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TOPIC 11 Triangles and Polygons

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TOPIC 11 Triangles and Polygons TOPIC 11 Triangles and polygons We can proceed directly from the results we learned about parallel lines and their associated angles to the ideas concerning triangles. An essential question is a question whose answer is not obvious, but it is a vitally important question that we continue to ponder. Essential question: Why are triangles so important in the study of geometry? We will spend more time studying triangles than any other single topic. Why? Let’s review the definitions: 1. What is a triangle? 2. What is an isosceles triangle? 3. What is an equilateral triangle? 4. What is a scalene triangle? 5. What is a scalene triangle? 6. What is an acute triangle? 7. What is an obtuse triangle? 8. What is a right triangle? Topic 11 (Triangles and Polygons) page 2 What can we discover that is true about a triangle? One well-known result concerns the sum of the angles of a triangle. We will do this with a paper triangle in class … a sort of kinesthetic proof. In a more formal way, we will work through a written proof. Given: Δ ABC Prove: m< 1 + m< 2 + m< 3 = 180 Proof: 1. Draw ECD || AB 1. 2. < 2 ______ 2. 3. < 3 ______ 3. 4. < ____ is supp to < ACD 4. 5. m < ____ + m < ACD = 180 5. Def supp 6. m< 1 +m < ____ = m < ACD 6. angle addition postulate 7. m < 1 + m < ____ + m < ____= 180 7 . 8. m< 1 + m < 2 + m < 3 = 180 8. QED Is the parallel postulate necessary in this proof? (We use this as the parallel postulate: If corresponding angles are congruent, then lines are parallel.) As an implication of this fact, we accept as true the fact that there is only one line through a point which is parallel to a given line. We will discuss this assumption in more detail in a few weeks. Next, we will look at other angles associated with triangles. An exterior angle is an angle which is adjacent to one of the interior angles of a triangle. It is formed by extending one of the line segment sides of a triangle. Topic 11 (Triangles and Polygons) page 3 Sketchpad Investigation: 1. Draw a triangle as shown. 2. Measure angles as needed to answer these questions, using Sketchpad evidence. 3. Questions: How is < 1 related to the interior angles of the triangle? What is the sum of the three exterior angles in a triangle? Can a triangle have ________ a) 3 obtuse interior angles b) 2 obtuse interior angles c) 1 obtuse interior angle d) 3 congruent, obtuse interior angles e) 2 congruent, obtuse interior angles f) 3 obtuse exterior angles g) 2 obtuse exterior angles h) 1 obtuse exterior angle i) 3 congruent, obtuse exterior angles j) 2 congruent, obtuse exterior angles Topic 11 (Triangles and Polygons) page 4 4. Draw a right triangle, ΔABC. Keep BC unchanged, but move A along AC . Describe what happens to m< 1 , m< 2, m < 3, and m< 4. 5. In the right triangles that you constructed above, can m< 1 in one of the right triangles be twice as large as m< 1 in another one, as you move A along AC ? If so, compare the lengths of AC in the two triangles? One Proof Look at the conclusion above about how one of the exterior angles of a triangle is related to the two remote interior angles. Prove your conclusion. Topic 11 (Triangles and Polygons) page 5 Polygons …what are they, and how do we use what we already know to learn about their properties? To produce polygons, we need to have some ground rules: The figure has to be bounded by line segments (but the segments do not all have to be congruent), and the line segments can only intersect at their endpoints. The figure must be closed. (There has to be an interior and an exterior? How do we know which is the interior?) Only two segments can intersect at each endpoint. (If more than two segments are drawn at a vertex, then the excess over two may be “diagonals”.) There are convex polygons and concave polygons. Do some research to find the names of polygons. They are labelled according to the number of sides that they have. N Name Second name (maybe) 1 2 3 4 5 6 7 8 9 10 11 12 13 20 40 52 73 n Here are some more names … 3 - triangle 4 - quadrilateral Topic 11 (Triangles and Polygons) page 6 5 - pentagon 6 - hexagon 7 - heptagon 8 - octagon 9 - nonagon or enneagon 10 - decagon 11 - hendecagon or undecagon 12 - dodecagon 13 - triskaidecagon 14 - tetrakaidecagon 15 - pentakaidecagon 16 - hexakaidecagon 17 - heptakaidecagon 18 - octakaidecagon 19 - enneakaidecagon 20 - icosagon 21 - icosikaihenagon or henicosagon Notice that “21” is ICOSA” KAI “HENAGON which means 20 (“icosa”) and (“kai”) 1 (“hen”) GON 22 - icosikaidigon or docosagon 23 - icosikaitrigon or tricosagon 24 - icosikaitetragon or tetracosagon 25 - icosikaipentagon or pentacosagon Notice that “25” is ICOSA” KAI “PENTAGON which means 20 (“icosa”) and (“kai”) 5 (“penta”) GON 26 - icosikaihexagon or hexacosagon 27 - icosikaiheptagon or heptacosagon 