Unit 6 Introduction to Polygons - Houston County High School
Unit 6 Introduction to Polygons This unit introduces Polygons. It defines polygons and regular polygons, and has the Polygon Angle Sum theorem. This unit also details quadrilaterals, special quadrilaterals, congruent polygons, similar polygons, and the Golden Ratio.
Standards SPIs taught in Unit 6: SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. SPI 3108.1.2 Determine areas of planar figures by decomposing them into simpler figures without a grid. SPI 3108.3.2 Use coordinate geometry to prove characteristics of polygonal figures. SPI 3108.4.3 Identify, describe and/or apply the relationships and theorems involving different types of triangles, quadrilaterals and other polygons. SPI 3108.4.7 Compute the area and/or perimeter of triangles, quadrilaterals and other polygons when one or more additional steps are required (e.g. find missing dimensions given area or perimeter of the figure, using trigonometry). SPI 3108.4.12 Solve problems involving congruence, similarity, proportional reasoning and/or scale factor of two similar figures or solids. CLE (Course Level Expectations) found in Unit 6: CLE 3108.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies. CLE 3108.4.2 Describe the properties of regular polygons, including comparative classification of them and special points and segments. CLE 3108.4.6 Generate formulas for perimeter, area, and volume, including their use, dimensional analysis, and applications. CFU (Checks for Understanding) applied to Unit 6: 3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized vocabulary (e.g. graphing calculators, interactive geometry software such as Geometers Sketchpad and Cabri, algebra tiles, pattern blocks,
tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polyhedrons, measurement tools, compasses, PentaBlocks, pentominoes, cubes, tangrams). 3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems. 3108.4.4 Describe and recognize minimal conditions necessary to define geometric objects. 3108.4.9 Classify triangles, quadrilaterals, and polygons (regular, non-regular, convex and concave) using their properties. 3108.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids). 3108.4.12 Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve contextual problems. 3108.4.28 Derive and use the formulas for the area and perimeter of a regular polygon. (A=1/2ap) Polygons A polygon is a closed plane figure with at least three sides that are segments The sides of a polygon must intersect only at the endpoints. They cannot cross. B B B A
C E D A E A C C D D E To name a polygon start at any vertex (corner) and go in order around the polygon, either
clockwise or counter clockwise Classifying Polygons Convex Polygon: No vertex is in -- all point out Concave Polygon: Has at least one vertex inside and two sides go in to form it Sides 3 4 5 6 8 9 10 12 n
Name Triangle Quadrilateral Pentagon Hexagon Octagon Nonagon Decagon Dodecagon n-gon Names of Polygons Generally accepted names Sides Name n N-gon 3 Triangle 4 Quadrilateral 5
Pentagon 6 Hexagon 7 Heptagon 8 Octagon 10 Decagon 12 Dodecagon Other names not normally used Sides 9 11 13 14 15 16 17 18
Enneadecagon, Enneakaidecagon Icosagon Triacontagon Tetracontagon Pentacontagon Hexacontagon Heptacontagon Octacontagon Enneacontagon Hectogon, Hecatontagon Chiliagon Myriagon Polygon Angle Sum Theorem The sum of the measure of the interior angles of an n-gon is (n-2)*180 Name Triangle Quadrilateral Pentagon Hexagon Octagon
7 8 10 n-2 (n-2)*180 180 360 540 720 1080 1260 1440 1800 (n-2)*180 Example Find the sum of the measure of the interior angles of a 15-gon (By the way, if you have a cool calculator, this is where you turn open apps, then A+ Geom, then A. Polygons then enter 15 for number of
sides) Sum = (n-2)* 180, or (15-2)* 180 or (13) * 180 Therefore, the sum of the interior angles is 2340 Example What if you are not told the number of sides, but are only told that the sum of the measure of the angles is 720? Can you determine the number of sides? If you have the cool calculator then use it now Otherwise, substitute into the equation: (n-2)*180 = 720, so (n-2) = 720/180 n-2 = 4 n=6 Example Find the measure of angle y This is a 5 sided object The sum of the interior angles is (n-2)*180 = 540 degrees Therefore, we have 540 90 90 90 136 = Y
