7. What is the hypothesis of a theorem? the conclusion? What is the hypothesis of Prop. III? the conclusion. If the hypothesis and conclusion of Prop. III were interchanged, how would it be stated? Do you think it is true? 54. Definition. The converse of a theorem is the theorem when the hypothesis and conclusion are interchanged. 55. PROPOSITION XIII. If two adjacent angles are supplementary, how are their exterior sides related? State and prove Prop. XIII. Is Prop. XIII the converse of any proposition? 56. PROPOSITION XIV. Given the isosceles ▲ A B C with A B and A C the equal sides. Join A with the middle point of the base B C. Prove the 2 As formed equal. How does the median meet the base? Why? How does it divide the triangle A B C? How does it divide the angle at the vertex? Write a formal statement of these three truths and call it Prop. XIV. Erect a 1 at the mid-point of the base of any isosceles A. Through what point must it pass? Let P be a point without the straight line A B, and P D be a to it. Suppose it is possible to draw another 1 from P and let P F be that 1. Can you think of any axiom which this supposition violates? Produce PD to P' making DPD P'. Join F and P'. Can you prove F P' equal to F P? If P F is 1 to A B, how large is x? What is the value of < x + < x? What kind of a line is PF P'? But what axiom is violated if our supposition is true? Can P F be L to A B? Can any other line than PD be drawn from P 1 to A B? Why? What then must we say of the supposition which led to this absurd conclusion? Write a formal statement of the truth discovered and call it Prop. XV. 58. PROPOSITION XVI. x M Let the two straight lines O P and M N intersect at I, forming the vertical angles x and x', y and y'. Prove in a similar manner that x Lx'. Write Prop. XVI. 22. Ify and true of the bisectors. 23. If the bisectors. duced through x and EXERCISES. = y' in § 58 are bisected, prove what is y are bisected, prove what is true of What can you say of the bisector of x prox'? SURVEYOR'S PROBLEM. Suppose a surveyor wishes to know the distance between A and B, with a lake between. Suppose that the distance A B is not over 300 feet and that the surrounding land is level; tell how to measure the distance required by the hint in diagram, and prove your work. H It is required to find the distance from A to B, with a river between. Supposing the surrounding land to be level and that the surveyor has a transit which measures angles and which enables him to run straight lines; can you tell from the suggestion in the diagram how to estimate the distance required? 59. PROPOSITION XVII. Given P a point outside of the line A B. What is the shortest line that can be drawn from P to A B? If P D is the shortest line, how should it be drawn? [Try to state and prove Prop. XVII. Use the above figure.] (If you fail, consult the "hint" given below.) [Hint.- Given: PDa L to A B at D.] Required: To prove P D the shortest line from P to A B. Proof: Draw any other line, P F. Produce P D its own length to P'. Draw P' F. PP' < PF P'. [Auth.] PD < PF. [Auth. Pupil must prove P' F = P F, etc.] |