Name: Geometry Honors Unit 5 Notes Packet Triangles Properties & More Proofs

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1 Name: Geometry Honors Unit 5 Notes Packet Triangles Properties & More Proofs 1

2 Negations, Contradictions, & Intro to Indirect Proof Writing an Indirect Proof: 1 state as an assumption the opposite (negation) of what you want to prove 2 show that this assumption leads to a contradiction 3 conclude that the assumption must be false The First Step writing the opposite (negation) of a statement Statement: y is less than 7 Negation: Statement: x is less than or equal to 10 Negation: Statement: The coat costs more than $40. Negation: Statement: Quadrilateral GEOM does not have four acute angles Negation: 2

3 The Second Step identify the contradiction Circle the two statements that contradict each other: 1. FG KL FG KL FG KL 2. In right ABC, the measure of angle A is 60. In right ABC, A C. In right ABC, the measure of angle B is Each of the two items that Rachel bought costs more than $10. Rachel spent $34 for the two items. Neither of the two items Rachel bought costs more than $ ABC is acute. ABC is scalene. ABC is equilateral. The Third Step conclude your assumption was wrong You need to be writing an actual proof to see this step in action: Steps: 1. Assume the opposite of what you want to prove. 2. Use that assumption as a fact and develop a proof. 3. Identify a logical inconsistency (contradiction). 3

4 Given: Line L is not parallel to Line M Prove: <1 is not congruent to <2 Given: Prove: A C AB CB BD BE 4

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6 Given: and Prove: 1. S T A T E M E N T S and 1. Given R E A S O N S Assumption leading to a contradiction Transitive property

7 Given: where Prove: S T A T E M E N T S 1. where 1. Given R E A S O N S Assumption leading to a contradiction If two angles of a triangle are congruent, the sides opposite them are congruent Reflexive property SSS CPCTC An angle bisector is a ray whose endpoint is the vertex of the angle and which divides the angle into two congruent angles Contradiction Steps 7 and 1 7

8 Analytic Geometry Proofs with Different Types of Triangles Process: 1 state what you are going to do 2 show necessary formulas & work labeled by sides! 3 explain what your work shows OR you can do a Statement/Reason format: 1. Given the points A(8,9), B(10,3), and C(3,4), prove that ΔABC is isosceles. 8

9 2. Show that the triangle with the vertices S(-4,-1), O(0,-5), and X (1,4) is a right triangle. 3. Triangle SUB has vertices S(-3, -1), U(0, 3), and B(4, -2). Classify the triangle as equilateral, isosceles, or scalene. 9

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11 Given R(0,0), S(2a,2b), and T(4a,0) 4. Find the midpoint of RS, call it L 5. Find the midpoint of ST, call it M 6. Find the midpoint of RT, call it N 7. Prove LM RT 8. Show SN is perpendicular to RT 9. Prove that RST is isosceles 10. Given A(0.0), B(4a,0), and C(0,4a), prove ABC is an isosceles right triangle 11

12 Homework: Analytic Geometry Proofs with different types of triangles Use the three step process in your proofs or the Statement Reason format 1) Triangle TRI has vertices T(15,6), R(5,1), and I(5,11). Use coordinate geometry to prove that triangle TRI is isosceles. 2) Triangle DAN has coordinates D(-10,4), A(-4,1), and N(-2,5) Using coordinate geometry, prove that triangle DAN is a right triangle. 12

13 3) The coordinates of the vertices of SUE are S(-2,-4), U(2,-1), and E(8,-9). Using coordinate geometry, prove that triangle SUE is scalene. 4) 13

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15 Interior/Exterior Angle Theorems Theorem: The sum of the angles in a triangle is 180º Given: ABC Prove: m 1 m 2 m Triangle Exterior Angle Theorem Proof 15

16 Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the non-adjacent (also called remote ) interior angles. A polygon s Exterior Angle is the angle formed by the extension of any side and its adjacent side. No part of this angle is in the interior of the triangle. In the diagram to the right, ACD is an exterior angle m ACD m A m B 16

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19 Pythagorean Theorem Pythagorean Theorem In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. (a 2 + b 2 = c 2 ) a c b The Pythagorean Theorem can be used to classify the triangle by type if a b 2 c then the triangle is a right triangle if a b 2 c then the triangle is an acute triangle if a b 2 c then the triangle is an obtuse triangle 19

