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96.

PROPOSITION XXXVIII.

B

Given: 1. The quadrilateral A B C D and the diagonal A C, formings x, m, n and y.

2. Also (1) A B D C, and (2) A D B C.

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Is the quadrilateral a parallelogram?

State and prove Prop. XXXVIII.
(If you fail, see "hint" below.)

[Hint.-Prove As equal; consequently Lm =

Ln and

A B parallel to D C. Then prove B C parallel to A D, and tell why the figure is a parallelogram.]

97.

PROPOSITION XXXIX.

BC

Given the quadrilateral A B C D, in which A D B C

and A D is parallel to B C.

Can you prove the figure a parallelogram?

(If you fail, see "hint" below.)

[Hint.-Draw either diagonal and use method given in the "hint" to $96.]

98.

PROPOSITION XL.

Draw two parallelograms ABCD and A' B' C' D', in which any two adjacent sides and the included angle of the one equal respectively two adjacent sides and the included angle of the other.

Are the parallelograms equal?

[N. B.-Equal figures can be made to coincide in all parts. Figures may be equal in area, or equivalent, which are not equal.]

State and prove Prop. XL.

(If you fail, consult "hint" below.)

[Hint.-Superpose A' B' C' D' on A B C D so that the equal sides and the included s will coincide. Then show that the opposite sides and the remaining vertex must coincide. See § 14, 12, and § 14, 10,]

EXERCISES.

46. If the diagonals of a quadrilateral bisect each other, what is the figure? Distinguish in the above when the diagonals are (1) equal, (2) unequal.

47. If the diagonals of a quadrilateral are (1) equal, (2) unequal, and bisect each other at right angles, what deductions can you make?

QUESTIONS.

1. If the diagonals of a quadrilateral are equal, can you make and prove any deduction?

2. If the opposite angles of a quadrilateral are equal, what deduction can you make and prove?

EXERCISES.

By drawing the diagonals A C and A' C' in § 98 can you prove Prop. XL. in another way?

48.

49. Given the ▲ A B C, in which A plus twice B minus three times C equals 110°, and ▲ A minus twice ZB plus ZC equals 90°; find s A, B, C.

50. Given two rectangles having the base and altitude of one respectively equal to the base and altitude of the other; show how they compare in area.

What is the value of the sum of all the angles of a parallelogram?

51.

Problem.

99.

PROPOSITION XLI.

To construct a parallelogram when two adjacent sides and the included angle are given.

EXERCISES.

52. Construct a rectangle when a side is given and the adjacent side is 234 times the given side.

53. Construct a square when a diagonal is given. [Show two ways.]

54. Construct a rhombus when the diagonals are given. 55. Construct a rhombus when one diagonal is given and the other is 3% times it.

56. Construct a "kite" trapezium when the diagonals are given. Discuss the possible lengths and position of the diagonals.

57. Construct an isosceles trapezoid when a diagonal and distance between the bases are given. Discuss.

POLYGONS.

Compare the "arrow"
What is a convex pol-

What is a polygon? its perimeter? trapezium with the "kite" trapezium. ygon? a re-entrant angled or concave polygon? What is a regular polygon? the angle of a regular polygon? the angle at the center? What is an exterior angle of a polygon? What are the diagonals of a polygon? How are polygons classified?

100.

Polygons are classified according to the number of sides. A polygon of three sides is a triangle, or trigon; of four sides is a quadrilateral; of five sides is a pentagon; of six sides is a hexagon; of seven sides is a heptagon; of eight sides is an octagon; of nine sides is a nonagon; of ten sides is a decagon; of eleven sides is an undecagon; of twelve sides is a dodecagon.

101.

The sum of the sides of a polygon is called the perimeter.

102.

A convex polygon is one in which no side produced will enter the polygon.

103.

A concave polygon is one in which at least two sides when produced enter the polygon. The angle whose sides produced enter the polygon is called a re-entrant angle.

104.

A regular polygon is both equilateral and equiangular. When the term polygon is used, a convex polygon is meant.

105.

PROPOSITION XLII.

Into how many triangles may we divide any polygon by joining any vertex to all other vertices when there are four sides? five sides? six sides? n sides?

What is the sum of all the interior angles of a polygon of four sides? five sides? six sides? n sides?

The sum of the interior angles of a polygon is equal to as many straight angles as

-

Finish the above statement and prove it. Call it Prop.

XLII.

106.

Cor. I. If the sides of a polygon are produced in order, the exterior angles thus formed are equal to two straight angles.

[Hint. What is the sum of the interior angles? of the interior and exterior angies taken together?]

107.

Cor. II. If a polygon is equiangular, each interior angle is equal to as many straight angles

Finish and prove.

[Hint. What is the sum of the /s of any triangle? Then what is the size of each angle of an equiangular triangle? What is each angle of an equiangular quadrilateral? etc.]

Write the fraction which represents the number of st. 4s in each of a polygon of n sides.

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