Unit 2 Properties of Triangles Unit Bundle Math 2 Spring 2017 1
Day Topic Homework Monday 2/6 Triangle Angle Sum Tuesday 2/7 Wednesday 2/8 Thursday 2/9 Friday 2/10 (Early Release) Monday 2/13 Tuesday 2/14 Wednesday 2/15 Thursday 2/16 Friday 2/17 Isosceles Triangles Midsegments of Trianlges Triangle Congruency Triangle Congruency cont. Triangle Similarity Proportions in Triangles Proportions in Triangles cont. Review Test Unit Vocabulary Triangle AA SAS ASA SSS Congruency Similarity Mid - Segments Proportions Side Splitter Triangle Angle Bisector Isosceles NC Standards Math 2 G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180o G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three dimensional objects generated by rotations of two-dimensional objects G-MG.1 Use geometric shapes, their measures, and their properties to describe objects G-MG.2 Apply concepts of density based on area and volume in modeling situations G-MG.3 Apply geometric methods to solve design problems 2
Contents Properties of Triangles and Triangle Angle Sum Theorems... 4 Practice Triangle Sum Theorem and Remote Interior Angles... 8 Isosceles Triangles... 9 Practice Isosceles Triangles... 11 Midsegments in a Triangle... 13 Practice Midsegments in a Triangle... 16 Congruence in Triangles... 17 Triangle Congruence Exploration Using AngLegs... 18 Triangle Congruence Notes... 21 Practice Triangle Congruence... 24 Practice 2 Triangle Congruence... 26 Triangle Similarity... 28 Practice Triangle Similarity... 31 Proportions in Triangles... 34 Practice 1 Proportions in Triangles... 36 Practice 2 Proportions in Triangles... 38 Unit 2 Review... 41 3
Properties of Triangles and Triangle Angle Sum Theorems Parts of a Triangle: Triangle Name Sides Vertices Angles Classifying Triangles by Angles: Acute Obtuse Right Equiangular Classifying Triangles by Sides: Scalene Isosceles Equilateral Example #1: Identify the indicated type of triangle in the figure. a.) two different isosceles triangles b.) at least one scalene triangle 4
Triangle Angle Sum Theorem Investigation: Go to www.geogebra.org/m/byh6f6ak and follow the directions in the activity. Answer the following questions (they are the same as the questions on the website): 1) What geometric transformations took place in the applet above? 2) When working with the triangle's interior angles, did any of these transformations change the measures of the blue or green angles? 3) From your observations, what is the sum of the measures of the interior angles of any triangle? 4) When working with the triangle's exterior angles, did any of these transformations change the measures of the maroon or green angles? 5) From your observations, what is the sum of the measures of the exterior angles of any triangle? Angle Sum Theorem: The sum of the measures of the interior angles of a is. Vertical Angles Linear Pairs Definition Picture Rule 5
Exterior and Remote Interior Angles in a Triangle Go to www.geogebra.org/m/sa5mb4vd and follow the directions for the investigation. Answer the following questions: 1) What can you conclude about the measure of an exterior angle of a triangle with respect to its 2 remote interior angles? 2) What other theorem is readily made obvious here? Checkpoint: The measure of any angle of a triangle is equal to the of the measures of the. Examples. Find the measure of each missing angle. 1. 2. Find 3. 4. m A= B D 20 C Find m DCB = A 56 C D A 55 27 B 6
You try these. Find the measure of the indicated angle: 7
Practice Triangle Sum Theorem and Remote Interior Angles 1. 1. 2. b= b= x= 4. 5. 6. x= x= x= 7. 8. 9. x= x= n= 10) Solve for x. 11) Solve for x. 12) 8
Isosceles Triangles Isosceles Triangle: A triangle with at least. The two sides are called the The third side is called the The two angles whose vertices are the endpoints of the are called the angles. The angle formed by the two is called the angle. Label the parts of the isosceles triangle below. Isosceles Triangle Properties Investigation: Go to www.geogebra.org/m/mxxysnzg and move the points in the triangle around. Pay attention to the measures of the sides and the angles in the triangle. What do you notices about the base angles of the triangle? Isosceles Triangle Theorem: If two sides of a triangle are, then the angles opposite those sides are. Converse of Isosceles Triangle Theorem: If two angles of a are congruent, then the sides opposite those angles are. Examples: 1. If DE CD, BC AC, and m CDE 120, what is the measure of BAC? 9
2. 1 4 5 a.) Name all of the pairs of congruent angles. 2 3 6 b) Name all of the pairs of congruent segments. 3. The vertex angle of an isosceles triangle is 40. What is the measure of one base angle? 4. The degree measure of the vertex angle is (3x - 8). The degree measure for each base angle is (6x - 41) What is the value of vertex angle? A triangle is if and only if it is. Each angle of an equilateral triangle measures. 5. EFG is equilateral, and EH bisects E. a.) Find m 1 and m 2. b.) Find x. 10
Practice Isosceles Triangles 11
12