20 Proving the Pythagorean Theorem If you visit you will find 81 different proofs of the Pythagorean Theorem along with the following introduction: The statement of the Theorem was discovered on a Babylonian tablet circa B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that a triangle whose sides satisfy a² + b² = c² is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact. W. Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in We are going to perform just a few proofs of the Pythagorean Theorem. Proof 1: Each side of the given square has been broken into2 non-congruent segments of lengths a and b. Represent the area of the square using the formula A = (side)(side) Represent the area of the square by adding the area of each little part (5 total parts) Set your two equations equal to each other and solve for c 2. 20

21 President James A. Garfield ( ) 20 th President of the United States While US Presidents are rarely known for their mathematical abilities, President Garfield was the exception. He graduated from Williams College in MA in 1856, was a math teacher, then a school principal. In 1859 he left his school to be a member of the Ohio Senate and joined the Union Army when the Civil War erupted in After the war, he served in Congress, and was elected President in In 1876 Garfield published a proof of the Pythagorean Theorem using the area of a trapezoid in the New England Journal of Education. Proof 2: Write 2 different expressions that represent the area of the trapezoid, then set them equal to each other Algebraically manipulate your equation to show a b c. 21

22 Proof 3: A different Proof using a trapezoid. Again, write 2 different expressions that represent the area of the trapezoid, then set them equal to each other. Then algebraically manipulate your equation to show a b c. 22

23 Triangle Inequality: A Geometry class did an activity to study the possible lengths of the sides in triangles. This is the data they compiled. What does this data tell you? Theorem: In any triangle, the sum of any two sides must be greater than the length of the 3 rd side. 23

24 Side and Angle relationships in Triangles: The same class did the following activity with the measures of the angles and the sides of triangles to try and develop a relationship between side lengths and angle measures. What do you notice? In any triangle, the longer side lies opposite the larger angle 24 In any triangle, the larger angle lies opposite the longest side.

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27 Find the possible values for x that will make the sides of the triangle: You will need to write and solve three inequalities. 27

28 Midsegments & Other Line Segments in a Triangle Midsegment of a Triangle Big Idea #1 Big Idea #2 1. Use ABC where L, M and N are midpoints of the sides of the triangles. a. LM b. AB B c. If AC = 20, then LN = d. If MN = 7, then AB= L N e. If NC = 9, then LM = A M C 2. Given BCD with G the midpoint of BC, F the midpoint of CD and E the midpoint of BD. If CD = 14, GF = 8 and GC = 5, find the perimeter of BCD and the perimeter of EFG. C G F B E D 28

29 Angle Bisector: Perpendicular Bisector: Altitude of a Triangle: Median of a Triangle: Given ABC with BAE EAC, BF FC, BDA is a right angle and GF BC. Match the term with the correct segment. 1. Median A. AD A 2. Altitude B. AE 3. Perpendicular Bisector C. AF G 4. Angle Bisector D. GF B D E F C Given ABC with altitude CD, angle bisector CE, and median CF. 5. State two congruent angles, each of which has a vertex at C. C 6. State two line segments that are congruent. A D E F B 7. State two line segments that are perpendicular to each other. 29

30 Name the type of segment shown in each triangle:

31 1. In the diagram, MK and LK are angle bisectors of MNL o and m MNL 110. Find the number of degrees in MKL. (A) 110 (B) 145 (C) 90 (D) 70 (E) 40 Use the diagram of ABC at the right for problems # Identify the median of ABC. (A) BF (B) GH (C) AD (D) CE (E) none of these 3. Identify the altitude of ABC. (A) BF (B) GH (C) AD (D) CE (E) CB 4 o In ABC, if m ABF 39 and BF is an angle bisector, find m BCE. (A) 90 (B) 45 (C) 39 (D) 51 (E) 12 31

32 Fill in the blanks & sketch a diagram for each problem. WORD BANK angle bisector altitude median 1. The of a segment is the center of the line segment and splits the segment into two congruent parts. 2. The of a segment intersects the segment at the center, splitting the segment into two congruent parts, and creating right angles. 3. An is a segment, ray, or line, that goes through the vertex of the angle and splits the angle into two congruent parts. 4. The of a triangle is a segment created by connecting the midpoints of two sides of the triangle. It is parallel to the third side of the triangle and half as long as the third side of the triangle. 5. The of a triangle is a segment that connects a vertex of a triangle to the midpoint of the side opposite from the chosen vertex. 6. The of a triangle is a segment that is drawn from a vertex perpendicular to the side across from the chosen vertex. This is sometimes called the height of the triangle. 32

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35 Proofs with Special Line Segments in a Triangle Ex 1: Given: BD is a median of ABC a. What conclusion(s) can you draw, if any? b. Reason: Ex 2: Given: AC is an altitude of ACE a. What conclusion(s) can you draw, if any? b. Reason: Ex 3: Given: DE is a mid-segment of ABC a. What conclusion(s) can you draw, if any? b. Reason: 35

