Parallels and Polygons By: Carly Salzberg and Courtney Marsh

Symmetry in Polygons 3.1 Polygon- a plane figure formed from 3 or more segments such that each segment intersects exactly two other segments, one at each endpoint, and no two segments with a common endpoint are collinear. Equiangular polygon- one in which all angles are congruent. Equilateral polygon- one in which all sides are congruent Regular Polygon- one that is both equiangular and equilateral. Center of a regular polygon- the point that is equidistant from all vertices of a polygon. Central angle- an angle whose vertex is the center of the polygon and whose sides pass through two consecutive vertices of a regular polygon.

3.1 continued Reflectional symmetry- a figure has reflectional symmetry if and only if its reflected image across a line coincides exactly with the preimage. This line is called the axis of symmetry. 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 of 360°, that coincides with the original image.

3.1 continued Polygons Classified by Number of Sides Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides 11-gon – 11 sides Dodecagon - 12 sides 13-gon – 13 sides N-gon – n sides Triangles Classified by Number of Congruent Sides Equilateral- three congruent sides Isosceles- at least two congruent sides Scalene- no congruent sides To Find The Central Angle 360÷n (n = the number of sides the polygon has.)

Properties of Quadrilaterals 3.2 Quadrilateral- any four sided polygon. Parallelogram- a quadrilateral with two pairs of parallel sides. Conjectures: 1) Opposite sides are congruent. 2) Opposite angles are congruent. 3) Consecutive angles are supplementary. 4) Diagonals bisect each other.

3.2 continued Rhombus- a quadrilateral with 4 congruent sides. Conjectures: 1) A rhombus is a parallelogram. 2) Diagonals are perpendicular. Rectangle- a quadrilateral with 4 right angles. Conjectures: 1) A rectangle is a parallelogram. 2) Diagonals are congruent.

3.2 continued Square- a quadrilateral with 4 congruent sides and 4 right angles. Conjectures: A square is a parallelogram, rectangle, and rhombus. Diagonals bisect each other, are perpendicular, and are congruent.

3.2 continued Trapezoid- a quadrilateral with one and only one pair of parallel sides. Isosceles

3.2 continued Kite- a quadrilateral with 2 pairs of consecutive congruent sides. Conjectures: Diagonals are perpendicular. One pair of opposite angles are congruent.

Parallel Lines and Transversals 3.3 Transversal- a line, ray, or segment that intersects 2 or more coplanar lines, rays, or segments, each at a different point. Postulates and Theorems: Corresponding Angles Postulate- if 2 lines cut by a transversal are parallel, then corresponding angles are congruent. Alternate Interior Angles Theorem- if 2 lines cut by a transversal are parallel, then alternate interior angles are congruent. Alternate Exterior Angles Theorem- if 2 lines cut by a transversal are parallel, then alternate exterior angles are congruent. Same-Side Interior Angles Theorem- if 2 lines cut by a transversal are parallel, then same-sides interior angles are supplementary.

3.3 continued Alternate Interior Angles: Angles 3 and 6; Angles 4 and 5 Alternate Exterior Angles: Angles1 and 8; Angles 2 and 7 Same-Side Interior Angles: Angles 3 and 5; Angles 4 and 6 Corresponding Angles: Angles 1 and 5; Angles 3 and 7; Angles 2 and 6; Angles 4 and 8

Proving That Lines Are Parallel 3.4 Theorems: Converse of the Corresponding Angles Postulate: if two lines are cut by a transversal in such a way that corresponding angles are congruent, then the two lines are parallel. Converse of the Same-Side Interior Angles Theorem: if two lines are cut by a transversal in such a way that same- side interior angles are supplementary, then the two lines are parallel. Converse of the Alternate Interior Angles Theorem: if two lines are cut by a transversal in such a way that alternate interior angles are congruent, then the two lines are parallel.

3.4 continued Theorems: Converse of the Alternate Exterior Angles Theorem: if two lines are cut by a transversal in such a way that alternate exterior angles are congruent, then the two lines are parallel. If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. If two coplanar lines are parallel to the same line, then the two lines are parallel to each other.

Questions What is the central angle for a hexagon? Decide whether the statement is true or false. If the statement is false, give a counterexample. 2a. If a figure is a parallelogram, then it cannot be a rectangle. 2b. If a figure is a parallelogram, then it cannot be a trapezoid. 2c. If a figure is a square, then it is a rhombus. 2d. If a figure is a rhombus, then it cannot be a rectangle.

Questions Indicate whether the pairs below are alternate interior, alternate exterior, same-side interior, or corresponding angles. (Diagram on next slide) a. <1 and <8 b. <7 and <3 c. <5 and <4 d. <3 and <5

Questions

StatementsReasons <1 and <2 are supplementary a. ? M<1 +m<2 = 180°b. ? M<1 + m<3 = 180°c. ? M<1 + m<2 = m<1 + m<3d. ? M<2 = m<3e. ? L1IIL2F. ?

Answers /n  360/6 = 60° 2. A. false; rectangle B. true C. true D. false; square 3. A. alternate exterior angles B. corresponding angles C. alternate interior angles D. same-side interior angles

Answers 4. A. Given B. Definition of supplementary angles C. Linear Pair D. Transitive Property E. Subtraction F. Converse of Corresponding Angles Postulate