4 Ready to Go On? Skills Intervention 4-1 lassifying Triangles Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle lassifying Triangles by ngle Measures. PQS Q 31 R right angle has a measure of. Since QPS is a angle, PQS is a triangle. 3 54. PRQ P 59 First, find mqrp. Since QRP and SRP form a pair, the angles are. To find mqrp, subtract 54 from. mqrp 54 What kind of angle is QRP?. So, PRQ is an triangle.. PRS First, find mrps. Since RPS and RPQ form a angle, the angles are. To find mrps, subtract 3 from. mrps 3 What kind of angle is RPS?. What kind of angles are SRP and PSR? So, PRS is an triangle. K lassifying Triangles by Side Lengths. JKL 9 How many sides are congruent in JKL? 8 What kind of triangle is JKL? J. KML 7 M 4 Find KL. How many sides are congruent, or have the same measure, in KML? So, what kind of triangle is KML? L S 41 Holt Geometry
4 Ready to Go On? Problem Intervention 4- ngle Relationships in Triangles omplementary angles have a sum of 90. roofer is making repairs on the roof of a house. In order to be safe, he sets his ladder so that it makes a 15 angle with the house. What angle will his ladder make with the ground? Understand the Problem 1. What angle does the ladder make with the house?. What angle does the house form with the ground? 3. What kind of triangle is? Mark the figure with the information given in the problem. ladder house ground Make a Plan 4. The acute angles of a right triangle are complementary, so the sum of the measures of the acute angles equals. 5. omplete: m m 90 6. Write an equation by substituting the known angle measures. m 90 Solve 7. Solve the equation you wrote in Exercise 6: m 90 Subtract 15 from both sides to isolate the variable. m 8. What angle does the roofer s ladder make with the ground? Look ack You can check your work in two ways. 9. What is the sum of the angles in a triangle? 10. From your answer in Exercise 9, you know that m m m. 11. Substitute the angle measures and check your work. 90 15 oes your answer check? 1. To check using a second method, substitute your solution from Exercise 8 into the equation you wrote in Exercise 6: 15 oes your answer check? 90 4 Holt Geometry
4 Ready To Go On? Skills Intervention 4- ngle Relationships in Triangles Find these vocabulary words in Lesson 4- and the Multilingual Glossary. Vocabulary uxiliary line corollary interior exterior interior angle exterior angle remote interior angle Finding ngle Measures in Triangles Find m. y the Triangle Sum Theorem, the sum of the angle measures in a triangle is. In this triangle, m m m 180. 108 m 180 Substitute known measures. m 180 dd. Subtract to isolate the variable. m Solve. 108 4 Finding ngle Measures in Right Triangles One of the acute angles in a right triangle measures 37.9. What is the measure of the other acute angle? Let the acute angles be T and U, with mt 37.9. Since the measures of the acute angles in a right triangle are complementary, mt mu. Substitute 37.9 for mt and solve for mu. mu Subtract to isolate the variable. mu Solve. pplying the Exterior ngles Theorem Find mq. Using the Exterior ngles Theorem, m m mprs. Substitute the given angle measures into the equation and solve for x. (5x 3) 5x 5x 8x 4 x dd. Subtract 47 from both sides. Subtract 8x from both sides. ivide both sides by 3. T S U P 44 (8x + 5) (5x x + 3) R Q Substitute the value of x into (5x 3) to find mq: (5x 3) (5)( ) 3 43 Holt Geometry
