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:

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 °
2
(n-2) • 180°
360°

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