28 - icosikaioctagon or octacosagon 29 - icosikaienneagon, enneacosagon or nonacosagon 30 - triacontagon 40 - tetracontagon 50 - pentacontagon 60 - hexacontagon 70 - heptacontagon 80 - octacontagon 90 - nonacontagon or enneacontagon 100 - hectagon 1000 - chiliagon 10000 - myriagon 100000 - decemyriagon 1000000 - hecatommyriagon Source(s):http://www.2dshapes.com/polygons.html Topic 11 (Triangles and Polygons) page 7 Look up the definitions of equilateral and equiangular. Look up the definition of a regular polygon. Draw an equilateral but non- equiangular quadrilateral. Draw an equiangular but non- equilateral quadrilateral. Draw an equilateral and equiangular quadrilateral. Draw an equilateral but non- equiangular hexagon. Topic 11 (Triangles and Polygons) page 8 Draw an equiangular but non- equilateral hexagon. Draw an equilateral and equiangular hexagon. Draw a regular triangle. Draw a regular quadrilateral. Draw a regular pentagon Draw a regular hexagon. Topic 11 (Triangles and Polygons) page 9 Looking at the angles in polygons: Draw a quadrilateral. Make it scalene and not regular for now. Pick one vertex, and draw all of the diagonals from that vertex. (A diagonal is a line segment which connects non-consecutive vertices.) How many triangles are formed? How many degrees are in the sum of the angles of each triangle? Therefore, what is the sum of the measures of the interior angles of the quadrilateral? Do you think that result would be different had you drawn a parallelogram? A square? A rhombus? Draw a pentagon. Make it scalene and not regular for now. Pick one vertex, and draw all of the diagonals from that vertex. Topic 11 (Triangles and Polygons) page 10 How many triangles are formed? How many degrees are in the sum of the angles of each triangle? Therefore, what is the sum of the measures of the interior angles of the quadrilateral? Do you think that result would be different had you drawn a regular pentagon? Continue the process for other polygons. You are thinking inductively, looking for a pattern. N Name Sum of interior angles 3 4 5 6 7 8 9 10 n Topic 11 (Triangles and Polygons) page 11 Open a Sketchpad document. Draw a convex quadrilateral. Try not to make it too special. Extend each side to form one exterior angle, as shown. Measure each of the four exterior angles, and calculate the sum of these exterior angles. Contort your quadrilateral and observe what happens to the sum of the exterior angles. Open a new sketch. Do the same investigation with a convex pentagon. In a regular polygon, since the angles all have equal measure, then the measure of each angle can be calculated by dividing the sum of the interior angles by the number of sides. 1. Calculate the size of each interior angle in a regular pentagon. Show your work. 2. Calculate the size of each interior angle in a regular hexagon. Show your work. Topic 11 (Triangles and Polygons) page 12 3. Calculate the size of each interior angle in a regular nonagon. Show your work. 4. Calculate the size of each interior angle in a regular 42-gon. Show your work. The same idea is used to find the size of each exterior angle in a regular polygon. 5. Calculate the size of each exterior angle in a regular pentagon. Show your work. 6. Calculate the size of each exterior angle in a regular hexagon. Show your work. 7. Calculate the size of each exterior angle in a regular nonagon. Show your work. Summarize the calculation of these angles for an n-gon: Sum of the measures of the interior angles = Measure of each interior angle (regular polygons) = Topic 11 (Triangles and Polygons) page 13 Sum of the measures of the exterior angles = Measure of each exterior angle (regular polygons) = Design an Excel spreadsheet to see the patterns in the angles in polygons. (You will be using your laptop to this.) The first rows should look like this, and you will be shown how to enter formulas in a spreadsheet. gles in Polygons GULAR of of m of m of ch int ch ext es angles <'s t <'s gle gle 3 1 180 360 60 120 4 2 360 360 90 90 5 3 540 360 108 72 6 4 720 360 120 60 7 5 900 360 128.5714 51.42857 8 6 1080 360 135 45 9 7 1260 360 140 40 10 8 1440 360 144 36 11 9 1620 360 147.2727 32.72727 12 10 1800 360 150 30 Observations: What happens in the sequence of the sum of the interior angles? What happens in the sequence of the sum of the exterior angles? Topic 11 (Triangles and Polygons) page 14 What does the size of each interior angle (regular polygons) sem to approach as n gets really large? What does the size of each exterior angle (regular polygons) sem to approach as n gets really large? How many different kinds of regular polygons have exterior angles which are integers? How many different kinds of regular polygons have interior angles which are all less than 1510? How Big are the ANGLES in TRIANGLES & POLYGONS Here are some triangles and polygons whose angles need to be calculated.
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