136 So Y = 134 Y Polygon Exterior Angle-Sum Theorem The sum of the measure of the exterior angles of a polygon, one at each vertex, is 360 ALWAYS It DOESNT MATTER HOW MANY SIDES THERE ARE, IT IS ALWAYS 360 DEGREES Angle 1 + 2 + 3 + 4 + 5 = 360 1 2 5 4 3 Regular Polygons An Equilateral polygon has all sides equal An Equiangular polygon has all angles equal A REGULAR Polygon has all sides and all
angles equal it is both equilateral and equiangular What are some examples of Regular Polygons in the real world? Example If you have a regular polygon, then you can determine the measure of each interior angle For example, determine the measure of the sum of the interior angles of a regular 11-gon, and the measure of 1 angle Sum = (11-2)*180 = 1620 Since all angles are exactly the same, we can divide our answer by the number of angles to find one angle 1620/11 = 147.27 degrees The generic form of this equation is this: Sum = [(n-2)*180]/n Regular Polygons Name n -Sides Triangle 3
Unit 6 Quiz 1 1. 2. 3. 4. 5. 6. 7. What is the sum of the measure of the interior angles of a 21-gon? What is the sum of the measure of the interior angles of a 18-gon? What is the sum of the measure of the interior angles of a 99-gon? What is the sum of the measure of the interior angles of a 55-gon? What is the measure of one interior angle of a 17-gon? What is the measure of one interior angle of a 28-gon? What is the equation used to solve the sum of the measure of the interior angles of a polygon? (all angles added together) 8. What is the name of a polygon with 12 sides? 9. Given: A REGULAR Pentagon has 5 sides, and the sum of the measure of the interior angles is 540 degrees. What equation would you use to find the measure of ONE angle
10. Calculate the measure of one exterior angle to a regular Pentagon Classifying Quadrilaterals This is what we already know about Quadrilaterals: Four sides Four corners vertices Sum of interior angles is 360 degrees Sum of exterior angles is 360 degrees If it is a regular quadrilateral, then each interior angle is 90 degrees, and each exterior angle is 90 degrees and each side is equal in length Now we will begin to look at some Special Quadrilaterals Review
What x, and what is the measure of the missing angle? x 129 23 1 X+10 y 75 2 3 X+20 X-25 4
140 z 5 X-30 x 6 75 140 7 y X+15 45 8 X+25
9 150 10 z 1. 2. 3. 4. 5. 6. 7. 8. 9. Parallelogram a
a quadrilateral has 4 sides Has 4 vertices b Sum of interior angles is 360 degrees Sum of Exterior angles is 360 degrees Has both pairs of opposite sides parallel Both pairs of opposite angles are congruent Both pairs of opposite sides are congruent Diagonals bisect each other If one pair of opposite sides are congruent and parallel, then it is a parallelogram 10. In a parallelogram, consecutive angles are supplementary as we reviewed c d Consecutive Angles
Angles of a polygon that share a side are consecutive angles For example, angle A and angle B share segment AB. Therefore they are consecutive angles. Which makes sense, because consecutive means in order and they are in order on the polygon shown On a parallelogram, consecutive angles are Same Side Interior angles, which means they are supplementary These angles are B supplementary: A and B B and C C and D D and A
A D C Example using Consecutive Angles Find the measure of angle C Find the measure of angle B Find the measure of angle A Angle D + Angle C = 180, Angle D = 112, therefore Angle C = 180 112, or 680 Angle B + Angle C = 1800, Angle C = 68, therefore Angle B = 180 68, or 1120 Angle D + Angle A = 180, Angle D = 112, therefore Angle A = 180 112, or 680 Note that Angle A and C are equal, and Angle B and D are equal, and weve just proved why A
680 B 1120 C Opposite corners of a parallelogram have equal measure 680 1120 D Example with Algebra Find the value of X in ABCD Then find the length of BC and AD Since opposite sides are congruent, set the values
