Geometry Unit 5 - Notes Polygons
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1 Geometry Unit 5 - Notes Polygons Syllabus Objective: The student will differentiate among polygons by their attributes. Review terms: 1) segment 2) vertex 3) collinear 4) intersect Polygon- a plane figure bounded by a closed path, composed of a finite sequence of straight line segments. These segments are called its sides, and the points where two edges meet are the polygon's vertices. Polygon Parts Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon. Polygon Type Number of Sides Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Hendecagon 11 Dodecagon 12 n-gon n Unit 5 Polygons Page 1 of 20
2 *Take the opportunity to relate parts of the polygon name to real life objects and concepts. For instance, a pentagon has five sides. Show a picture of the Pentagon. The building is called the Pentagon due to its shape. Show a picture of an octopus. It has eight legs just like an octagon has eight sides. A decade is ten years and a decagon has ten sides. A tricycle has three wheels just like the triangle has three sides! A quadrilateral has four sides just like a four-wheeler is called a quad for short. onvex Polygon - a polygon for which there is no line containing a side of the polygon that also contains a point in the interior of the polygon. Polygon AD is a convex quadrilateral. A Interior D oncave Polygon - a polygon for which there is a line containing a side of the polygon that also contains a point in the interior of the polygon. Polygon EFGHIJKL is a concave octagon. L E F Interior G H I I always draw a teddy bear hiding in the cave. I m not a good artist, but they remember. Probably because I m so bad!! K J Examples: Determine which of the following figures is a polygon. If it is a polygon, state whether it is a convex or concave polygon. A D E Solutions: A and are polygons. A is concave and is convex. is not a polygon because some of its sides intersect more than two other sides. D is not a polygon because it has a side that is not a segment. E is not a polygon because it is not closed, two of its sides intersect only one other side. Unit 5 Polygons Page 2 of 20
3 Equilateral - a polygon with all sides congruent to each other. Equiangular - a polygon with all of its interior angles congruent to each other. Regular - a polygon that is both equilateral and equiangular. Examples: Determine if the polygon is convex or concave; appear to be equiangular, equilateral, or regular and state the name of the polygon. A Solutions: A is an equilateral, convex pentagon. It has five segments that are congruent to each other, but it is not equiangular. Therefore, it is not a regular polygon. is equiangular and equilateral. Thus, it is a regular, convex triangle. is an equilateral, concave decagon. It is not equiangular, so it is not a regular decagon. Syllabus Objectives: The student will classify polygons by their special properties. Diagonal - a segment that joins two nonconsecutive vertices in a polygon. The green (internal) segments in the three polygons above are examples of diagonals. Interior Angles of a Quadrilateral Theorem: The sum of the four angles of any quadrilateral is 360. Unit 5 Polygons Page 3 of 20
4 Types of Quadrilaterals: Quadrilateral Parallelogram Trapezoid Kite Rectangle Rhombus Isosceles Trapezoid Square Quadrilateral four sided polygon When referring to the flow chart above, the quadrilateral at the top center is a four-sided polygon. The four polygons on the left side all have opposite sides parallel and congruent. Therefore, the parallelogram, rectangle, rhombus, and square are all parallelograms. All parallelograms have opposite sides congruent, opposite angles are congruent, the diagonals bisect each other, and consecutive angles are supplementary. The quadrilaterals on the right side are not parallelograms. The trapezoid is a quadrilateral that has one pair of opposite sides parallel. If the legs, or base angles, are congruent, then it is an isosceles trapezoid. {Link concept to isosceles triangles.