Given the right As A B C and D E F having the hypotenuse. A C of the first equal to the hypotenuse D F of the second A, and x = x'. If not, superpose DEF upon ▲ A B C so that D shall fall on A and D F take the direction of A C. Where will F fall? Why? What direction will F E take? Why? Will E fall on B? Why? If not, how many Is would be drawn from point A to the same line? Suppose y = y' and that we know nothing of angles x and x'; prove the As equal. Write the formal statement of the truth proved and call it Prop. XVIII. Given the right As A B C and D F F with A CDF and A B DE. Can you prove the As equal? Place DEF on ▲ A B C so that D will fall on A and let D E take the direction of A B and let F fall on F'. Must E fall on B? Why? Will C B and F E form one straight line? form a A? Why? Prove F = C. any point in the 1 x D; (4) P B and P A lines cutting off of y z the unequal distances D B and D A, D B being less than D A. Can you prove PB less than PA? Produce PD and make DQ=DP. Make D B' = D B. Join Q to B' and to A. Produce QB' to F. Prove B'Q=B'P, and AQ=A P. How does the broken line P B'Q compare with PDQ? Prove P F B' > PB'. Prove PF B'Q > P B'Q. Can you prove QAP> QFP? How does Q A P compare with Q B' P? How does A P compare with B' P? How does A P compare with B P? Write the general truth proved and call it Prop. XX. EXERCISES. Given: Construct the following triangles: 25. Two sides and the angle opposite one of them. Discuss, showing under what conditions the construction is possible. 27. Hypotenuse and one adjacent angle of a rt. A. Discuss. 28. Two legs of a rt. A. - .29. One side and an adjacent acute angle of art. A. 30. Hypotenuse and one leg of a rt. A. Discuss. 31. Prove the side of a ▲ is greater than the difference of the other two sides. 32. How are the altitudes to the equal sides of an isosceles triangle related? Prove your answer. 63. PROPOSITION XXI. B Given ▲ A B C and P any point within the A. Compare xy with a. Compare x + z with b. Compare y + z with c. z> Can you now prove that x+y+> } (a+b+c)? PARALLELS. 64. PROPOSITION XXII. How many straight lines can be drawn through a given point parallel to a given straight line? Is your answer selfevident? [See § 14, 12.] If two straight lines in the same plane are 1 to to the same line, can you prove these two lines ||? If they are not ||, can you show that any P. P. is violated? Write the formal statement of the truth proved and call it Prop. XXII. 65. PROPOSITION XXIII. In how many ways can you draw a line through a given point parallel to a given line? Show them and try to prove them. Given m and n 2 || lines, and A B 1 to m at P. Is A B to n? If we suppose that n is not to A Bat Q, draw H I through QL to A B How is H I related to m? But how is n related to m? What then is true of the line H I and n? proposition and call it Prop. XXIII. EXERCISE. 33. Draw from any point a 1 to 2 || lines. 66. PROPOSITION XXIV. Write the Can you prove E F and C D to each other? [Hint.-Draw a 1 to A B.] Write the general statement and call it Prop. XXIV. Given any 2 lines, A B and C D, cut by a third line, E F How many angles are formed? If there were three lines cut by E F, how many s would be formed? E F is called a transversal or secant line. Write carefully a definition of a transversal. |