sequently PR, which is equal to PD+D R, is equal to PD+DQ. Therefore E C=P D+D Q.

2. The line joining the middle points of two sides of a triangle is equal to half the third side, and is parallel to it.

Let ABC be the triangle, D and E the middle points of AC and A B. Produce ED to F, and draw CF parallel to A B. The triangles A E D, CF D, are equal, for the angle A= the angle D CF, because they are alternate; the angle ADE the angle C D F, and A D D C. Therefore D F=D E, and CF=A E, and therefore C F= Since C F and B E are parallel and equal, EF and BC are parallel and equal; consequently ED, which is half E F, is half B C.

BE.

Fig. 122.

Questions for Examination.

1. Define parallel straight lines.

2. Prove that two straight lines perpendicular to a third are arallel.

3. Every straight line perpendicular to one of two parallel straight lines is perpendicular to the other.

4. Show how to draw through a given point a straight line paralled to a given straight line,—

Ist. On a drawing board.

2nd. On land.

5. Give the names of each of the eight angles formed when one straight line cuts two others.

6. Two parallel straight lines are cut by a third; prove the equality,

Ist. Of the interior alternate angles.

2nd. Of the exterior alternate angles.

3rd. Of the corresponding angles.

7. Prove that the sum of the two interior angles on the same side of the transversal is equal to two right-angles.

8. Prove that two angles are equal when their sides are parallel and directed the same way from the vertex.

9. Two parallel lines are at the same distance from one another throughout. Explain the use and construction of the carpenter's guage.

10. Define the terms quadrilateral, parallelogram, rhombus, rectangle, square, and trapezoid.

II. Prove that the parts of two parallel straight lines intercepted between two other parallel straight lines are equal.

12. Prove that if the parts of two straight lines which are intercepted between two parallel straight lines are equal, they are parallel.

13. The diagonals of a parallelogram bisect each other.

14. Any line drawn through the centre of a parallelogram to the opposite side is bisected in the centre.

15. If two such lines be drawn, the lines joining their extremities will be parallel.

16. Construct a parallelogram having given two sides and included angle.

17. Prove that the sum of the angles,

Ist. Of a parallelogram is equal to four right-angles.

2nd. Of a triangle is equal to two right-angles.

3rd. Of any quadrilateral to four.

4th. Of a figure with five sides to six.

5th. Of a figure with six sides to eight.

18. The diagonals of a rhombus are at right-angles, and are axes of symmetry of the figure.

19. What is meant by centre of symmetry. Prove that the intersection of the diagonals of rhombus is a centre of symmetry.

20. The diagonals of a square intersect in a centre of symmetry.

21. The figure formed by joining the extremities of two straight lines at right-angles, and intersecting in the centre of a square, is a square.

22. Describe a tenon and mortise joint.
23. Give applications of the parallelogram.

Problems and Theorems.

1. Find the locus of points,

Ist. Equally distant from a given line.

2nd. Equally distant from two given straight lines.

3rd. Equally distant from a given circle.

4th. Of the middle points of lines drawn from a given point

to a given straight line.

2. From a given point draw to a straight line another line making with it a given angle.

3. If two angles have their sides parallel, their bisectors are parallel or perpendicular. The same if the sides are perpendicular. 4. Show how to draw a straight line of given length, such that it shall be parallel to a given straight line and shall have its two extremities on two other given straight lines.

5. Find the locus of points, the sum of whose distances from two given lines is equal to a given length.

6. The same when the differences of the distances are equal to the given length.

7. Prove that whatever point be taken within an equilateral triangle, the sum of its distances from the three sides is the

same.

8. One side of a triangle being given, as well as the length of the middle line from one of its extremities, find the locus of the vertex opposite the given side.

9. Through a given point draw a straight line of a given length, so that its extremities are on two parallel straight lines.

10. Through a given point draw a straight line which shall be bisected in that point, and have its extremities on two given straight lines which intersect.

II. Draw a straight line through a given point, and through three straight lines, which intersect so that the parts of it intercepted between these straight lines shall be equal.

12. The sum of the four sides of a quadrilateral is greater than the sum and less than twice the sum of the diagonals.

13. Construct a parallelogram, having given two adjacent sides and half the diagonal through their point of intersection. 14. Prove that

Ist. If the diagonals of a quadrilateral bisect one another, the figure is a parallelogram.