} The diagonals of an isosceles trapezoid are congruent just like those of the rectangle; however they do not bisect each other. The diagonals of a kite are perpendicular like those in the rhombus; however they do not bisect each other. Only one pair of opposite angles is congruent to each other. The other pair of opposite angles is bisected by the diagonal that passes through them. ************************************************************************************ Parallelogram A quadrilateral with both pairs of opposite sides parallel. The parallelogram symbol is. Properties of eing a Parallelogram Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent. {Students often confuse this with the definition of parallelogram. The definition (above) does not indicate congruence of sides.} Theorem: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other. Unit 5 Polygons Page 4 of 20
5 ************************************************************************************ Rectangle A parallelogram with four right angles. Properties of eing a Rectangle The rectangle is equiangular. Its diagonals not only bisect each other but they are also congruent to each other. ************************************************************************************ Rhombus A parallelogram with four congruent sides. Properties of eing a Rhombus The rhombus is equilateral. Its diagonals not only bisect each other but they intersect perpendicularly. The diagonals of a rhombus bisect a pair of opposite angles. ************************************************************************************ Square A parallelogram with four congruent sides and four right angles. Properties of eing a Square The square is a regular quadrilateral. It has all the properties of the quadrilateral, parallelogram, rectangle, and rhombus. ************************************************************************************ Trapezoid A quadrilateral with exactly one pair of parallel sides, called bases. The nonparallel sides are legs. Midsegment of a trapezoid a segment that connects the midpoints of the legs of a trapezoid. The midsegment of a trapezoid is half the sum of the bases. {Link concept to the midsegment of a triangle, imagine a trapezoid with one base length equal to 0. What would be the best name for that figure? A triangle.} Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. ************************************************************************************ Kite a quadrilateral that has two pairs of consecutive congruent sides, but in which opposite sides are not congruent. In a kite, the diagonals are perpendicular, one pair of opposite angles are congruent and the other pair are bisected by the diagonal. ************************************************************************************ Unit 5 Polygons Page 5 of 20
6 *elow is a table that shows the different types of quadrilaterals and the properties that they have. Property Quad ogram Rectangle Rhombus Square Trapezoid Isos. Trap. Kite 4 sides X X X X X X X X Interior angle sum 360. oth pairs of opp. sides. Exactly 1 pair of opp. sides. X X X X X X X X X X X X oth pairs of opp. sides. X X X X Exactly 1 pair of opp. sides. All sides are. X X All angles are. X X oth pairs of opp. angles are. X X X X Exactly 1 pair of X opp. angles is. Diagonals are. X X X Diagonals are. X X X Diagonals bisect each other. X X X X X X X Given: RS TW Prove: ΔRST ΔTWR R 4 2 W S 1 3 T Statements 1) RSTW is a ogram 1) Given 2) RS WT 2) Def. of ogram Reasons 3) 1 2 3) 2 lines cut by trans., alt. int. 's 4) RT RT 5) RW ST 4) Reflexive 5) Def. of ogram 6) 3 4 6) 2 lines cut by trans., alt. int. 's 7) ΔRST ΔTWR 7) ASA Unit 5 Polygons Page 6 of 20
7 Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more relationships. o The opposite sides of a parallelogram are congruent. o The opposite angles of a parallelogram are congruent. oth of those can be proven by adding PT to the last proof. Many students mistakenly think the opposite sides of a parallelogram are equal by definition. That s not true, the definition states the opposites sides are parallel. This proof allows us to show they are also equal or congruent. **Inner Rectangle** The Problem: Let AD be a convex quadrilateral and let the angle bisectors of the angles at A,,, D form the sides of a second quadrilateral