4 Ready To Go On? Skills Intervention 4-3 ongruent Triangles Find these vocabulary words in Lesson 4-3 and the Multilingual Glossary. Vocabulary corresponding angles corresponding sides congruent polygons Naming ongruent orresponding Parts Given PQR. Identify the congruent corresponding parts to and. In a congruence statement, vertices are written in corresponding. corresponds with, so. corresponds with, so. Using orresponding Parts of ongruent Triangles Given EF WXY.. Find the value of m. x 9 87 corresponds with, so. Since, m m. E 6 Substitute values for the angle measures and W. Solve to find the value of m. 87 Subtract from both sides. 85. Find E. 85 5 5 ivide both sides by 5. m Solve for m. First find the value of x. XY corresponds with, so XY and XY. XY EF Substitute values for XY and EF and solve for x. 3x 7 7 7 dd 7 to both sides. 3x 3x 3 3 x Solve for x. ivide both sides by 3. Substitute the value of x into E and simplify. E x 9 ( (5m + ) W F X ) 9 3x 7 Y 44 Holt Geometry
4 Ready to Go On? Quiz 4-1 lassifying Triangles lassify each triangle by its angle measures. Q 1. QPR 30 T. SRQ 3. TRQ P 60 60 R 30 S lassify each triangle by its side lengths. M 4. QNM 5. MPQ 8 = = 8 6. NLM 4- ngle Relationships in Triangles Find each angle measure. 7. mgf 8. m 6 L P Q N 8 (9x 8) (15x + 14) (1x 7) G F (8x + 19) 9. high school baseball team is designing a pennant with the school logo. The pennant is an isosceles triangle and the measure of the vertex angle is 46. Find the measure of the base angles. 46 45 Holt Geometry
4 Ready to Go On? Quiz continued 4-3 ongruent Triangles Given MNO GHI. Identify the congruent corresponding parts. 10. MO 11. GH 1. N 13. G Given LMN. Find each value. 14. LM L 7t 1 15. x 33 (3x) M 5t + 4 9 N 16. Given: RS UT, UR TS, RS UT, UR TS Prove: URT STR R S omplete the proof. U T Statements 1. RS UT 1. Reasons. SRT UTR. 3. UR TS 3. 4. 4. lt Int. Thm. 5. RUT RST 5. Third Thm. 6. RS UT 6. 7. 7. ef. segments 8. UR TS 8. 9. 9. Reflex. Prop of 10. URT STR 10. 46 Holt Geometry
4 Ready to Go On? Enrichment Exploring Exterior ngles For Exercises 14, find the angle measures. 1. m. m 3. m 4. me 5. What is the sum of the measures of the exterior angles of the triangle? E 47 118 For Exercises 69, find the angle measures. 6. m1 7. m 38 8. m3 9. m4 3 1 4 10. What is the sum of the measures of the exterior angles of the triangle? P For Exercises 1117, find the indicated values. 11. x 1. mqsr Q (7x 3) (15x 7) R 13. mqsu U S (x 13) T 14. mqrs 15. msrt 16. msqr 17. mpqr 18. What is the sum of the measures of the exterior angles of the triangle? 19. Make a conjecture about the sum of the measures of the exterior angles of a triangle. 47 Holt Geometry
4 Ready to Go On? Skills Intervention 4-4 Triangle ongruence: SSS and SS Find these vocabulary words in Lesson 4-4 and the Multilingual Glossary. Vocabulary triangle rigidity included angle JK Using SSS and SS to Prove Triangles ongruent ML and JK ML. Use SS to explain why JKM LMK. It is given that JK ML. This means that segment JK is to segment ML. Mark this information on the figure. It is given that JK ML M. This means that segment JK is to segment ML. Mark this information on the figure. J K L Since JK ML, you know that LMK because of the Theorem. y the Reflexive Property of ongruence, you know that MK. Therefore, by. Proving Triangles ongruent Given:, bisects. Prove: It is given that and bisects. Mark this information on the figure. Since bisects, you know that because of the definition of an. Enter this information in Step of the proof. y the Reflexive Property of ongruence, you know that. Enter this information in Step 3 of the proof. Therefore, you know that by. Enter this information in Step 4 of the proof. Statements 1., bisects 1. Given Reasons.. 3. 3. 4. 4. 48 Holt Geometry