2nd. If they bisect one another and are equal, the figure is a rectangle.

3rd. If they bisect one another at right-angles and are equal, the figure is a square.

15. Through a given point to draw a straight line bisecting a given parallelogram.

16. Trisect a right-angle.

17. Trisect an angle of 60°.

18. In a right-angled triangle if one acute angle be twice the other, the hypothenuse is double of the shorter side.

19. The diagonal of a rectangle is longer than any other straight line intercepted by the sides of the rectangle.

20. A straight line is drawn at the extremity of the base of an isosceles triangle at right-angles to the side. Prove that it makes

with the base an angle equal to half the vertical ungle of the triangle.

21. Two quadrilaterals are equal when they have one angle and the four sides of the one respectively equal to one angle and the four sides of the other.

22. Two trapezoids are equal when the four sides of the one are respectively equal to the four sides of the other.

23. If through the angular points of a quadrilateral straight lines be drawn parallel to the diagonals, the figure thus formed is double the quadrilateral.

24. The straight lines joining the middle points of the sides of a quadrilateral form a parallelogram.

25. The bisectors of the angles of a quadrilateral form a quadrilateral, of which the opposite angles are supplementary. When the first quadrilateral is a parallelogram the second is a rectangle; when the first is a rectangle the second is asquare.

26. In every isosceles trapezoid the opposite angles are supplementary.

27. A ball is struck from a point on a rectangular billiard board, and after rebounding from each side reaches the point of departure. The path of the ball before and after impact makes equal angles with the side; draw the path described.

28. Two equal straight lines, AB, CD, which are drawn between two parallels, A C, D B, intersect in the point O. Prove that A O CO and BO=DO.

29. Prove that in any triangle,—

Ist. The three perpendiculars drawn at the middle points of the sides meet in a point.

2nd. The three lines from the middle points of the sides to the opposite angles meet in a point.

3rd. The three perpendiculars drawn from the angular points to the opposite sides meet in a point.

30. The three points of intersection in the above question are in the same straight line, and the distance of the first from the second is half that from the second to the third.

31. If from points in the base of an isosceles triangle perpendiculars be drawn to the sides, their sum is the same for each of the points.

32. Construct a parallelogram having given,

Ist. Two adjacent sides and a diagonal.

2nd. One side and the two diagonals.

3rd. The two diagonals and the angle between them.

4th. The perimeter, one side, and one angle.

33. Construct a rectangle having given,—

Ist. The diagonal and a side.

2nd. The diagonal and the angle it makes with a side.

3rd. A side and the angle between the diagonals.

34. Construct a rhombus, having given the two diagonals. 35. Construct a square having its diagonal given.

36. Construct a quadrilateral, having the four sides and the straight line which joins the middle of the opposite sides.

37. In a triangle place a rhombus, one of the angles of which coincides with an angle of the triangle.

38. Find a single square which shall be equivalent to two equal squares.

Arithmetical Questions.

1. In a right-angled triangle one acute angle is double of the other give the magnitude of the angles. Ans. 30° and 60°. 2. In a triangle two angles are equal and the third is double either of the first; find the angles. Ans. 45°, 45°, and 90°. 3. One angle of a parallelogram is 25°: find the others.

4. Two angles of a triangle are 25° 13′ 15′′ and 56° 14′ 13′′: 'find the third.

5. Two angles of a triangle are 57° 13′ 45′′ and 95° 46′ 15′′ : find the three exterior angles.

6. In an isosceles trapezoid one angle is 39586": find the others. Ans. 169°, 14", and 10° 59′ 46′′. 7. The angle at the vertex of an isosceles triangle is 25° 15′ 46′′: find the angle at the base. Ans. 77° 22′ 7". 8. In a five sided figure the angles are equal: find them. Ans. 108°.

9. What is the magnitude of the angle in a six sided figure, the angles of which are equal.

10. The same for a seven sided figure.

II. The same for an eight sided figure.

CHAPTER VI.

AREAS.

Equivalent Figures.

Ans. 120°.

Ans. 128.

Ans. 135°.

139. The amount of surface presented by a figure is termed its area.

140. Figures which are equal in area but differ in shape are termed equivalent figures.

Suppose any figure to be cut into pieces, and the parts to be placed in contact in different ways, the new figures formed will be equivalent to the first.