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23. Given the sum of the diagonal and a side of a square, construct the square.

24. If BE is parallel to the base AC of the triangle ABC, and also bisects the exterior angle CBD, prove that the triangle ABC is isosceles.

25. Given the difference between the diagonal and a side of a square, construct the square.

26. Draw DE parallel to the base of the triangle ABC so that DE = DA + EC.

Two constructions.

longed sides.

A

-E

E

DE may cut the pro

Through E,

27. ABCD is a trapezoid.

the middle of CD, draw FG parallel to BA

and meeting BC produced at F.

Prove the parallelogram ABFG equal in

area to the trapezoid ABCD.

A

28. The angle formed by the bisectors of two angles of an equilateral triangle is double the third angle.

B

29. In the isosceles triangle ABC draw DE parallel to the base AC, so that DA = DE EC. =

A

30. If the diagonals of a parallelogram are equal and perpendicular to each other, the figure is a square.

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B

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35. Construct an equilateral triangle, having given its altitude.

36. The quadrilateral formed by the bisectors of the angles of a quadrilateral has its opposite angles supplementary.

B

[See Exercise 13.]

H

A

B

37. If the quadrilateral ABCD (see figure of Ex. 36) is a parallelogram, EFGH is a rectangle.

38. If the quadrilateral ABCD (see figure of Ex. 36) is a rectangle, EFGH is a square.

39. The bisectors of the exterior angles of a quadrilateral form a second quadrilateral whose opposite angles are supplementary.

40. The altitudes of a triangle meet in a common point.

Suggestion. Through the three vertices of the ABC draw parallels to the opposite sides, forming AGHI. Show that the altitudes of ▲ ABC are Is to the sides of A GHI, at their middle points.

G

B

>H

41. If the number of sides of an equiangular polygon is increased by four, each angle is increased by of a right angle. How many sides has the polygon? [§ 158.]

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46. What is the greatest number of acute angles a convex polygon can have?

Suggestion. Show that if there were more than three acute angles the sum of the exterior angles of the polygon would exceed 4 R.A.'s.

47. Given two lines that would meet if sufficiently produced, draw the bisector of their angle, without prolonging the lines.

48. Construct a triangle, having given one angle, one of its including sides, and the sum of the other two sides.

D

A

49. Construct a triangle, having given one angle, one of its including sides, and the difference of the other two sides.

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The side opposite the given angle may be less than the other unknown side (see Fig. 1), or it may be greater than the other unknown side (see Fig. 2).

50. BE is the bisector of ZABC, and BD is an altitude of the triangle ABC. Prove that 1 is one half the difference between the base angles A and C.

A

B

ED

51. Through a point draw a line that shall be equally distant from two given points. [Two ways.]

52. The line joining the middle points of two opposite sides of a quadrilateral bisects the line joining the middle points of the diagonals.

Suggestion. Prove that EGFH is a parallelogram.

B

G

E

H

A

53. Of all triangles having the same base and equal altitudes the isosceles triangle has the least perimeter. [See Ex. 20.]

54. Construct a triangle, having given the perimeter and the two base angles.

A

B

55. Construct a triangle, having given the lengths of the three medians. [§§ 244 and 245.]

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