CHAPTER VI.

QUADRILATERALS.

338. IN a polygon, two angles which immediately succeed each other in going round the figure, are called adjacent angles. The student will distinguish adjacent angles of a polygon from the adjacent angles defined in Article 85.

B

A DIAGONAL of a polygon is a straight line joining the vertices of any two angles which are not adjacent. Sometimes a diagonal is exterior, as the diagonal BD of the figure ABCD.

A CONVEX polygon has all its diagonals interior.

A CONCAVE polygon has at least one diagonal exterior, as in the above diagram.

Angles, such as BCD, are called reëntrant.

339. A QUADRILATERAL is a polygon of four sides. 340. Corollary.—Every quadrilateral has two diagonals.

341. Corollary.-An interior diagonal of a quadrilateral divides the figure into two triangles.

EQUAL QUADRILATERALS.

342. Theorem.-Two quadrilaterals are equal when they are each composed of two triangles, which are respectively equal, and similarly arranged.

For, since the parts are equal and similarly arranged, the wholes may be made to coincide (40).

343. Corollary.-Conversely, two equal quadrilaterals may be divided into equal triangles similarly arranged. In every convex quadrilateral this division may be made in either of two ways.

344. Theorem.-Two quadrilaterals are equal when the four sides and a diagonal of one are respectively equal to the four sides and the same diagonal of the other.

By the same diagonal is meant the diagonal that has the same position with reference to the equal sides. For, since all their sides are equal, the triangles AEI

E.

B

and BCD are equal, also the triangles AIO and BDF (282). Therefore, the quadrilaterals are equal (342).

345. Theorem.-Two quadrilaterals are equal when the four sides and an angle of the one are respectively equal to the four sides and the similarly situated angle of the other.

By the similarly situated angle is meant the angle included by equal sides.

D F

equal (284); and AI is equal to BD. But since the

three sides of the triangles AIO and BDF are respectively equal, the triangles are equal (282). Hence, the quadrilaterals are equal (342).

SUM OF THE ANGLES.

346. Theorem.-The sum of the angles of a quadrilateral is equal to four right angles.

For the angles of the two triangles into which every quadrilateral may be divided, are together coincident with the angles of the quadrilateral. Therefore, the sum of the angles of a quadrilateral is twice the sum of the angles of a triangle.

Let the student illustrate this with a diagram.

In applying this theorem to a concave figure (338), the value of the reëntrant angle must be taken on the side toward the polygon, and therefore as amounting to more than two right angles.

INSCRIBED QUADRILATERAL.

347. Problem.-Any four points of a circumference may be joined by chords, thus making an inscribed quadrilateral.

This is a corollary of Article 47.

348. Theorem.-The opposite angles of an inscribed quadrilateral are supplementary.

For the angle A is measured by half of the arc EIO (222), and the angle I by half of the arc EAO. Therefore, the two together are measured by half of the whole circumference, and their sum is equal to two right angles (207).

Geom.-11

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TRAPEZOID.

349. If two adjacent angles of a quadrilateral are sup

A TRAPEZOID is a quadrilateral which has two sides parallel. The parallel sides are called its bases.

350. Corollary-If the angles adjacent to one base of a trapezoid be equal, those adjacent to the other base must also be equal. For if A and D are equal, their supplements, B and C, must be equal (96).

APPLICATION.

351. The figure described in the last corollary is symmetrical. For it can be divided into equal parts by

a line joining the middle points of the bases.

The symmetrical trapezoid is used in

architecture, sometimes for ornament, and sometimes as the form of the stones of an arch.

EXERCISES.

352.-1. To construct a quadrilateral when the four sides and one diagonal are given. For example, the side AB, 2 inches; the side BC, 5; CD, 3; DA, 4; and the diagonal AC, 6 inches.

2. To construct a quadrilateral when the four sides and one angle are given.

3. In a quadrilateral, join any point on one side to each end of the side opposite, and with the figure thus constructed demonstrate the theorem, Article 346.

4. The sum of two opposite sides of any quadrilateral which is

circumscribed about a circle, is equal to the sum of the other two sides.

5. If the two oblique sides of a trapezoid be produced till they meet, then the point of meeting, the point of intersection of the two diagonals of the trapezoid, and the middle points of the two bases are all in one straight line.

PARALLELOGRAMS.

353. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel.

B

354. Corollary.-Two adjacent angles of a parallelogram are supplementary. The angles A and B, being between the parallels AD and BC, and on one side of the secant AB, are supplementary (126).

355. Corollary.-The opposite angles of a parallelogram are equal. For both D and B are supplements of the angle C (96).

356. Theorem.-The opposite sides of a parallelogram are equal.

A

B

For, joining AC by a diagonal, the triangles thus formed have the side AC common; the angles ACB and DAC equal, for they are alternate (125); and ACD and BAC equal, for the same reason. Therefore (285), the triangles are equal, and the side AD is equal to BC, and AB to CD.

357. Corollary. When two systems of parallels cross each other, the parts of one system included between two lines of the other are equal.

D

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