Polygons and Quadrilaterals Journal Chapter 6. Polygons and Quadrilaterals Kirsten Erichsen 9-5 INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms 6-3: Conditions for Parallelograms 6-4: Properties of Special Parallelograms 6-5: Conditions for Special Parallelograms 6-6: Properties of Kites and Trapezoids

POLYGONS.

WHAT IS A POLYGON? Definition: a closed figure formed by 3 or more segments, whose endpoints have to touch another 2 endpoints. Types of Polygons (sides): Triangle = 3 Quadrilateral = 4 Pentagon = 5 Hexagon = 6 Heptagon = 7 Octagon = 8 Nonagon = 9 Decagon = 10 Dodecagon = 12 For the rest of the polygons with more sides than 12 (including 11) you just place the number and add “gon” to it = n-gon.

HOW DO YOU KNOW IT’S A POLYGON? A polygon must not have curved sides, they all have to be straight. The sides of the polygon must not intersect, meaning they must not intersect the other sides of the polygon.

Example 1. What type of polygon is this, based on its sides? HEXAGON

Example 2. What type of polygon is this, based on its sides? DODECAGON

Example 3. What type of polygon is this, based on its sides? PENTAGON

Example 4. What type of polygon is this, based on its sides? IT IS NOT A POLYGON BECAUSE IT HAS A CURVED SIDE.

PARTS OF A POLYGON. Side of a Polygon: each one of the segments that forms the sides of any polygon. Vertex of a Polygon: it is the common point where any of the 2 endpoint of the polygon meet. Diagonal: a segment that connects 2 non-consecutive vertices. Diagonal Vertex Side

Example 1. Tell me the parts of the polygon in this shape. Diagonal Vertex Side

Example 2. Tell me the parts of the polygon in this shape. Side Diagonal Vertex

Example 3. Tell me the parts of the polygon in this shape. Vertex Side Diagonal

CONVEX AND CONCAVE. Convex Polygons: If the polygon contains all of the angles facing the outside (exterior), then it is considered convex. Concave Polygons: If any of the angles in the polygon face the inside of the shape.

Example 1. Identify the polygon as a convex or concave and name the polygon by its sides. CONCAVE, 11-gon

Example 2. Identify the polygon as a convex or concave and name the polygon by its sides. CONCAVE, DODECAGON

Example 3. Identify the polygon as a convex or concave and name the polygon by its sides. CONVEX, QUADRILATERAL

REGULAR POLYGONS. Regular Polygon: a polygon that is both equilateral or equiangular. EQUILATERAL: All of the sides are congruent in the polygon. EQUIANGULAR: All of the angles are congruent in the polygon. Each side is 4 inches. Each angle is 90°.

Example 1. Classify the polygon as equilateral or equiangular. Equiangular, because each angle measures 90°.

Example 2. Classify the polygon as equilateral or equiangular. 5 cm. Equilateral, because each side measures 5 centimeters. 5 cm. 5 cm. 5 cm.

Example 3. Classify the polygon as equilateral or equiangular. 6 cm. Both equilateral and equiangular because the sides measure 6 centimeters and the angles measure 90°. 6 cm. 6 cm. 6 cm.

INTERIOR ANGLES THEOREM. This theorem states that the sum of the interior angle measures of a regular, convex polygon follows the equation of (n-2)180°. After using the equation above, you have to divide by the number of sides of the polygon to get the angle measures. It is also called Theorem 6-1-1 or Polygon Angle Sum Theorem.

Example 1. Find the interior angle measures of a regular decagon using the Interior angles theorem. (n − 2)180° (10 − 2)180° 8 × 180° = 1440 1440 ÷ 10 = 144° Each angle measures 144°.

Example 2. Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (12 − 2)180° 10 × 180° = 1800 1800 ÷ 12 = 150° Each angle measures 150°.

Example 3. Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (8 − 2)180° 6 × 180° = 1080 1080 ÷ 8 = 135° Each angle measures 135°.

Example 4. Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (5 − 2)180° 3 × 180° = 540 540 ÷ 5 = 108° Each angle measures 108°.

EXTERIOR ANGLES THEOREM. This theorem states that the sum of the exterior angles of a regular, convex polygon is always going to be 360°. To find the angle measurements you just divide 360° by the number of sides. This theorem is also called 6-1-2 (Polygon Exterior Angle Sum Theorem).

Example 1. Find the exterior angle measures of a regular decagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 10 = 36° Each angle measures 36°.

Example 2. Find the exterior angle measures of a regular hexagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 6 = 60° Each angle measures 60°.

Example 3. Find the exterior angle measures of a regular dodecagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 12 = 30° Each angle measures 30°.

PARALLELOGRAM THEOREMS. Parallelogram: a quadrilateral with two pairs of opposite sides.