equal to each other 3X 15 = 2X + 3 3x - 15 B 3X = 2X + 18 X = 18 If X = 18, then 3X 15 = 39 If BC = 39, then AD = 39, since opposite sides are congruent A 2x + 3 D C Another Algebra Example Find the value of Y Then find the measure of all angles Since opposite angles are equal in a parallelogram, then
set the values equal 3y + 37 = 6y + 4 37 = 3y + 4 B C 3y = 33 (3y + 37) y = 11 If y = 11, then angle A = 6(11) + 4, or angles A and C = 70 Angle B and D = 110 6y + 4 A D An Example with Algebra
Find the value of X and Y, and the value of AE, CE, BE, and DE Set each side (value) equal to each other Y=X+1 3Y 7 = 2X Choose a value to substitute for (well use Y) Therefore 3 (X + 1) 7 = 2X 3X + 3 7 = 2X 3X 4 = 2X
X=4 Now solve for Y B 3Y 7 = 2(4) 3Y = 7 + 8 X + 1 3Y = 15 Y=5 AE = 3(5) 7, or 8 E 7 CE = AE, or 8 3Y DE = Y, or 5 BE = DE, or 5 A C 2X
Y D Another Algebra Example Solve for m and n m=n+2 n + 10 = 2(n+2) 8 n + 10 = 2n + 4 8 n + 10 = 2n 4 n = 14 m = 14 + 2 m = 16
B C m 1 n+ A 2m 8 0 n + 2
D Transversal Theorem If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal BD = DF, therefore A B H AC = CE We could draw a new transversal And we know the segments it makes are congruent to each other as well HJ = JK C E
J K D F Assignment Page 364 9-27 (guided practice) Page 365 29,30,38-40 (guided practice) Page 372 7-15 Worksheet 6-2 (independent practice) Worksheet 6-3 (independent practice) Unit 6 Quiz 2 There are ten (or more) characteristics of a parallelogram
Name five of the ten characteristics (2 points each) 2 points each question (10 points) 1 point extra credit for each additional characteristic Total possible score: 10 + 5 points = 15 1. 2. 3. 4. 5. 6. 7. 8. 9. Answers to Quiz a quadrilateral has 4 sides Has 4 vertices Sum of interior angles is 360 degrees
Sum of Exterior angles is 360 degrees Has both pairs of opposite sides parallel Both pairs of opposite angles are congruent Both pairs of opposite sides are congruent Diagonals bisect each other If one pair of opposite sides are congruent and parallel, then it is a parallelogram 10. In a parallelogram, consecutive angles are supplementary as we reviewed Narcissist A person who is overly self-involved, and often vain and selfish. Deriving gratification from admiration of his or her own physical or mental attributes. See also: Nolen (named changed to protect the innocent) Rhombus Rhombus: a parallelogram with four congruent sides
Or We can draw the same conclusions about Same Side Interior angles here they are also corresponding angles, and any 2 in a row add up to 180 degrees (supplementary angle pairs) Rhombus Theorems Each diagonal of a rhombus bisects 2 angles of the rhombus The diagonals of a rhombus are perpendicular If we remember the Perpendicular Bisector theorem, we know that if 2 points are equally distant from the endpoints of a line segment, then they are on the perpendicular bisector. That is the case here. Points R and S are equally distant R from points P and Q. Therefore they are on the perpendicular bisector made by P the diagonal used to connect them S Q
Finding Angle Measure Example MNPQ is a rhombus Find the measure of the numbered angles Angle 1 = Angle 3 N Angles 1 + 3 + 120 =180 120 2 x Angle 1 = 180-120 2 x Angle 1 = 60 1 M
2 Angle 1 = 30 Angle 3 = 30 Angle 2 and 4 = 30 0 3 4 P Q Another Find the Measure Example
What is the measure of Angle 2? 50 degrees (alternate interior angles are equal) What is the measure of Angle 3? 50 degrees (the diagonal is an angle bisector, so if angle 2 is 50 degrees, angle 3 is 50 degrees) What is the measure of Angle 1? 90 degrees (the diagonals of a rhombus are perpendicular bisectors) What is the measure of Angle 4? 500 40 degrees (180 degrees minus 90 degrees 1 minus 50 degrees) 4 2 3 Rectangle
Rectangle: a parallelogram with four right angles Or Diagonals on a rectangle are equal Note: All four sides do not have to be equal, but opposite sides are (because its a parallelogram) Square Square: a parallelogram with four congruent sides and four right angles NOTE: Diagonals on a square are equal too Why? Is a square a rectangle? Is a rectangle a square? Check on Learning
