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Chapter 4 Geometry Homework Answers Sec 4.1 2) 3) 4) 5) 6) 7) 8) 9) 73 60 110 24 3 x 360 – 180 = 900 3 x 180 – 180 = 360 a = 69, b = 47, c = 116, d = 93, e = 86 m = 30, n = 50, p = 82, q = 28, r = 32, s = 78, t = 118, u = 50 14-16) See description. 17) False. See diagram. 18) False. See diagram. 19) False. See diagram. 20) False. See diagram. 21) True. • Answers in degrees unless labeled otherwise. Sec 4.2 1) 2) 3) 4) 5) 6) 7) 79 16) Parallel 54 17) Parallel 107.5 18) Neither 44; 35 cm 21) New: (6, -3), (2, -5), (3, 0). Triangles are congruent. 76; 3.5 cm 72; 10 cm a = 124, b = 56, c = 56, d = 38, e = 38, 22) New: (3, -3), (-3, -1), (-1, -5). Triangles are congruent. f = 76, g = 66, h = 104, k = 76, n = 86, p = 38 8) a = 36, b = 36, c = 72, d = 108, e = 36; none 10) NCA • Answers in degrees unless labeled 11) IEC otherwise. 15) Perpendicular Using Your Algebra Skills 4: Writing Linear Equations 1-3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) See graphs. y = -x + 2 y = (-6/13)x + (74/13) y = x+1 y = -3x +5 y = (2/5)x – (8/5) y = 80 + 4x y = -3x + 26 y = (-1/4)x – 3 y = (6/5)x y=x+1 y = (-2/9)x + (43/9) Sec 4.3 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) Yes 14) No. See diagram. 15) No. See diagram. 16) Yes 17) a, b, c 19) c, b, a 20) b, a, c 21) a, c, b 22) a, b, c 23) v, z, y, w, x 6 < length < 102 By the Triangle Inequality Conjecture, the sum of 11 cm and 25 cm should be greater than 48 cm. • 13) By the Triangle Sum Conjecture, the third angle must measure 36° in the small triangle, but it measures 32° in the large triangle. These are the same angle, so they can’t have different measures. 135 72 72 See description. a = 52, b = 38, c = 110, d = 35 a = 90, b = 68, c = 112, d = 112, e = 68, f = 56, g = 124, h = 124 ABE FNK cannot be determined Answers in degrees unless labeled otherwise. Sec 4.4 1) 2) 3) 4) 5) 6) SAS not correspond. SSS 12)Cannot be determined. Parts do not correspond. cannot be determined 13)SAO by SAS SSS 14)AIN by SSS or SAS SAS 15)RAY by SAS SSS (and the converse of the Isosceles Triangle Conjecture) 16)The midpoint of SD and PR is (0, 0). Therefore, ΔDRO ≈ ΔSPO by SAS. 8) b = 55, but 55 + 130 > 180, which is impossible by the Triangle Sum 20)a = 37, b = 143, c = 37, d = 58, e = Conjecture. 37, f = 53, g = 48, h = 84, k = 96, m = 26, p = 69, r = 111, s = 69 9) FLE by SSS 10)Cannot be determined. SSA is not a congruence conjecture. • Answers in degrees unless labeled otherwise. 11)Cannot be determined. Parts do Sec 4.5 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) ASA ASA SAA Cannot be determined. Cannot be determined. Cannot be determined. FED by SSS WTA by ASA or SAA SAT by SAS PRN by ASA or SAS; SRE by ASA Cannot be determined. Parts do not correspond. 12) MRA by SAS 13) Cannot be determined. AAA does not guarantee congruence. 14) WKL by ASA 15) Yes, three exterior triangles are congruent by SAS. Sec 4.6 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) Yes; AAS Yes; ASA Cannot be determined. Yes; AAS Yes; SSS Yes; SAS Yes; SSS Cannot be determined. Cannot be determined. Yes. The triangles are congruent by SAS. Yes. The triangles are congruent by SAS, and the angles are congruent by CPCTC. See description. Cannot be determined. KEI by ASA UTE by SAS 18) 23) a = 112, b = 68, c = 44, d = 44, e = 136, f = 68, g = 68, h = 56, k = 68, l = 56, m = 124 x = 3, y = 10 • Answers to # 18 are in degrees. Sec 4.7 1) 2) 3) 4) 5) 6) 8) The angle bisector does not go to the midpoint of the opposite side in any triangle, only in an isosceles triangle. 