36 Ex 4) Given: ID is an angle bisector of KIM a. What conclusion(s) can you draw, if any? b. Reason: Ex 5) Given: AC is a perpendicular bisector of ABD a. What conclusion(s) can you draw, if any? b. Reason: 36

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40 Prove: If the triangle is isosceles, then the medians to two sides of a triangle are congruent. Prove that the altitude to the base of an isosceles triangle is a median. Prove: If the triangle is isosceles, then the bisectors of two angles of the triangle are congruent. Prove: The medians to two sides of an isosceles triangle are congruent. 40

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42 Circumcenter: the Point of Concurrence of the perpendicular bisectors of a triangle: Activity: 1.) Each member of the group will choose one of the triangles on the next page. Each student chooses one triangle and CONSTRUCTS the perpendicular bisectors of all three sides on their sheet. Extend the perpendicular bisectors to the edges of the page. 2.) Compare your results. What do you notice about the intersection of the three perpendicular bisectors? Does it change when the type of triangle changes? 3.) Open your compass so that the point is on the point of concurrence and the pencil is on one of the vertices of the triangle. Using this center and radius, draw a circle. 4.) What do you notice about the circle? Why does this happen? 42

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44 1. The name of the point of concurrency of the three perpendicular bisectors of the sides of a triangle is the A) orthocenter B) incenter C) circumcenter D) centroid 2. What of the following always describes the circumcenter of a triangle? (Circle all that apply) A) Equidistant from each side of the triangle B) Equidistant from each vertex of the triangle C) Point where the perpendicular bisectors of a triangle intersect D) Point where the angle bisectors of a triangle intersect 3. Which describes the point where three or more lines intersect? (Circle all that apply) A) point of concurrency B) point of perpendicularity C) point of intersection D) point of parallelism 4. Which of the following properties always applies to the Circumcenter? (Circle all that apply) A.) It is equidistant from the vertices of the triangle B.) It is equidistant from the sides of the triangle C.) It is the center of the circle that circumscribes the triangle D.) It is the center of the circle that is inscribed in the triangle 44

45 Practice with Circumcenter: 45

46 Centroid: The Point of Concurrence of the three medians of a triangle. A B C Special Properties of the Centroid: 46

47 1.) Name the point of concurrency of the medians of a triangle 2.) Name the point that is equidistant from the three vertices of a triangle 3.) In the triangle below, Addie found centroid P by constructing the 3 medians. a. What is the relationship between the length of PF and the length of CP? b. If the length of CF is 6 inches, find the length of PF and the length of CP. 47

48 Points of Concurrency: The Incenter Use your compass and straightedge to construct the angle bisector of each angle of the triangle below. Do this carefully and with precision! A B C Incenter: Extra! Extra! Construct a perpendicular segment from the incenter to one side of the triangle. 1. With your compass, draw a circle that has the center at the incenter and the radius should be the perpendicular distance from the incenter to the point where your perpendicular segment meets the side of the triangle. 2. What do you notice about the circle? Why does this happen? 48

49 1.) The incenter is the point of concurrency for what segments of a triangle? a. Perpendicular bisectors b. Angle bisectors c. Medians d. Altitudes 3.) Which of the following properties always applies to the incenter? (Circle all that apply) e. It is equidistant from the vertices of the triangle f. It is equidistant from the sides of the triangle g. It is the center of balance of the triangle h. It is the center of the circle that circumscribes the triangle i. It is the center of the circle that is inscribed in the triangle 49

50 Points of Concurrency: The Orthocenter Use your ruler and compass to construct the altitude to each side of the triangle below. Do this carefully and with precision! A B C Orthocenter: 50

51 1.) The orthocenter is the point of concurrency for what segments of a triangle? a. Perpendicular bisectors b. Angle bisectors c. Medians d. Altitudes 2.) The incenter is the point of concurrency for what segments of a triangle? a. Perpendicular bisectors b. Angle bisectors c. Medians d. Altitudes 3.) The cicumcenter is the point of concurrency for what segments of a triangle? a. Perpendicular bisectors b. Angle bisectors c. Medians d. Altitudes 4.) The centroid is the point of concurrency for what segments of a triangle? a. Perpendicular bisectors b. Angle bisectors c. Medians d. Altitudes 5.) Which of the following properties always applies to the orthocenter? (Circle all that apply) a. It is equidistant from the vertices of the triangle b. It is equidistant from the sides of the triangle c. It is the center of balance of the triangle d. It is the center of the circle that circumscribes the triangle e. It is the center of the circle that is inscribed in the triangle f. None of the Above 51

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