Plane and solid geometry . Given Z ABC in plane MN and Z DEF in plane RS with BAand BC II respectively to ED and EF, and lying on the same sideof line BE. To prove Z. ABC=Z. DEF. Argument 1. Measure off BA = ED and BC = EF 2. Draw AD, CF, AC, and DF, 3. BA 11 ED and BC II EF. 4. Then ADEB and CFEB are Z17. 5. .. ^i) = ^^ and Ci^= jBJ^. 6. .-. AD= CF, 7. Also AD 11 5^ and Ci^ li BE 8. .*. ^2) II CF. 9. .*. vlCi^D is a O. 10. .-.^(7=2)^. 11. But BA = ED and BC== EF. 12. .-. A ABC = A DEF. 13. .-. Z ABC= Z D^i^. 1 2 3 4 5 6 7 8 9 10 11 12 13 Reasons§ 122.§ 54, hyp.§240.§ 232.§ 54, 1.
Plane and solid geometry . Given Z ABC in plane MN and Z DEF in plane RS with BAand BC II respectively to ED and EF, and lying on the same sideof line BE. To prove Z. ABC=Z. DEF. Argument 1. Measure off BA = ED and BC = EF 2. Draw AD, CF, AC, and DF, 3. BA 11 ED and BC II EF. 4. Then ADEB and CFEB are Z17. 5. .. ^i) = ^^ and Ci^= jBJ^. 6. .-. AD= CF, 7. Also AD 11 5^ and Ci^ li BE 8. .*. ^2) II CF. 9. .*. vlCi^D is a O. 10. .-.^(7=2)^. 11. But BA = ED and BC== EF. 12. .-. A ABC = A DEF. 13. .-. Z ABC= Z D^i^. 1 2 3 4 5 6 7 8 9 10 11 12 13 Reasons§ 122.§ 54, hyp.§240.§ 232.§ 54, 1.§ 220.§ C18.§ 240.§ 1.§ 116.§110. Ex. 1167. Prove Prop. IX if the angles lie on opposite sides of BE* It will ulso be seen (§ C40) that the (ilaues of those angles are parallel. BOOK VI 311 Proposition X. Theorem 636. If one of two -parallel lines is perpendicular to aplane, the other also is perpendicular to the Given AB II CD and AB ± plane prove CD _L plane MN. Argument 1. Through D draw any line in plane MN, as DF, 2. Through B draw BE in plane MN 1! DF. 3. Then Z ABE = Z CZ)i^. 4. But Z ^^^ is a rt. Z CDF is a rt. Z; Ci) ± Di^, any line in plane MN through D,.\ CD ± plane MN. 5. 6. Reasons 1. § 54, 15. 2. § 179. 3. § 635. 4. § 619. 5. § 54, 1. 6. § 619. Ex. 1168. In the accompanying diagram AB and CD lie in the sameplane. Angle CBA^So^, angle BCD = S6°,angle ABE = 90°, ^£ lying in plane MN. IsCD necessarily perpendicular to plane MN?Prove your answer. Ex. 1169. Can a line be perpendicular toeach of two intersecting planes ? Prove. Ex. 1170. If one of two planes is per-pendicular to a given line, but the otheris not, the planes are not parallel. Ex. 1171. If a straight line and a plane are each perpendicular to thesame straight line, they are parallel to each other.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