PQRS. What can you say about the original quadrilateral AD, if PQRS is to be a rectangle? Solution: AD must be a parallelogram. For suppose PA, P are the bisectors of the angles at A and. Then m AP = 90 if and only if m PA + m PA = 90,that is, if and only if 2m PA + 2m PA 2( 90 ) = m DA + m A = 180, which in turn happens if and only if the lines DA and are parallel. Similarly at the other vertices. Syllabus Objective: The student will find the sum of the measures of the interior angles of a polygon. Review the Triangle Sum Theorem. {Show students how to derive the formula from constructing diagonals in a polygon from a single vertex and applying the Triangle Sum Theorem.} Polygon Interior Angle Theorem: The sum of the measures of the interior angles of a convex n-gon is (n - 2)(180 ). Unit 5 Polygons Page 7 of 20
8 orollary to Polygon Interior Angle Theorem: The measure of each interior angle of a regular ( n ) n-gon is n Table of Polygon values: Polygon Type Number of Sides (n) Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Number of s Inside the Polygon Sum of the Measures of the Interior Angles 1(180 ) = 180 2(180 ) = 360 3(180 ) = 540 4(180 ) = 720 Heptagon 7 5 5(180 ) = 900 Octagon 8 6 6(180 ) = 1080 Nonagon 9 7 7(180 ) = 1260 Decagon (180 ) = 1440 Hendecagon (180 ) = 1620 Dodecagon (180 ) = 1800 n-gon n n - 2 (n - 2)(180 ) Measure of each Interior Angle (regular polygon) 1(180 ) 3 2(180 ) 4 3(180 ) 5 4(180 ) 6 5(180 ) 7 6(180 ) 8 7(180 ) 9 8(180 ) 10 9(180 ) 11 10(180 ) 12 (n - 2)(180 ) n Example: Find the measure of each interior angle. First, the sum of the interior angles is found to be (3)180 = 540. x 8 + 3x 11 + x x + 2x + 7 = 540 U (x - 8) 8x 4 = 540 (3x - 11) W (x + 8) 8x = 544 x = 68 (2 x + 7) Solutions: m U = 60, m V = 193, m W = 76, m Y = 68, and m Z = 143. Z V x Y Unit 5 Polygons Page 8 of 20
9 Syllabus Objective: The student will solve problems involving properties of special quadrilaterals. Review the special properties of all quadrilaterals and special angles created when parallel lines are cut by a transversal. Example: Find the value of each variable in a parallelogram. a x b y x =? y =? a =? b =? Solutions: x = 11 y = 8 a = 85 b= 95 Example: Find the angle measurements of trapezoid AD. A D {Review AIA, AEA, IA, and A.} D 45 A 60 m DA =? m A=? Solutions: m DA = 135 m A= 120 Example: If KLMN is a parallelogram, find the value of x and y. Opp. sides, so 9x = 36 x = 4. Diag. bisect each other, so 3y 7 = y y = 22 y = 11 K 36 y+15 3y-7 J L 9x N M **The Sliced ube** The picture shows a cube of side length 2 units. M and N are the mid-points of sides DE and G. Unit 5 Polygons Page 9 of 20
10 The Problem: What sort of shape is ANHM? alculate the lengths of AH and MN. Any comment about your previous answer now? Find the area of the shape ANHM. Solution: This concerns the equal-diagonal property of a square. learly all four sides of the required shape are equal in length, so the choice is between a square (the obvious choice!) and a rhombus. We can use Pythagoras to find that AH is 2 3units in length, while MN is 2 2units in length. ANHM cannot be a square if it has unequal diagonals, so it must be a rhombus. Finally, the area of a rhombus is half the product of its diagonals = 2 6 units². Syllabus Objective: The student will write proofs involving special quadrilaterals. F G Example: Write a two-column proof. Given: FGHJ Prove: F H, J G J H Statements 1) FGHJ 1) Given Reasons 2) FG JH, FJ GH 2) Def. of Parallelogram F and J are supplementary, 3) lines cut by a transversal, IA 3) J and H are supplementary, are supp. H and G are supplementary 4) F H, J G 4) supp. Th. Syllabus Objective: The student will solve problems involving properties of polygons. Review the Types of Quadrilaterals chart from p. 5 of these notes. Reasoning with Quadrilaterals The flow chart on page 3 of these notes can help students answer these. Movement from the bottom towards the top will always result in a true statement. Movement from the top towards the bottom will only sometimes be true. Finally, movement sideways across the chart is never true. Unit 5 Polygons Page 10 of 20
