsegment conjecture__- A midsegment of a triangle is parallel to a side of the triangle and has a measure equal to half of that side. __Trapezoid midsegment conjecture__- A midsegment of a trapezoid is parallel to the bases of the trapezoid and has a measure equal to half of base 1 + base 2. = = = = =Analyzing Polygons with Coordinates=


 * Objectives:**
 * Develop and use theorems about equal slopes and slopes of perpendicular lines.
 * Solve problems involving perpendicular and parallel lines in the coordinate plane by using appropriate theorems.


 * Vocabulary:**

__Definition of a slope__- The slope of a nonvertical line that contains the points (**X**, **Y**) and (x, y) is equal to the ratio __y -__ **__Y__** x - **X**. __Parallel lines theorem__-Two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. __Perpendicular lines theorem__- Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Any vertical line is perpendicular to any horizantal line. __Midpoint formula__- The midpoint of a segment with endpoints (**X**, **Y**) and (x, y) has the following coordinates: __**X** + x__, __**Y** + y__ ....2.......2