Of Parallelograms. parallels are cut by the diagonal DB, the alternate angles will be equal (Th. xii): that is the angle ADB DBC and BDC ABD. IIcnce the two triangles ADB BDC, having two angles in the one equal to two angles in the other, will have their third angles equal (Th. xvii. Cor. 1), viz. the angle A equal to the angle C, and these are two of the opposite angles of the parallelogram. Also, if to the equal angles ADB, DBC, we add the equals BDC, ABD, the sums will be equal (Ax. 2): viz. the whole angle ADC to the whole angle ABC, and these are the other two opposite angles of the parallelogram. Again, since the two triangles ADB, DBC, have the side DB common, and the two adjacent angles in the one equal to the two adjacent angles in the other, each to each, the two triangles will be equal (Th. v): hence, the diagonal divides 'he parallelogram into two equal triangles. Cor. 1. If one angle of a parallelogram be a right angle, each of the angles will also be a right angle, and the parallelo. gram will be a rectangle. Cor. 2. Hence, also, the sum of either two adjacent angles of a parallelogram, will be equal to two right angles. If the opposite sides of a quadrilateral, are equal, each to each, the equal sides will be parallel, and the figure will be a parallelogram. Of Parallelograms. Let ABCD be a quadrilateral, having its opposite sides respectively equal, viz. hen will these sides be parallel, and the Α figure will be a parallelogram. D B For, draw the diagonal BD. Then, the two triangles ABD, BDC, have all the sides of the one equal to all the sides of the other, each to each: therefore, the two triangles are equai (Th. viii); hence, the angle ADB, opposite the side AB, is equal to the angle DBC opposite the side DC; therefore, the sides AD, BC, are parallel (Th. xiii). For a like reason DC is parallel to AB, and the figure ABCD is a parallelogram. THEOREM XXV. If two opposite sides of a quadrilateral are equal and paralle the remaining sides will also be equal and parallel, and the figur will be a parallelogram. Let ABCD be a quadrilateral, having D the sides AB, CD, equal and parallel: then will the figure be a parallelogram. A C For, draw the diagonal DB, dividing the quadrilateral into two triangles. Then, since AB is parallel to DC, the alternate angles, ABD and BDC are equal (Th. xii): moreover, the side BD is common; hence the two triangles have two sides and the included angle of the one, equal to two sides and the included angle :f the other: the triangles are therefore equal, and consequently, AD is equal to BC, and the angle ADB to the angle DBC; and consequently, AD is also parallel to BC (Th xin) Therefore, the figure ABCD is a parallelogram. Of Parallelograms. THEOREM XXVI. The two diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other Let ABCD be a parallelogram, and AC, BD its two diagonals intersecting at E. Then will D E B BEC, we find the side AD=BC (Th. xxiii), the angle ADE EBC and EAD=ECB: hence, the two triangles are equal (Th. v): therefore, AE, the side opposite ADE, is equal to EC, the side opposite EBC; and ED is equal to EB Sch. In the case of a rhombus (Def. 48), the sides AB, BC being equal, the triangles AEB and BEC have all the sides of the one equal to the corresponding sides of the other, and are therefore equal. A Whence it follows that the angles AEB D E and BEC are equal. Therefore, the diagonals of a rhombus Lisect each other at right ar gles. ) THE circumference of a circle is a curve line, all the pots of which are equally distant from a certain point within called the centre. 2. The circle is the space bounded by this curve line. 3. Every straight line, CA, CD, CE, drawn from the centre to the circumference, is called a radius or semidiameter. Every line which, like AB, passes through the centre and terminates in the circumference, is called a diameter. 4. Any portion of the circumference, 23 EFG, is called an arc. 5. A straight line, as EG, joining the E extremities of an arc, is called a chord. 6 A segment is the surface or portion of a circle included between an arc and its chord. Thus EFG is a segment. E D Б C G |