11 Examples: Decide if the statement is Always, Sometimes, or Never true. a) A rhombus is a kite. a) Never b) A parallelogram is a rectangle. b) Sometimes c) A square is a quadrilateral. c) Always Syllabus Objective: The student will perform constructions involving polygons. (supplemental material required) Example: onstruct an equilateral triangle. Mark a center, O, and draw a circle with a given radius. Draw the diameter, A. Using A as the center and AO as the radius, draw an arc that intersects the circle at and D. onnect,, and D to form a triangle. Example: onstruct a square. Mark a center, O, and draw a circle with a given radius. Draw the diameter, A. Draw another diameter, D, which is the perpendicular bisector of A. onnect A, D,, and to form a square. Syllabus Objective: The student will find the measure of interior, exterior, and central angles of a given regular polygon. O A A O D D Discuss how to find the measure of each central angle. Guide students into examples that show how the central angles create a full circle, so you can divide 360 by how many central angles the polygon has. Show an example with a triangle and have students create and measure exterior angles. Total the three angles up and then do another example with a rectangle. Students will be able to come to the conclusion that the sum of the exterior angles of a polygon is 360. ompare and contrast interior, exterior, and central angle measurements of polygons in general. It is very important to stress that students read carefully to understand exactly what they are being asked to find; a total or a single angle measurement. They must also be aware if the polygon is regular or not. Unit 5 Polygons Page 11 of 20
12 Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360. I always point out what a hard formula this one is! orollary to Polygon Exterior Angles Theorem: The measure of each exterior angle of a regular n-gon is 360 n. Or 180 ( interior angle) It depends what you already know! There s no reason to make it harder than necessary. entral angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. The central angle of a regular polygon can be found by 360 n. What if students are told the measure of an interior angle and asked to find how many sides the polygon has? Example: The measure of an interior angle of a regular polygon is 156. Find the number of sides in the polygon. ( n ) 156 = n = ( n 2) n = 180n n = 360 n n =15. OR If an interior angle of a regular polygon is 156, then the exterior angle will measure 24. Since the sum of the exterior angles will measure 360 and all exterior angles are equal, one can divide 360 by 24 to get an answer of 15 sides. Syllabus Objective: The student will utilize the distance, slope, and midpoint formulas to classify a given quadrilateral. Review: 1) distance formula 2) slope formula 3) midpoint formula Students must be told that although a graph may look like a particular quadrilateral, it should only give them a hint of the direction their work should follow. There MUST be mathematical proof to justify their answer. Review the distance, slope, and midpoint formulas. Also, discuss how you know if two lines are parallel. Unit 5 Polygons Page 12 of 20
13 **What conditions are required to prove that a given quadrilateral is a parallelogram?** Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Rhombus orollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle orollary: A quadrilateral is a rectangle if and only if it has four right angles. Square orollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle. **Ok, I know I have a parallelogram. How do I get more specific?** Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent. Unit 5 Polygons Page 13 of 20
14 **Now I don t have a parallelogram. an I classify these quadrilaterals?** Theorem: If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. Theorem: A trapezoid is isosceles if and only if its diagonals are congruent. Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. Example 12: Use the figure below to prove that quadrilateral AD is an isosceles trapezoid, given that AD. 4 2 D A - - Proof: It must be shown that two sides are parallel and the two legs are congruent. Given AD, it must only be proven that A D but not parallel. egin by identifying the ordered pair for each point on quadrilateral AD. A(-1,-1), (-3,2), (-1,4), and D(2,2). Next, use the distance formula to determine the length of A and D. Use the slope formula to show that the two segments have different slopes. Unit 5 Polygons Page 14 of 20
