Problem 11.

To draw a line from a given point D, outside a given line AB, making with the given line, an angle equal to a given angle C.

1. From the point D draw a line DE parallel to AB (Pr. 9).

2. At the point D make an angle EDF equal to the angle C (Pr. 10).

3. Produce DF to meet AB in F. Then the angle DFB will be equal to the angle EDF, which is equal to the given angle C (constr.)

NOTE. We know from Euclid I. 29, that "if a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another," hence angle EDF is equal to angle DFB.

Problem 12.

To bisect the angle made by any two given converging lines AB and CD, when the angular point is inaccessible.

1. Draw any two parallel lines EF and GH, across AB and CD (Pr. 8).

2. Bisect EF and GH in the points K and L respectively (Pr. 1).

3. Draw a line through K and I, the points of bisection. This line, if produced, will bisect the angle made by the produced given lines ÁB, CD.

Problem 13.

To draw straight lines from any two given points A and B, outside a given straight line CD, so as to make equal angles with the given line CD.

E

1. From A, let fall a perpendicular AE on CD (Pr. 3), and
produce it indefinitely towards F.

2. Make EF equal to AE, and join BF, cutting CD in G.
3. Draw the line AG. Then the angles AGC and BGD are
equal, and AG and BG are the required lines.

Problem 14.

To draw straight lines from any two given points A and B outside a given straight line CD, and to meet CD, so that they may be equal in length.

1. Draw the straight line AB, and bisect it in E (Pr. 1).

2. Produce the bisecting line to meet CD in F.

退

F

D

3. Join AF and BF, which will be the two required lines.

Problem 15.

To divide a given line AB into any number of equal parts (say in this case five).

1. Draw a line AC at any angle with AB, and draw BD, making the angle ABD equal to the angle at A (Pr. 10).

2. Commencing at A, mark off on AC the number of points, less one, that AB is to be divided into, i.e., set off four equal parts of any length, as 1, 2, 3, 4.

3. From B, mark off on BD the same number of equal parts, as 1, 2, 3, 4.

4. Join 1 4, 2 3, 3 2, &c., and the given line AB will be divided into five equal parts.

1. Draw a line AC of any length, and making any angle with AB.

2. From A mark off any five spaces on AC.

3. From the end of the last equal space, draw a line to B, as 5B.

4. From the remaining points of division between A and 5, draw lines to AB, but parallel to 5B, as 4 4, 3 3, &c., then the given straight line AB will be divided into five equal parts.

Problem 16.

To divide any line AB proportionally to a given divided line CD.

1. Draw a line similarly divided to CD, parallel to AB, and

at any distance from it (Pr. 8).

B

2. From A and B, draw AE and BE through the ends of this line to meet in E.

3. Draw lines from E, through the points of division, 1, 2, 3, 4, cutting AB in 1, 2, 3, 4, then AB is divided proportionally to the given divided line CD.

Another Method.

2

23 4
23 4

1. Draw CE equal to AB, and at any angle with CD, and join ED.

2. Draw 4 4, 3 3, &c., parallel to ED.

3. Transfer the divisions on the line CE to the given line AB, and then the divisions on the given line AB will bear the same proportion to each other that the divisions on CE bear to each other.

Problem 17.

To draw lines from any two given points A and B, which shall go to the same point to which any two given lines CD and EF converge, when the point of convergence is inaccessible.

1. From A, through B, draw AL. From any point G in the line EF, draw GH of unlimited length, parallel to AL (Pr. 9), cutting CD in K.

2. From L, draw LM of unlimited length, and at any

angle.