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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE 4-1 PERIOD NAME DATE 4-1 Practice Word Problem Practice Classifying Triangles Classifying Triangles 1. MUSEUMS Paul is standing in front of a museum exhibition. When he turns his head 60° to the left, he can see a statue by Donatello. When he turns his head 60° to the right, he can see a statue by Della Robbia. The two statues and Paul form the vertices of a triangle. Classify this triangle as acute, right, or obtuse. Classify each triangle as acute, equiangular, obtuse, or right. 1. 2. 100° 40° 40° obtuse 3. 85° 30° 65° 90° 60° 30° acute right 4. BOOKENDS Two bookends are shaped like right triangles. obtuse Classify each triangle in the figure at the right by its angles and sides. 5. ABC equiangular; equilateral right; scalene 6. EDC E A D 2. PAPER Marsha cuts a rectangular piece of paper in half along a diagonal. The result is two triangles. Classify these triangles as acute, right, or obtuse. C 7. BDC right; scalene equilateral obtuse; isosceles 9. LMN is an isosceles triangle, with LM = LN, LM = 3x - 2, LN = 2x + 1, and MN = 5x - 2. x = 3; LM = 7, LN = 7, MN = 13 Find the measures of the sides of KPL and classify each triangle by its sides. 10. K(-3, 2), P(2, 1), L(-2, -3) , LK = √26 ; isosceles KP = √ 26 , PL = 4 √2 11. K(5, -3), P(3, 4), L(-1, 1) 53 , PL = 5, LK = 2 √ 13 ; scalene KP = √ 12. K(-2, -6), P(-4, 0), L(3, -1) KP = 2 √ 10 , PL = 5 √ 2 , LK = 5 √ 2 ; isosceles Glencoe Geometry 13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there? 5 Chapter 4 001_018_GEOCRMC04_890513.indd 8 8 Answers Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. x = 7; FG = 12, GH = 12, FH = 12 5. DESIGNS Suzanne saw this pattern on a pentagonal floor tile. She noticed many different kinds of triangles were created by the lines on the tile. They are both right triangles. 8. FGH is an equilateral triangle with FG = x + 5, GH = 3x - 9, and FH = 2x - 2. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A3 ALGEBRA For each triangle, find x and the measure of each side. 3. WATERSKIING Kim and Cassandra are waterskiing. They are holding on to ropes that are the same length and tied to the same point on the back of a speed boat. The boat is going full speed ahead and the ropes are fully taut. B G A H F C I J E D Kim a. Identify five triangles that appear to be acute isosceles triangles. Sample answers: EBD, CAE, ADC, BGH, CHI, DIJ, EJF, AGF Cassandra Kim, Cassandra, and the point where the ropes are tied on the boat form the vertices of a triangle. The distance between Kim and Cassandra is never equal to the length of the ropes. Classify the triangle as equilateral, isosceles, or scalene. b. Identify five triangles that appear to be obtuse isosceles triangles. Sample answers: AFE, EJD, CID, BHC, BGA, AHD, CGE, ACJ isosceles Glencoe Geometry Chapter 4 4/12/08 001_018_GEOCRMC04_890513.indd 12:34:17 AM 9 9 Glencoe Geometry 4/12/08 12:34:26 AM Answers (Lesson 4-1) The bottom side of each triangle is exactly half as long as the slanted side of the triangle. If all the books between the bookends are removed and they are pushed together, they will form a single triangle. Classify the triangle that can be formed as equilateral, isosceles, or scalene. B 4. ABD PERIOD Lesson 4-1 Chapter 4 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE 4-2 PERIOD Study Guide and Intervention NAME DATE 4-2 (continued) Angles of Triangles Find the measure of each numbered angle. At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m∠1 = m∠A + m∠B Example 1 Find m∠1. Example 2 1. S TIGER 1 2. 