15 A = ( x x ) + ( y y ) = ( 3 ( 1)) (2 ( 1)) = = 13. D = ( x x ) + ( y y ) = (2 ( 1)) + (2 4) = 9+ 4 = 13. Therefore, A = D, or A D! y y Slope of A = 2 1 x x 2 1 = 2 ( 1) 3 ( 1) 3. = 2 y y 2 4 Slope of D = 2 1 = = x x 2 ( 1) Since they have different slopes, the two segments will intersect. It has now been shown that there is exactly one pair of opposite sides parallel and that the legs, A and D, are congruent. It can now be determined that quadrilateral AD is an isosceles trapezoid. Unit 5 Polygons Page 15 of 20
16 This unit is designed to follow the Nevada State Standards for Geometry, SD syllabus and benchmark calendar. It loosely correlates to hapter 6 of McDougal Littell Geometry 2004, sections & The following questions were taken from the 1 st semester common assessment practice and operational exams for and would apply to this unit. Multiple hoice # Practice Exam (08-09) Operational Exam (08-09) 19. Use the triangle below. Use the triangle below. 45 x ( 3x + 3) What is the value of x? A D How many sides does a nonagon have? A D Which figure is a polygon? ( 3x + 10) What is the value of x? A D. 70 How many sides does a heptagon have? A D. 17 Which figure is a polygon? x 70 A. A..... D. D. Unit 5 Polygons Page 16 of 20
17 38. A hexagon is shown below. a What is the value of a? A D Use the figure below. 130 x 40 What is the value of x? A D Parallelogram AD is given below. A 11x + 9 D 6(x + 4) What is the value of x? A D What is the measure of each exterior angle of a regular hexagon? A D A pentagon is shown below. What is the value of a? A D. 150 The figure below is a hexagon with an external angle. x What is the value of x? A D. 160 Rectangle AD is given below. A 10x 4 D 4x + 8 What is the value of x? A a D. What is the measure of each exterior angle of a regular octagon? A D Unit 5 Polygons Page 17 of 20
18 42. Which statement is true about a kite? A. A kite has 4 congruent sides.. A kite has 2 pairs of parallel sides.. A kite has perpendicular diagonals. D. A kite has congruent diagonals. 43. Which statement below is true about an isosceles trapezoid? A. oth pairs of opposite sides are parallel.. oth pairs of opposite sides are congruent.. One pair of opposite sides is congruent and the other is parallel. D. One pair of opposite sides is both parallel and congruent. 44. In the figure below, ΔKLM ΔA. L A Which statement is true about a trapezoid? A. A trapezoid has 1 acute angle and 3 obtuse angles.. A trapezoid has 2 pairs of parallel sides and 4 sides.. A trapezoid has 4 right angles and 2 pairs of parallel sides. D. A trapezoid has 4 sides and exactly 1 pair of parallel sides. Which statement below is true? A. All rhombi are squares.. All squares are rectangles.. All quadrilaterals are parallelograms. D. All rectangles are rhombi. In the figure below, ΔA Δ XYZ. X 8 cm 4 cm 80 K cm M 53 A 45 5 cm Y Which statement must be true? A. A = 8cm. = 6cm. m A= 53 D. m = Use the rhombus below. 65 A Which statement must be true? A. m X = 45. m Z = 45. YZ = 3cm D. XY = 3cm In trapezoid GHIJ below, GH JI. G H ( x + 10) Z E What is m DE? A D. 115 D x J What is the value of x? A D. 105 I Unit 5 Polygons Page 18 of 20
19 46. A regular polygon has interior angles that measure 144. How many sides does this polygon have? A D Use the figure below. 75 A regular polygon has interior angles that measure 150. How many sides does this polygon have? A D. 10 Use the figure below. x 41 x What is the value of x? A D. 126 What is the value of x? A D. 57 Unit 5 Polygons Page 19 of 20
20 Free Response Practice Exam (08-09) 3. Show that the quadrilateral QUAD, having vertices Q( 1, 1), U(3,2), A(6, 2), and D(2, 5); is a square. Operational Exam (08-09) 3. Prove that the quadrilateral QUAD, having vertices Q( 1,0), U( 3,6), A(0,7), and D(2,1) is a rectangle. (A blank coordinate grid is provided.) y x Unit 5 Polygons Page 20 of 20
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