Unit 4 ay by ay ay Sections and Objectives Homework Monday October 26 U41 4.2 and 4.9 Packet Pages 1-3 Types of triangles, isosceles and equilateral triangles Page 228 (23-31, 35-37) Page 288 (5-10, 17-20, 22-26) Wednesday October 28 ½ ay U42 Friday October 30 U43 4.3 and 4.4 Packet Pages 4-7 Sum of interior angles of a triangle Exterior ngel Theorem ongruent Triangles 4.5-4.7 Packet Pages8-15 SSS, SS, S, S, HL, PT Page 236 (19-24, 41-44) Page 242 (11, 17-19, 23-25, 31-34) Finish Packet Pages 11-15 Tuesday November 3 U44 Wednesday November 4 Quiz 4.2-4.4 and 4.9 Review Page 247 (5-17 Page 288 (13-16, 42-44) Review Packet Pages8-15 U45 Friday November 6 Quiz 4.2-4.9 Review U46 Tuesday November 10 U47 Test Unit 4 None 1
hapter 4 ongruent Triangles 4.2 and 4.9 lassifying and ngle Relationships within Triangles. Isosceles triangles are triangles with two congruent sides. The two congruent sides are called legs. The third side is the base. The two angles at the base are called base angles. 2
Match the letter of the figure to the correct vocabulary word in Exercises 1 4. 1. right triangle 2. obtuse triangle 3. acute triangle 4. equiangular triangle lassify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two classifications for Exercise 7.) 5. 6. 7. For Exercises 8 10, fill in the blanks to complete each definition. 8. n isosceles triangle has congruent sides. 9. n triangle has three congruent sides. 10. triangle has no congruent sides. lassify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two classifications in Exercise 13.) 11. 12. 13. Find the side lengths of the triangle. 14. 15. Given: is isosceles with base ; E E 16. In isosceles PQR, P is the vertex angle. If m Q = 8x 4 and m R = 5x + 14, find the m P. 17. In isosceles triangle T, is the vertex angle. If = 8x 4 and m T = 5x + 14, then what is the measure of? 3
3. m X 4. m 5. PQ 6. m K t 4
4.3 and 4.4 ngle Relationships and ongruent Triangles. The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. n interior angle is formed by two sides of a triangle. n exterior angle is formed by one side of the triangle and extension of an adjacent side. Each exterior angle has two remote interior angles. remote interior angle is an interior angle that is not adjacent to the exterior angle. 5
ongruent Triangles: Two s are if their vertices can be matched up so that corresponding angles and sides of the s are. ongruence Statement: RE FOX List the corresponding s: R E corresponding sides: RE E R Examples: 1. The two s shown are. a) O b) c) O d) O = O K S E 2. The pentagons shown are. a) corresponds to b) LK c) = m E d) K = cm e) If L, name two right s in the figures. R 4 cm H 3. Given IG T, I = 14, IG = 18, G = 21, T = 2x + 7. Find x. L O The following s are, complete the congruence statement: X Y 4. YWZ W Z M N O 5. MQN 6. WT W Q P T Parts of a Triangle in terms of their relative positions. 7. Name the opposite side to. H 8. Name the included side between and. 9. Name the opposite angle to. 6
10. Name the included angle between and. State whether the pairs of figures are congruent. Explain. Exterior ngles: Find each angle measure. 37. m 38. m PRS 39. In LMN, the measure of an exterior angle at N measures 99. 1 m L x 3 2 and m M x. Find m L, m M, and m LNM. 3 40. m E and m G 41. m T and m V 42. In and EF, m m and m m E. Find m F if an exterior angle at measures 107, m (5x 2), and m (5x 5). 43. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle. 44. One of the acute angles in a right triangle measures 2x. What is the measure of the other acute angle? 7
