TOPIC 11. Triangles and polygons

Transcription

1 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?

2 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 < < < is supp to < ACD m < + m < ACD = Def supp 6. m< 1 +m < = m < ACD 6. angle addition postulate 7. m < 1 + m < + m < = m< 1 + m < 2 + m < 3 = 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.

3 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

4 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< 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.

5 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) n Here are some more names 3 - triangle 4 - quadrilateral

6 Topic 11 (Triangles and Polygons) page 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 hectagon chiliagon myriagon decemyriagon hecatommyriagon Source(s):

7 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.

8 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.

9 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.

10 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 n

11 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.

12 Topic 11 (Triangles and Polygons) page 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) =

13 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 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?

14 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 151 0? How Big are the ANGLES in TRIANGLES & POLYGONS Here are some triangles and polygons whose angles need to be calculated. Use the new formulas that you have learned and verified. 1. Verify that the sum of the exterior angles, taken one at each vertex, of any n-gon is (This can be done in an algebraic flavor.)

15 Topic 11 (Triangles and Polygons) page The measures of the three angles of a triangle are in the ratio 2:4:9. Find the measure of each angle. (Hint: Call the measures of the angles 2x, 4x, and 9x). 3. The measures of the three angles of a triangle are in the ratio 2:3:5. Find the measure of the smallest exterior angle. 4. Four of the angles in a pentagon measure 45, 65, 128, and 147. What is the measure of the fifth angle? 5. Two of the interior angles of a quadrilateral have measure 36 0 and The other two angles are in the ratio 3:5. Find the measure of the other two interior angles as well as the second to largest exterior angle. 6. The measure of one of the acute angles of a right triangle is 15 0 less than twice the measure of the other acute angle. Find the measure of each of the acute angles.

16 Topic 11 (Triangles and Polygons) page Find the measures of the numbered angles in the drawing. It is given that AB BC. m < 1 = m < 2 = m < 3 = m < 4 = 8. Okay, not all numerical problems.. Given: The plane figure Prove: r + s = t + q

17 Topic 11 (Triangles and Polygons) page Each of the five angles of a convex hexagon measures What is the measure of the sixth interior angle? 10. How many sides does a regular convex polygon have if one exterior angle has measure 24 0? 11. What is the measure of an interior angle of a regular polygon with 12 sides? 12. If each interior angle of a regular polygon has a measure of 170 0, how many sides does it have? 13. Is it possible for a regular polygon to have interior angles whose measure is? (Describe why or why not.) a) 10 0 b) 45 0 c) d) 50 0

18 Topic 11 (Triangles and Polygons) page If each exterior angle of a regular polygon has measure 30 0, what is the sum of the measure of the interior angels of that polygon? 15. Discover a formula for the total number of diagonals that can be drawn from all vertices in a convex n-gon. 16. One angle of a triangle has measure 45 0, and the other two angles are in the ratio 1:2. Find the other two angles. 17. Find the angle between the two bisectors of the acute angles of a right triangle. 18. An exterior angle of a triangle has measure One remote interior angle is 36 0 more than the other remote inter angle. Find the measures of those remote interior angles.

19 Topic 11 (Triangles and Polygons) page The sum of an angle and twice its complement is Find the measure of the angle. 20. Find the first instance where an n-gon and an n+1-gon contribute one interior angle each whose sum exceeds (Assume that the polygons are sitting adjacent to each other, and they have congruent sides.) 21. Write three examples of combinations of polygons whose interior angles share a vertex and yet do not intersect so that the sum of one each of their interior angles does not exceed ISOSCELES TRIANGLES.. Two angles of an isosceles triangle are congruent. (Accept this right now without proof. The verification will come in a few weeks.). The congruent angles are called base angles, and the third angle is called the vertex angle. Vertex angle The congruent sides are called legs, and the third side is called the base. leg leg base angle base base angle

20 Topic 11 (Triangles and Polygons) page m < C = m< A = m< B = 23. m < C = 46 0 m< A = m< B = 24. m < 1 = m< 2 = m< 3 = m < 4 = 25. m < 4 = 82 0 m< 1 = m< 2 = m < 3 = 26. m < 2 = 3x + 10 m < 4 = 2x Solve for x.