The quadrilateral has congruent diagonals and one angle of 60 0. Can it be a parallelogram? No. A parallelogram with congruent diagonals is a rectangle with four 90 0 angles. The quadrilateral has perpendicular diagonals and four right angles. Can it be a parallelogram? Yes. Perpendicular diagonals means that it is a rhombus, and four right angles means it would be a rectangle. Both properties together describe a square. A diagonal of a parallelogram bisects two angles of the parallelogram. Is it possible for the parallelogram to have sides of lengths 5,6,5 and 6? No. If a diagonal (of a parallelogram) bisects two angles then the figure is a rhombus, and rhombuses have all sides the same size (congruent). Assignment Page 379-80 7-39 odd (guided practice)
Worksheet 6-4 Unit 6 Quiz 3 1. 2. Name 2 corresponding angles Are corresponding angles congruent or supplementary? 3. Name 2 same side interior angles 4. Are same side interior angles congruent or supplementary? 5. Name 2 alternate interior angles 6. Are alternate interior angles congruent or supplementary? 7. If angle C is 70 degrees, what is the measure of angle E? 8. If angle B is 120 degrees, what is the measure of angle F? 9. If angle D is 125 degrees, what is the measure of angle E? 10. If angle E is 130 degrees, what is the measure of
angle F? A B C D E F GH Kite Kite: a quadrilateral with two pairs of adjacent sides congruent, and no opposite sides congruent Kites Remember, a Kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. The diagonals of a kite are perpendicular Perpendicular Example Find a Measure of an Angle in a Kite
Find the measure of Angle 1, 2 and 3 Angle 1 = 90 degrees (diagonals of a kite are perpendicular) 900 + 320 + Angle 2 = 1800, therefore Angle 2 = 580 B 320 3 A 1 D 2 C
Trapezoid Trapezoid: a quadrilateral with exactly one pair of parallel sides. On an Isosceles trapezoid, the diagonals are congruent The isosceles trapezoid is one whose nonparallel opposite sides are congruent Again, we can conclude Supplementary Angles Name the Quadrilateral 1. What is this? Trapezoid 2. What is this? Parallelogram 3. What is this?
Square 4. What is this? Rectangle 5. What is this? Rhombus Classifying Quadrilaterals These quadrilaterals that have both pairs of opposite sides parallel Parallelograms Rectangles Rhombuses Squares These quadrilaterals that have four right angles Squares Rectangles These quadrilaterals that have one pair of parallel sides
Trapezoid Isosceles Trapezoid These quadrilaterals have two pairs of congruent adjacent sides Kites Assignment Page 394-95 7-24, 28-36 Worksheet 6-5 6-1 Trapezoid Worksheet Congruent Polygons Congruent Figures have the same size and shape When figures are congruent, it is possible to move one over the second one so that it covers it exactly Congruent polygons have congruent corresponding parts the sides and angles that match up are exactly the same Matching vertices (corners) are corresponding vertices. When naming congruent polygons, always list the corresponding vertices in the same order Example
Polygon ABCD is congruent to polygon EFGH Notice that the vertices that match each other are named in the same order Imagine a mirror here A E B C D F G H Example Polygon ABCDE is congruent to Polygon LMNOP Determine the value of angle P A E
125 B P L 135 M C N O UsingDthe Polygon Angle Sum Theorem, we know that a 5 sided polygon has (5-2)180, or 540 total interior degrees180, or 540 total interior degrees Because the polygons are congruent, we know that Angle B is congruent to Angle M. To solve for Angle P, we take 540 -90 -90 -125 -135 = 100 degrees Congruent Triangles
We are going to learn many, many ways to prove triangles are congruent Here is the first part of many of these proofs: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent A C D B F E If Angle A is congruent to Angle D, and Angle C is congruent to Angle F, then Angle B is congruent to Angle E. Why?