THEOREM ONE (6-2-1) If a quadrilateral is a parallelogram, then its opposite sides are congruent. CONVERSE: If its opposite sides are congruent, then the quadrilateral is a parallelogram. This means that the quadrilateral has to have a pair of congruent sides to be considered a parallelogram. It is considered a parallelogram if the sides are congruent and parallel.

Example 1. This is a parallelogram because the opposite sides are congruent.

Example 2. This is a parallelogram because the opposite sides are congruent.

Example 3. This is not a parallelogram because we don’t know it the opposite sides are congruent.

THEOREM TWO (6-2-2) If a quadrilateral is a parallelogram, then its opposite angles are congruent. CONVERSE: If the opposite angles are congruent, then the quadrilateral is considered a parallelogram. This theorem is used only when the parallelogram is proved to be when it has 2 pairs of congruent angles.

Example 1. This is a parallelogram because the opposite angles are congruent.

Example 2. This is a parallelogram because the opposite angles are congruent. 55° 125° 55° 125°

Example 3. This is not a parallelogram because the opposite angles are not congruent and they do not add up to 360°. 55° 125° 80° 130°

THEOREM THREE (6-2-3) If a quadrilateral is a parallelogram, then its consecutive angles or same side angles are supplementary. CONVERSE: If the consecutive or same side angles are supplementary, then it is considered a parallelogram. It is used when the same side angles add up to 180° to make up a linear pair.

Example 1. This is a parallelogram because the same side angles add up to 180° and both pairs add up to 360°. 120° 60° 60° 120°

Example 2. This is a parallelogram because the consecutive angles add up to 180° and both linear pairs add up to 360°. 55° 125° 55° 125°

Example 3. This is not considered a parallelogram because one pair of consecutive angles does not add up to 180°. 55° 125° 55° 130°

THEOREM FOUR (6-2-4) If a quadrilateral is a parallelogram, then its diagonals bisect each other. CONVERSE: If the diagonals bisect each other, then it is considered to be a parallelogram. The diagonals have to bisects right through the middle of reach other (meaning the midpoint).

Example 1. This is a parallelogram because the diagonals bisects at the midpoints. 5 cm. 7 cm. 7 cm. 5 cm.

Example 2. This is a parallelogram because the diagonals bisect exactly at the midpoints. 8 cm. 9 cm. 9 cm. 8 cm.

Example 3. This is not considered a parallelogram because both of the diagonals do not intersect at the midpoint. 9.5 cm. 7 cm. 7.5 cm. 9 cm.

PROOVING PARALLELOGRAMS

REMEMBER ABOUT PARALLELOGRAMS. Both pairs of opposite sides are parallel. One pair of opposite sides are parallel and congruent. Both pairs of opposite sides are congruent. Both pairs of angles are congruent. One angle is supplementary to both of the consecutive angles. The diagonals bisect each other.

THEOREM 6-3-1 If one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram. Tell me if it is a parallelogram and why. Yes Yes No

THEOREM 6-3-2 If both pairs of opposite sides of a quadrilateral are congruent, then it is considered a parallelogram. Tell me if it is a parallelogram and why. Yes Yes No

THEOREM 6-3-3 If both pairs of opposite angles of a quadrilateral are congruent, then it is considered a parallelogram. Tell me if it is a parallelogram and why. Yes Yes No

THEOREM 6-3-4 If an angle is supplementary to both of its consecutive angles, then it is a parallelogram. Tell me if it is a parallelogram and why. 130° 50° 110° 70° YES 130° 50° YES 130° 40° NO 70° 110° 40° 130°

THEOREM 6-3-5 If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram. Tell me if it is a parallelogram and why. 4.5 YES 5 5 4.5 3 4 2 NO 4 YES 3 4 5 3

EXAMPLE 1. Prove that JKLM is a parallelogram. K L 2 1 3 4 J M JK ≅ LM, KL ≅MJ KM ≅ KM Triangle MJK ≅ KLM <1 ≅ <3, <2 ≅ <4 JK II LM, KL II JM JKLM is a Parallelogram 1. Given 2. Reflexive Property 3. SSS 4. Alternate Interior Angles 5. Theorem 6-3-1 6. Definition of Parallelogram

EXAMPLE 2. Prove that ABCD is a parallelogram. C B 1 2 4 3 D A AB ≅ CD, AD ≅ CB AC ≅ AC Triangle ABC ≅ CDA <1 ≅ <3, <2 ≅ <4 AB II CD, AD II CB JKLM is a Parallelogram 1. Given 2. Reflexive Property 3. SSS 4. CPCT 5. Theorem 6-3-1 6. Definition of Parallelogram

EXAMPLE 3. Prove that WXYZ is a parallelogram. W Z 1 2 4 3 X Y WZ ≅ XY, WX ≅ ZY WY ≅ WY Triangle WXY ≅ YZW <1 ≅ <3, <2 ≅ <4 WZ II XY, WX II ZY WXYZ is a Parallelogram 1. Given 2. Reflexive Property 3. SSS 4. CPCT 5. Theorem 6-3-1 6. Definition of Parallelogram

RHOMBUSES AND THEOREMS.