Midsegments in a Triangle Go to www.geogebra.org/m/nfcwzehu and explore the applet. Make sure to move the vertices and try different triangles. A midsegment of a triangle is a that connects the of two sides of a triangle. In the figure D is the midpoint of and E is the midpoint of. So, is a of ABC because it connects two midpoints. In the figure D is the midpoint of and E is the midpoint of. So, is a midsegment because it connects two midpoints. The Triangle Midsegment Theorem A midsegment connecting two sides of a triangle is to the third side and is as long as the third side. If AD = DB and AE = EC, then and. Examples: 1. Find the value of x and list 1 pair of parallel segments. x = 13
**Hint: You need to use Pythagorean Theorem to find JL in the example below** You Try #3 14
12. 13. 15
Practice Midsegments in a Triangle 16
Congruence in Triangles Congruent Figures: have the same & same. Each ( matching ) side and angle of congruent figures will also be. Example #1: ABCDE Congruent Angles Congruent Sides (Points can be named in any consecutive order, but each corresponding vertex must be in the same order for each figure). You Try #1: Given: ABCD EFGH. Complete the following a) Rewrite the congruence statement in at least 2 more ways. b) Name all congruent angles c) Name all congruent sides We will deal mostly with congruent triangles. Two triangles are congruent if and only if their vertices can be matched up so that the (both angles & sides) are congruent. 17
Triangle Congruence Exploration Using AngLegs During this activity, you will compare combinations of sides and angles that may be used to prove two triangles are congruent. You have been given a bag containing AngLegs that represent segments that can be used to form a triangle. The measurements are as follows: Orange: 5 cm Purple: 7.07 cm Green: 8.66 cm Yellow: 10 cm Blue: 12.24 cm Red: 14.14 cm 1. Given: 3 side measures of 8.66 cm, 10 cm and 12.24 How many different triangles can you create? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 2. Given: 3 angle measures of 45,45,90 How many different triangles can you create? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 3. Given: 2 angle measures of 55 and 55, and one included side measuring 14.14 cm How many different triangles can you create? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 4. Given: 2 sides measuring 14.14 cm and 7.07 cm, and one included angle measuring 60 How many different triangles can you create? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 18
5. Given: 2 angles measuring 79 and 59, and one non-included side measuring 14.14 cm How many different triangles can you create? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 6. Given: One triangle with side measures of 5 cm, 8.66 cm and 10 cm and A second triangle with side measures of 5 cm, 5cm, and 8.66 cm. What do the triangles have in common? Sketch and label your triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? Right Triangle Congruence Exploration 7. Given: A hypotenuse with a measure of 12.24 cm and one leg measuring 10 cm. How many different right triangles can you create? Sketch and label your right triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 8. Given: One hypotenuse measuring 8.66 cm, and one acute angle measuring 35 How many different right triangles can you create? Sketch and label your right triangle(s): What combination of sides and angles did you use? Was this combination of sides and angles enough to establish congruence? 19
45 90 55 79 59 35 45 90 55 79 59 35 20
Triangle Congruence Notes Because triangles only have three sides, we can take some shortcuts in proving them congruent I. If all three sides are given, we call this. Postulate: If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. II. If 2 sides and the angle BETWEEN those sides are given, we call this. Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent. Included means III. If 2 angles and the side BETWEEN those angles are given, we call this. Postulate: If 2 angles and the included side are congruent to 2 angles and the include side of another triangle, then the triangles are congruent. IV. If 2 angles and the side NOT BETWEEN those angles are given, we call this. Postulate: If 2 angles and their non-included side are congruent to 2 angles and the nonincluded side of another triangle, then the triangles are congruent. 21
Example #3: Are the triangles congruent? If so, why? Anytime that 2 triangles share a side, think property! Example #4: EF HF and F is the midpoint of GI. Are the triangles congruent? If so, why? There are 3 ways that triangles are not congruent: 1. 2. 3. 22
You Try #2: State whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If none of the method work, write Not Congruent 23
Practice Triangle Congruence If the triangles can be proven congruent, give the reason (SSS, SAS, ASA, or AAS). If there is not enough information to prove the triangles congruent, write none. 1. 5. 9. 2. 6. 10. 3. 7. 11. 4. 8. 12. 24
13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 25
Practice 2 Triangle Congruence Determine whether you can conclude that another triangle is congruent to ABC. If so, complete the congruence statement and give the reason (SSS, SAS, ASA, or AAS). If not, write none. A 1. A B C 2. B D 3. K C P B N Y C A ABC ABC ABC by by by 4. A B S 5. X A 6. B A C C Y Z B C J ABC ABC ABC by by by 7. 8. 9. A P A P C 61 60 B B C Q B C Q D 60 A ABC ABC ABC by by by 26