Example Is triangle ABC congruent to triangle ADC? We need all 3 sides congruent, and all 3 angles congruent Side AD is congruent to side AB Side DC is congruent to side CB Side AC is congruent to itself Angle DAC is congruent to Angle BAC Angle ADC is congruent to Angle ABC All we need is the 3rd angle Angle DCA must be congruent to Angle BCA Since we have 2 angles congruent, we know the 3rd angle is congruent A D B C
Assignment Worksheet 4-1 Unit 6 Quiz 5 Ratios and Proportions Proportion: a statement that two ratios are equal. It can be written either as A/B = C/D A:B = C:D Extended proportion: When 3 or more ratios are equal, such as 6/24 = 4/16 = 1/4 Properties of Proportions A/B = C/D is equivalent to 1) AD = BC (cross-product Property) Also written A:B = C:D, it can be referred to the Product of the extremes (the two outside numbers) is equal to the product of the means (the two inside numbers) 2) B/A = D/C (flip both sides)
3) A/C = B/D (ratio of the tops is equal to the ratio of the bottoms 4) (A + B)/ B = (C + D)/D Example If X/Y = 5/6, Then? 6X = ? 5Y Y/X = ? 6/5 X/5 = ?
Y/6 (X + Y)/Y = ? 11/6 There is always a pattern Assignment Page 461 6-9 Page 462 33-47 John Wayne as The Shootist "I won't be wronged, I won't be insulted, and I won't be laid a hand on. I don't do these things to other people and I expect the same from them." Similar Polygons Two figures that have the same shape but not necessarily the same size are similar This is the symbol for similarity: ~ Two polygons are similar if 1) Corresponding angles are congruent and 2) Corresponding sides are proportional.
The ratio of the lengths of corresponding sides is the similarity ratio. Example of Similarity F G 127 E H B C 53 A D
If ABCD ~ EFGH then: m of angle E = m of angle ___ A So m of E = m of A = 53 because corresponding angles are congruent. AB/EF = AD/___ EH So AB/EF = AD/EH because corresponding sides are proportional. We can conclude that the m of angle B = 127 _____ 0 We can conclude that GH/CD = FG/___ BC Determining Similarity B 15 A
12 18 C E 16 D 20 24 F Determine whether the triangles are similar. If they are, write a similarity statement and give the similarity ratio 1. All three pairs of angles are
congruent. 2. Check for proportionality of corresponding angles: 1. AC/FD = 18/24 = 3/4 2. AB/FE = 15/20 = 3/4 3. BC/ED = 12/16 = 3/4 Triangle ABC~Triangle FED with a similarity ratio of 3/4 or 3 : 4. Using Similarity N 2 O 3.2 L 5 T
Q M X 6 S R LMNO~QRST Find the value of X
LM/QR = ON/TS 5/6 = 3.2/SR 5/6 = 2/X 5(SR) = 6x3.2 5X = 12 SR = 6x3.2/5 X = 2.4 SR = 3.8 Using this figure, find SR to the nearest tenth. Golden Rectangle / Ratio Numbers And so on 1/1.618 = x/30 X = 30/1.618
X = 18.54 cm wide A Golden Rectangle is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. In any golden rectangle, the length and width are in the Golden Ratio, which is about 1.618 : 1. The golden rectangle is considered pleasing to the human eye. It has appeared in architecture and art since ancient times. It has intrigued artists including Leonardo da Vinci (1452 - 1519). Da Vinci illustrated The Divine Proportion, a book about the golden rectangle. An artist plans to paint a picture. He wants the canvas to be a golden rectangle, with the longer horizontal sides to be 30 cm wide. How high should the canvas be? ASSIGNMENT Page 475 2-7 (skip 4)
Page 476 15-21 Page 477 24-27, 29,30, 39-42 Unit 6 Final Extra Credit Solve for the missing variables (2 points each, show work as required) 1. A=_______ 5*a Isosceles Trapezoid 650 12 9 + b 3* 33 Rectangle
9*d 2. 3. 4. 5. B=_______ C=_______ D=_______ E=_______ 13.5*e Regular Octagon c (round to nearest Right whole number) Triangle
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