4 Ready to Go On? Problem Solving Intervention 4-4 Triangle ongruence: SSS and SS Engineers often use triangles in designing structures because of their rigidity. The figure shows a radio tower supported by cables of equal length. M is the midpoint of LN. Use SSS to explain why PML PMN. P Understand the Problem 1. Why do you think a radio tower needs to be supported by cables? L M N. Why do the cables form triangles with the tower and the ground? 3. The problem asks you to Use SSS to explain why PML PMN. When you explain something in Geometry, you must essentially write a paragraph proof. For every statement you make about the situation, you must also provide a. Make a Plan The problem gives you information about the triangles that are formed by the tower, the cables, and the ground. Mark the figure with the given information as you answer each of the questions. 4. The sentence The figure shows a tower supported by cables of equal length, tells you that PN, and therefore,. 5. The phrase M is the midpoint of LN, tells you that. 6. The segment is congruent to itself. Solve Write a paragraph using the information you found in Exercises 46. Include justifications in your paragraph. 7. It is given that, so by the definition of segments. y of a midpoint,. y the Property of ongruence,. Therefore, PML PMN by. Look ack 8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle must be congruent to sides of a second triangle. 9. Have you proven that three sides of PML are congruent to three sides of PMN? How? 49 Holt Geometry
4 Ready to Go On? Skills Intervention 4-5 Triangle ongruence: S, S, and HL Find this vocabulary word in Lesson 4-5 and the Multilingual Glossary. Vocabulary included side pplying HL ongruence etermine if you can use the HL ongruence Theorem to prove the triangles congruent. Explain.. QPR and SRP ccording to the diagram, QPR and SRP are triangles that share leg. Q P S R by the Reflexive Property of ongruence. Is any information given to you about the hypotenuse of the right triangles? This conclusion be proven by HL. You need to know that the of the triangles are.. E and E ccording to the diagram, E and E are triangles that share hypotenuse. by the Reflexive Property of ongruence. It is given that, therefore by HL. E Using S to Prove Triangles ongruent Given: J L, JK ML Prove: JKM LMK Mark the given information on the figure. Since it is given that JK ML M, you know that. ecause of the Property of ongruence, you know that. Therefore, you know that because of S. omplete the flow-chart. 1. JK ML. Given. J L 3. Given 4. J L S K 3. 50 Holt Geometry
4 Ready to Go On? Skills Intervention 4-6 Triangle ongruence: PT Find this vocabulary word in Lesson 4-6 and the Multilingual Glossary. Proving orresponding Parts ongruent Given: is the midpoint of ; E Prove: E Mark the given information on the figure: is the midpoint of and E. Fill the given information into Step 1 and Step 3 of the flow-chart proof below. Vocabulary PT E Since is the midpoint of, you know that, because of the definition of a. Fill this information into Step of the proof. Since E, you know that and E because of the ngles Theorem. Fill this information into Step 4 of your proof. Therefore, E by and by PT. Fill this information into Steps 5 and 6 of your proof. omplete the flow-chart: 1.. Given. 5. 6. 3. Given 4. 5. 6. 4. 51 Holt Geometry
4 Ready to Go On? Skills Intervention 4-7 Introduction to oordinate Proof Find this vocabulary word in Lesson 4-7 and the Multilingual Glossary. Vocabulary coordinate proof Positioning a Figure in the oordinate Plane Position a right triangle with legs of 7 units and units in the coordinate plane. Use the origin as the vertex of the right angle. ount ount units spaces to the right to find a second vertex. from the origin to find the third vertex. onnect the vertices to form a right triangle. Label the vertices with their coordinates. y 4 6 x ssigning oordinates to Vertices Position square LMNO in the coordinate plane and give the coordinates of each vertex. Use the origin as one vertex of the square. Label it L. raw another vertex on the x-axis, to the right of origin. Label this vertex M(a, 0). Move the same distance up from the origin on the y-axis and label this vertex O(0, a). escribe where to place vertex N. What are the coordinates of this vertex? onnect the vertices to form a square. Writing a oordinate Proof Use the square LMNO you drew above to prove that LN MO. omplete and use the distance formula: d ( x 1 ) ( y ) Substitute the coordinates of L and N into the distance formula to find LN. Simplify. LN ( x x 1 ) ( y y 1 ) (a ) ( 0 ) Substitute the coordinates of M and O into the distance formula to find MO. Simplify. MO ( x x 1 ) ( y y 1 ) (0 ) (a ) oes LN MO? So, MO because of the definition of congruent segments. 