For Problems #10-15, ΔPQR ΔABC. Find the values of x and y. 10. m R = 5x + 70, m C = 24x 25, QR = 4y + 2, BC = 6y 8 11. m R = 90 y, m C = 13, PR = 5x 10, AC = 32 x 12. PQ = 5x 31, AB = 3x + 1, QR = 3y 1, BC = 2y 2 13. m A = 15y 3, m P = 12y + 30, PQ = 11 x, AB = 11x 1 14. AB = 2x, PQ = 18, BC = 11, QR = 4x + 3 15. ΔXYZ ΔMNO, m X = x + 10, m M = 4x 47, m Y = 2y, and m N = y + 56. 27
Triangle Similarity Determining if Triangles are Similar (created by S. Harris sharris10@wcpss.net) We have already determined that similar figures have corresponding pairs of angles that are congruent and corresponding pairs of sides that are proportional. Today we are going to test ways of determining if triangles are similar when only given certain combinations of parts. For this activity, each A stands for a pair of corresponding angles and each S stands for a pair of corresponding sides. You will need a ruler, a protractor, patty paper, and dry spaghetti. SAS ~ (Side-Angle-Side Similarity): The angles below are congruent. For each angle, use your ruler to measure from the vertex along each ray and mark the length of the two sides. Label the lengths. a. Side 1: 2 cm Now multiply by a magnitude of 2 b. New Side 1: Side 2: 3 cm New Side 2: Connect the endpoints of side 1 and side 2 to form a third side for both triangles. Use your ruler to measure the third side of each triangle. Use your protractor to measure all of the angles. Label your measurements in the pictures. Are the triangles similar? How do you know? AA ~ (Angle-Angle Similarity): Use your ruler as a straightedge to help you copy the angle on the right onto patty paper. Slide your patty paper so that one of the rays is on top of the other, and the other two rays are intersecting to form a triangle. Use your ruler to help you copy the angle from your patty paper so that a triangle is formed. Copy this angle Slide it over that angle to make a triangle Measure all of the sides and angles of your triangle and label your measurements on the picture. Compare your triangle to a person nearby. Is your triangle similar to their triangle? How do you know? 28
SSS ~ (Side-Side-Side Similarity): The side lengths of the triangle below are 2.5 cm, 4.5 cm, and 6 cm. Measure each side to verify these lengths and label each with the correct measurement. Multiply by a scale factor of 3, what are the new side lengths?,, Now, use your ruler and pencil to mark your spaghetti for each of the new lengths. Use your thumbnail to break your spaghetti at each mark. Use the spaghetti to create a new triangle in the space below. Mark the vertices of your triangle, remove the spaghetti, and use your ruler to draw in the sides of the triangle. Label the measures of the sides in your picture. Measure the angles of the original triangle and the new triangle; round to the nearest degree. Label the angle measures in both pictures. Are the triangles similar? How do you know? In conclusion, you do not need to know that all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are proportional to determine that two triangles are similar. At minimum, you need only one of the following combinations of corresponding parts:,, or. Why do we not need to check and see if ASA ~ (Angle-Side-Angle Similarity) is an appropriate method for determining if two triangles are similar? 29
Examples: Are triangles similar? If so, write the similarity statement and justify. 1. D A 3 10 C B 5 6 E 2. X A 4 12 C B 7 21 Y 3. B R A 70 T 70 C 4. H B 3 4 10 8 A 5 C W 6 Y 5. B 1 P 4 5 A 13 Q 2 C 30
Practice Triangle Similarity 31
32
33
Proportions in Triangles Side Splitter Theorem: Draw the two similar triangles, set up the proportion, and solve. Example 1: x Example 2: x You Try 1: x 5 I 8 D E V 20 O Angle Bisector Theorem: Set up a proportion using the sides and the divided base. Example 3: Example 4: 34
You Try 2: Parallel Lines Proportions: Create a proportion using corresponding parts, then solve for the indicated value. Example 5: 1. Example 6: You Try 3: 35
Practice 1 Proportions in Triangles 1. 2. 3. 4. 5. 6. 7. 8. 9. 36
10. 11. 12. 13. 14. 15. 37
Practice 2 Proportions in Triangles 38
39
40
Unit 2 Review 1. List the characteristics of an isosceles triangle. 2. List the characteristics of an equilateral triangle. 3. List the characteristics of a scalene triangle. 4. Name the congruent triangle and the congruent parts.. FGH EFI FG G GH FH H 5. Use the congruency statement to fill in the corresponding congruent parts. ABC QST ACB BC AC AB A B C 6. For which value(s) of x are the triangles congruent? A x = 4x + 8 7x - 4 C Directions: For each pair of triangles, tell which postulates, if any, make the triangles congruent. 8. ABC CDA by 9. ADC by C B R B C D A A D B 41
10. ABE by 11. ACB by D C C E A B A B D Directions: For each pair of triangles, tell: (a) Are they congruent (b) Write the triangle congruency statement. (c) Give the postulate that makes them congruent. O 12. 13. L U E L V G E a. a. b. b. c. c. 14. Write a congruency statement for the two triangles at right. C G O A R E Solve for x in each of the following. 15. 16. 17. 18. 42
19. 20. 21. 22. 23. 24. 43
25. 26. 27. 28. 44
29. 30. 31. 32. x= z= 45