146° 1 2 73° m∠1 = 27 B m∠1 = m∠2 = 17 Find each measure. D 1 A C Find x. P 78° Q 80° 80° 3. m∠1 55 4. m∠2 55 5. m∠3 70 85° 55° 1 2 40° 3 1 60° T x° S Exterior Angle Theorem Substitution Simplify. 55° R Find each measure. m∠PQS = m∠R + m∠S 78 = 55 + x 23 = x Exterior Angle Theorem Substitution side. Find the measures of each numbered angle. X 1. A 2. 35° 50° Y 1 65° Z m∠1 = 115 1 3 Q O M R 4. 60° 2 C D m∠1 = 60, m∠2 = 120 N 3. 2 1 25° B W 80° V 1 60° m∠1 = 60, m∠2 = 60, m∠3 = 120 3 2 35° U P S 36° T m∠1 = 109, m∠2 = 29, m∠3 = 71 Find each measure. Glencoe Geometry 5. m∠ABC 6. m∠F A B 2x° x° Chapter 4 001_018_GEOCRMC04_890513.indd 12 145° C D 50 H 12 Answers 58° 125 7. m∠2 55 8. m∠3 1 2 55° 150° 70° 95 Find each measure. 9. m∠1 140 10. m∠2 40 11. m∠3 65 12. m∠4 75 13. m∠5 115 40° 80° 60° 1 4 105° 2 5 3 Find each measure. B E 95° Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises 6. m∠1 3 Subtract 55 from each Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A5 m∠1 = m∠R + m∠S = 60 + 80 = 140 G x° F 29 Glencoe Geometry 14. m∠1 27 15. m∠2 27 Chapter 4 4/12/08 001_018_GEOCRMC04_890513.indd 12:34:41 AM 13 1 A 13 2 D 63° C Glencoe Geometry 5/22/08 8:04:02 PM Answers (Lesson 4-2) S R Skills Practice Angles of Triangles Exterior Angle Theorem Exterior Angle Theorem PERIOD Lesson 4-2 Chapter 4 NAME DATE 4-6 Study Guide and Intervention PERIOD NAME DATE 4-6 (continued) Isosceles and Equilateral Triangles Skills Practice Isosceles and Equilateral Triangles Properties of Equilateral Triangles Refer to the figure at the right. An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem leads to two corollaries about equilateral triangles. C −− −−− 1. If AC AD, name two congruent angles. 1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measures 60°. P 1 Reasons 1. 2. 3. 4. 5. 1. 2. 3. 4. 5. A −− −− EB EA C 4. If ∠CED ∠CDE, name two congruent segments. −− −− Given Each ∠ of an equilateral measures 60°. If lines, then corres. are . Substitution If a is equiangular, then it is equilateral. CE CD Find each measure. 5. m∠ABC 60 6. m∠EDF 70 # Exercises & ALGEBRA Find the value of each variable. 10 2. 6x - 5 6x° E 5 3. 20 - 5x J H . 4. " # 4x 10 X 5. 12 6 6. 15 3x + 8 60° 40 5 60° Z $ 4x - 4 Y Given: ABC is equilateral; ∠1 ∠2. Prove: ∠ADB ∠CDB 4x° 60° 7 7. PROOF Write a two-column proof. A D 1 2 B Reasons 1. Given 2. An equilateral has sides and angles. 3. Given 4. ABD CBD 5. ∠ADB ∠CDB 4. ASA Postulate 5. CPCTC C 38 Glencoe Geometry 037_054_GEOCRMC04_890513.indd 38 $ ' % ALGEBRA Find the value of each variable. 14 7. 2x + 4 3x - 10 Proof: Statements −− −− 1. CD CG 2. ∠D ∠G −− −− DE GF CDE CGF −− −− CE CF Chapter 4 4/12/08 037_054_GEOCRMC04_890513.indd 12:35:43 AM 39 21 8. (2x + 3)° 9. PROOF Write a two-column proof. −−− −−− Given: CD CG −−− −−− DE GF −−− −− Prove: CE CF 3. 4. 5. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Geometry Proof: Statements 1. ABC is equilateral. −− −− 2. AB CB; ∠A ∠C 3. ∠1 ∠2 Chapter 4 60° " , 3x° Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F 40° G Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A18 1. E 3. If ∠EBA ∠EAB, name two congruent segments. 2 Q B D ∠BEC ∠BCE Answers (Lesson 4-6) Proof: Statements −−− −−− 2. If BE BC, name two congruent angles. A Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle. D B ∠ACD ∠CDA Example −−− −−− ABC is equilateral; PQ BC. m∠A = m∠B = m∠C = 60 ∠1 ∠B, ∠2 ∠C m∠1 = 60, m∠2 = 60 APQ is equilateral. PERIOD Lesson 4-6 Chapter 4 NAME D E C F G Reasons 1. Given 2. If 2 sides of a are , then the opposite those sides are . 3. Given 4. SAS 5. CPCTC 39 Glencoe Geometry 5/22/08 8:06:48 PM