21 Topic 11 (Triangles and Polygons) page m < 1 = 28. x = y = (regular octagon) 29. Find the measures of the numbered angles. (regular pentagon)

22 Topic 11 (Triangles and Polygons) page The supplement of an angle is 10 0 more than three times its complement. Find the angle. 31. Two angles in a triangle sum to 2 of the measure of the third angle. One of those two 3 angles is 20 more than the other. Find the measure of the largest angle in the triangle. Problems are from Mathematics Review Exercises (Smith and Fagan, Ginn and Company, 1961), Modern School Mathematics: Geometry (Jurgenson, Donnelly, Dolciani,Hougling Mifflin, 1972), and your teacher on a sunny, cold Friday afternoon. Definition: A tessellation is a plane filling combination of shapes. (There are no gaps nor any overlaps of the shapes). This means that the interior angles that surround a point have to sum to 360 exactly. For example, four squares can surround a point. Since each square has an interior angle with measure 90 0, then the sum of four of these with be exactly It is also true that two regular octagons and one square will surround a point. This is because the interior angles are 135, 135, and 90, and = 360. A tessellation is called a regular

23 Topic 11 (Triangles and Polygons) page 23 tessellation if only one kind of regular polygon is used (such as four squares), and the symbol is (for square-square-square-square). A tessellation is called a semi-regular tessellation if different kinds of regular polygons are used, but the configuration at every vertex is identical throughout the tiling. The tessellation with two octagons and one square above is an example, and its symbol is (or or 4-8-8: all are the same). A demi-regular tessellation is one made of different kinds of regular polygons are used, but the configuration at various vertices is not identical throughout the tiling. Make a list of all of the regular tessellations, using symbols like Use the cut out tiles to experiment. Make a list of all of the semi-regular tessellations. Use the cut out tiles to experiment. Make a list of all of the demi-regular tessellations. Use the cut out tiles to experiment.

24 Topic 11 (Triangles and Polygons) page 24 Here are some more complex tessellations. Describe what happens at various vertices.

25 Topic 11 (Triangles and Polygons) page 25 Tessellations are from Remember that a tessellation is a tiling of the plane. The shapes that make up the tessellation have to be able to cover the plane with go gaps and no overlaps. (The tiling has to be able to continue indefinitely.) Our previous work used only regular polygons to do the tilings. Now, we will examine whether there are other shapes that can tile. QUESTION 1: QUESTION 2: QUESTION 3: QUESTION 4: Will all triangles tile? If not, which ones will tile? Will all quadrilaterals tile? If not, which ones will tile? Will all pentagons tile? If not, which ones will tile? What kinds of irregular shapes will tile?

26 Topic 11 (Triangles and Polygons) page 26 QUESTION 1: Will all triangles tile? If not, which ones will tile? Use the example of this one triangle, from which we will generalize to other triangles. Use a template of the triangle, and figure out how to position it, repeatedly, on the grid to tile. USE PENCIL!! You might need to erase! QUESTION 2: Will all quadrilaterals tile? If not, which ones will tile? Try one example of various kinds of quadrilaterals to see if they will tile. If they do indeed tile, look for a strategy to determine how they do so. (Is it already clear that rectangles and squares will tile???) USE PENCIL!! You might need to erase! Parallelograms

27 Topic 11 (Triangles and Polygons) page 27 Trapezoids Non-Convex Quadrilaterals

28 Topic 11 (Triangles and Polygons) page 28 QUESTION 3: Will all pentagons tile? If not, which ones will tile? Choose a pentagon not a regular pentagon (because we already know what happens) and try to tile with it. Another pentagon example

29 Topic 11 (Triangles and Polygons) page 29 QUESTION 4: What kinds of irregular shapes will tile? Escher style tessellations M. C. Escher was a Dutch artist whose specialty was tessellations. Born June 17, 1898 in the Dutch province of Friesland. He began studying tilings around He created his most famous tessellation, Reptiles in Escher drew his last tessellation in July of He died at the age of 73 in March of You would enjoy seeing some of his work. Look it up online! Here are some instructions to produce an Escher style tessellation From: The Geometer's Sketchpad and Tessellations Cathi Sanders Punahou School, Honolulu, Hawaii Construct a parallelogram, using Parallel Line in the Construct menu. Construct a random shape on the left side of the parallelogram.

30 Topic 11 (Triangles and Polygons) page 30 Select D and then C, and Mark Vector "D->C" in the Transform menu. Then select all of the random shape and Translate it By Marked Vector using the Transform menu. (Vector is the name for a segment with a given length and a given direction.) Construct a 2nd random shape on the top of the parallelogram. Mark D to A as a vector, and translate the 2nd random shape. Select the vertices of your polygon in consecutive order and Construct Polygon Interior

31 Topic 11 (Triangles and Polygons) page 31 Translate the whole polygon (vertices, sides and interior) by a vector DC and then by a vector DA to create a tessellation. Use different colors for adjacent polygons. Drag any point to change your tessellation. Finally. Experiment with Sketchpad to produce a creative tiling. Make it colorful, perhaps intricate, certainly interesting and visually appealing. Use the attached instructions from the Exploring Geometry with The Geometer s Sketchpad 2002 Key Curriculum Press to help you.