45. The measure of one of the acute angles in a right triangle is 63.7. What is the measure of the other acute angle? 46. The measure of one of the acute angles in a right triangle is x. What is the measure of the other acute angle? 47. Find m 48. Find m< 49. Find m K and m J 50. Find m<p and m<t Use the figure at the right for problems 1-3. 1. Find m 3 if m 5 = 130 and m 4 = 70. 2. Find m 1 if m 5 = 142 and m 4 = 65. 3. Find m 2 if m 3 = 125 and m 4 = 23. 5 2 4 3 1 Use the figure at the right for problems 4-7. 4. m 6 + m 7 + m 8 =. 11 8 5. If m 6 = x, m 7 = x 20, and m 11 = 80, then x =. 6. If m 8 = 4x, m 7 = 30, and m 9 = 6x -20, then x =. 7. m 9 + m 10 + m 11 =. 6 9 7 10 For 8 12, solve for x. 8. x x 140 9. (5x) 120 35 8
4.5 4.7 Proving Triangles are congruent Ways to Prove s : SSS Postulate: (side-side-side) Three sides of one are to three sides of a second, Given: S bisects PW ; P W P S W SS Postulate: (side-angle-side) Two sides and the included angle of one are to two sides and the included angle of another. X Given: PX bisects XE; X XE P E S Postulate: (angle-side-angle) Two angles and the included side of one are to two angles and the included side of another. M // TH Given: T // MH T M H S Theorem: (angle-angle-side) Two angles and a non-included side of one are to two angles and a non-included side of another. Given: UZ bi sec ts UZ U ; UZ Z R U Z HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right are to the hypotenuse Given: T F Isosceles F with legs F, and leg of another right. PT: orresponding parts of congruent triangles are congruent F T 9
s SSS, SS, S, S, or HL State which congruence method(s) can be used to prove the s. If no method applies, write none. ll markings must correspond to your answer. 10
Fill in the congruence statement and then name the postulate that proves the s are. If the s are not, write not possible in second blank. (Leave first blank empty) *Markings must go along with your answer** 11
#1 Given: SR UT; SR // UT; S U ST // UV 1. SR UT; SR // UT; S U 1. 2. 1 4 2. 3. RST TUV 3. 4. 3 2 4. 5. ST // UV 5. #2 Given: is the midpoint of ; bisects. 1. is the midpoint of ; 1. 2. 2. 3. 3. 4. 4. 5. 1 2 5. 6. bisects. 6. #3 Given: R Q; RS QT S T 1. R Q; RS QT 1. 2. <R <Q 2. 3. RS QT 3. 4. S T 4. 12
Fill in Proofs: #1 Given: Δ Δ 1. 1. 3 4 1 2 2. 2. 3. 1 & 2 are right s. 3. 4. 1 2 4. 5. 5. 6. Δ Δ 6. #2 Given: bisects 3 4 1. 1. 1 2 2. 1 & 2 are right s 2. 3. 1 2 3. 4. 4. 5. bisects 5. 6. 3 4 6. 7. Δ Δ 7. 8. 8. 13
ongruent Triangles Proofs R S 1. Given: P S ; O is the midpoint of PS O is the midpoint of RQ O P Q 2. Given: ; is the midpoint of K 3. Given: SK // NR; SN // KR SK NR; SN KR S 1 2 3 4 R E 4. Given: // ME; ME M is the midpoint M // E N M 5. ONUS Given: ; is isosceles 7. ONUS Given: QRM QRP ; MR PR Q MQ PQ M R P 14
E 8. Given: E E E 9. Given: MK is the midpoint of MK x y x y M K 10. Given: FM 1 2 bisects MF 1 F M S 2 11. Given: PS // QS PV // QV x y x P V Q Y 1 2 12. Given: E E 1 2 E E 13. Given: E E 15
14. Given: and are supplementary and are supplementary, F E x y x y E F S 15. Given: SL SK, TS RS K L T R L K 16. Given: R, R X RX X R 17. Given: YO YX, ZO ZX YO ZO YX ZX O Y X 18. Given: F, GE F EF, GF F is isosceles G Z E F 19. Given: is a median of 20. Given: PR bisects QPS Q PR is an altitude of PQS is isosceles QPS R P 16 S