1. Given, 2. Given, 3. Vertical Angles Conjecture, 4. Δ ESM ≈ Δ USO, 5. CPCTC 2. Given, 4. Definition of midpoint, 5. Vertical Angles Conjecture, 6. Δ CIL ≈ Δ MIB by SAS, 7. CL ≈ MB 5. Given, 6. Δ ESN by SSS, 7. ‹ E by CPCTC 2. ‹ 2 is an angle bisector, 3. Given, 4. Same segment, 5. ΔWNS ≈ ΔENS by SAA, 6. CPCTC, 7. Definition of isosceles triangle 1. Given, 2. Given, 4. ‹ 1 ≈ ‹ 2 by AIA Conjecture, 6. Δ ESN ≈ Δ ANS by ASA, 7. SA ≈ NE by CPCTC – This proof shows that in a parallelogram, opposite sides are congruent. NE, because it is across from the smallest angle in Δ NAE. It is shorter than AE, which is across from the smallest angle in Δ LAE. 9) 10) 11) 12) 13) • The triangles are congruent by SSS, so the two central angles cannot have different measures. PRN by ASA, SRE by ASA Cannot be determined. Parts do not correspond. ACK by SSS a = 72, b = 36, c = 144, d = 36, e = 144, f = 18, g = 162, h = 144, j = 36, k = 54, m = 126 Answers to # 13 are in degrees. Sec 4.8 1) 2) 3) 4) 5) 6 = 52, j = 70, k = 70, l = 40, m = 110, n = 58 90° 45° 2. ‹ 1 ≈ ‹ 2, 5. SAS 4. CPCTC, 7. Definition of perpendicular, 8. CD is an • Answers to # 11 are in altitude degrees. 6) See diagram. 11)a = 128, b = 128, c = 52, d = 76, e = 104, f = 104, g = 76, h Ch 4 Review 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) Their rigidity gives strength The Triangle Sum Conjecture states that the sum of the measures of the angles in every triangle is 180°. Possible answers: It applies to all triangles; many other conjectures rely on it. The angle bisector of the vertex angle is also the median and the altitude. The distance between A and B is along the segment connecting them. The distance from A to C to B can’t be shorter than the distance from A to B. Therefore AC + CB > AB. Points A, B, and C form a triangle. Therefore the sum of the lengths of any two sides is greater than the length of the third side. SSS, SAS, ASA, or SAA In some cases two different triangles can be constructed using the same two sides and non-included angle. Cannot be determined ΔTOP ≅ ΔZAP by SAA ΔMSE ≅ ΔOSU by SSS Cannot be determined ΔTRP ≅ ΔAPR by SAS Cannot be determined ΔCGH ≅ ΔNGI by SAS ΔABE ≅ ΔDCE by SAA or ASA ΔACN ≅ ΔRBO or OBR by SAS ΔAMD ≅ ΔUMT by SAS, AD ≅ UT by CPCTC Cannot be determined Ch 4 Review cont. 18) 19) 20) 21) 22) 23) 24) 25) Cannot be determined ΔTRI ≅ ΔALS by SAA, TR ≅ AL by CPCTC ΔSVE ≅ ΔNIK by SSS, EI ≅ KV by overlapping segments property Cannot be determined Cannot be Determined ΔLAZ ≅ ΔIAR by ASA, ΔLRI ≅ ΔIZL by ASA, ΔLRD ≅ ΔIZD by ASA ΔPTS ≅ ΔTPO by ASA or SAA ΔANG is isosceles, so ∠𝐴 ≅ ∠𝐺. However, the sum of m∠𝐴 + m∠𝑁 + m∠𝐺 = 188°. The measures of the three angles of a triangle must sum to 180°. 26) ΔROW ≅ ΔNOG by ASA, implying that OW ≅ OG. However, the two segments shown are not equal in measure. 27) a = g < e = d = b = f < c. Thus, c is the longest segment and a and g are the shortest. 28) X = 20°