WHAT IS A RHOMBUS? It is another special quadrilateral. Definition: it is a quadrilateral with 4 congruent sides. A rhombus

THEOREM 6-4-3 A I H YES If a quadrilateral is a rhombus, then it is a parallelogram. YES D B J K C N Q NO No because we don’t know if the other 2 sides are congruent to the other pair. O P

THEOREM 6-4-4 A I H If a parallelogram is a rhombus, then its diagonals are perpendicular. D B J YES YES K C NO No because we don’t know if the bisectors actually bisect.

THEOREM 6-4-5 A I H If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. D B J YES YES K C NO No because we only know that one pair of angles are being bisected.

SQUARES AND THEOREMS.

WHAT IS A SQUARE? It is another special quadrilateral. Definition: it is a quadrilateral with 4 right angles and 4 congruent sides. A square has the properties of a parallelogram, a rectangle and a rhombus.

EXAMPLE 1. Do you consider this a square? No because we only know that it has 4 right angles and no 4 congruent sides.

EXAMPLE 2. Do you consider this a square? Yes because we have the 4 right angles and the 4 congruent sides.

EXAMPLE 3. Do you consider this a square? No because we only know that there are 4 congruent sides, but we don’t know anything about the angles.

RECTANGLES AND THEOREMS.

WHAT IS A RECTANGLE? It is another special quadrilateral. Definition: it is a quadrilateral or parallelogram with 4 right angles. The diagonals in a rectangle are congruent.

THEOREM 6-4-2 NO If a parallelogram is a rectangle, then its diagonals are congruent. Tell me if these are rectangles. YES NO 3 4 4 3

THEOREM 6-5-1 YES If one angle is a parallelogram is a right angle, then the parallelogram is a rectangle. Tell me if these are rectangles. YES NO 3 4 4 3

RECTANGLES, RHOMBUS AND SQUARES All of these shapes are quadrilaterals and parallelograms. A rectangle has the properties of a parallelogram and a few more. A rhombus has the properties of a rectangle, parallelogram and its individual properties. A square has the properties of a rectangle, parallelogram, rhombus and a pair of its own properties.

TRAPEZOIDS AND THEOREMS.

WHAT IS A TRAPEZOID? Definition: it is a quadrilateral with exactly one pair of parallel sides. Isosceles Trapezoid

PARTS OF A TRAPEZOID. Base Base: each one of the parallel sides of the trapezoid. Legs: the pair of non- parallel sides. Base Angles: they are 2 consecutive angles whose the common side is a base. Base Angles Legs Base Angles Base

ISOSCELES TRAPEZOID. Definition: it is when the legs of the trapezoid are congruent.

THEOREM 6-6-3 If a quadrilateral is an isosceles, then each pair of base angles are congruent.

EXAMPLES. Tell me if each one of the isosceles trapezoid are really isosceles. NO YES YES

THEOREM 6-6-5 A trapezoid is isosceles only if its diagonals are congruent.

EXAMPLES. Tell me if each one of the isosceles trapezoid are really isosceles. 5 11 NO YES 7 7 YES 4 11

TRAPEZOID MISEGMENT THEOREM The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the bases. LM = 1/2 (HI + KJ) H I L M K J

EXAMPLE 1. Find the missing measurement. The distance between EF and DC is 6 cm, so since EF is in the middle of the trapezoid you subtract 6 from 32. AB = 26 cm. A B 26 cm. 32 cm. E F D 38 cm. C

EXAMPLE 2. Find the missing measurement. The distance between GH and JK is 8 cm, so since LI is in the middle of the trapezoid you divide 8 by 2, which is 4. Then you add 4 to 16, or subtract 4 from 24. LI = 20 cm. G H 16 cm. 20 cm. L I K 24 cm. J

EXAMPLE 3. Find the missing measurement. The distance between MN and RO is 7 cm, so since QP is at the bottom of the trapezoid you add 7 to 13 cm or RO. PQ = 20 cm. M N 6 cm. 13 cm. R O Q 20 cm. P

KITES AND THEOREMS.

WHAT IS A KITE? Definition: it is a quadrilateral with exactly two pairs of congruent consecutive sides. It has 2 pairs of congruent adjacent sides, diagonals are perpendicular, and it has non- congruent adjacent angles

THEOREM 6-6-1 B If a quadrilateral is a kite, then its diagonals are perpendicular. A C D

EXAMPLES. Tell me if the following kites are really showing that they are kites. E 8 8 6.5 6.5 D F 3.5 2.5 NO YES YES G

THEOREM 6-6-2 B If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. NOTE: only one of the perpendicular lines is being bisected. A C D

EXAMPLES. Tell me if the following kites are really showing that they are kites. E 35° 95° 95° 110° D 110° F 90° NO YES YES G