5 Holt Geometry
4 Ready to Go On? Skills Intervention 4-8 Isosceles and Equilateral Triangles Find these vocabulary words in Lesson 4-8 and the Multilingual Glossary. Vocabulary legs of an isosceles triangle vertex angle base base angle Finding the Measure of an ngle Find ml. Look at the diagram. What type of triangle is JKL? From the Isosceles Triangle Theorem, you know that (7x + 4) (9x 1) L. Therefore, m m. L K ml mk Substitute the given values and solve to find x. 7x 4 7x 16 dd 1 to both sides. 16 Subtract 7x from both sides. x ivide both sides by. Substitute the value of x into ml and simplify. J ml 7x 4 7( ) 4 4 L(0, 4b) Using oordinate Proof Given: Isosceles JKL has coordinates J(a, 0), K(a, 0) and L(0, 4b). M is the midpoint of JL, N is the midpoint of KL, and O is the midpoint of JK. Prove: MNO is isosceles. J(a, 0) K(a, 0) Use the Midpoint Formula M x 1 + x, y 1 + y to find the coordinates of M, N, and O. oordinates of M oordinates of N oordinates of O M 0, 4b N 0, 0 O a, 0 (a, ) (a, ) (0, ) raw MNO on the diagram above. Substitute the coordinates into the istance Formula and simplify to find OM and ON. OM ( x x 1 ) ( y y 1 ) ( 0 ) (b 0 ) ON ( x x 1 ) ( y y 1 ) (a 0 ) ( 0 ) oes OM ON? Since OM ON, by definition, ON. Therefore, MNO is an triangle. 53 Holt Geometry
= Name ate lass 4 Ready to Go On? Quiz 4-4 Triangle ongruence SSS and SS 1. The figure shows the logo used for a department store. Given that KI bisects HKJ and KH KJ, use SS to explain why KIH KIJ. K H I J. Given: UV TW, UV TW Prove : VUW TWU T U W V Statements 1. 1... 3. 3. 4. 4. Reasons 4-5 Triangle ongruence S, S, and HL etermine if you can use the HL ongruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 3. and 4. NMO and PMO O = N M P 5. Use S to prove the triangles congruent. Given: K is the midpoint of OM, ON LM Prove: LMK NOK O K L M 1.. N Given. 5. 3. Given 4. 4. 5. 54 Holt Geometry
4 Ready to Go On? Quiz continued 4-6 Triangle ongruence PT 6. Given TU RS, TU RS Prove: QS QT T R 4-7 Introduction to oordinate Proof Position each figure in the coordinate plane. Q S U Statements 1. 1... 3. 3. 4. 4. 5. 5. 7. a square with length 5 units 8. a right triangle with legs 5 units in length. Reasons O O 9. ssign coordinates to each vertex and write a coordinate proof Given: rectangle WXYZ Prove: WX YZ O 4-8 Isosceles and Equilateral Triangles Find each angle measure. 10. mq Z Q 11. me (x + 4) (x 17) Y = = E 48 1. Given: Isosceles triangle LMN has coordinates L(0, b), M(a, 0), and N(0, b). X is the midpoint of LM and Y is the midpoint of NM. Prove: XMY is isosceles. O 55 Holt Geometry
= Name ate lass 4 Ready to Go On? Enrichment Trying Triangles y 8 N 1. In the figure at the right, X is the midpoint of. Write a paragraph to explain whether or not XM XN. M 6y y 5 X 4y + 3 y 5. In the figure at the right, ML NO, and mmol (x ). Find NOM. M (15x + 4y) N (5x 4y) 3. Figure has coordinates (, 5), (5, 1), (1, ) and (, ). m m m m. What type of figure is? oes? Explain how you got your answers. L 4 y O x 4 O 4 4 4. What kind of triangle is formed by the lines y 9x 3, x y, and x 9y 3? Explain your answer. 4 y x 4 O 4 4 5. In the figure at right,. Is this enough information to show that? Explain your reasoning. = 56 Holt Geometry