Properties of the Triangle. 11. If a line be drawn parallel to the base of a triangle, it will cut the two other sides proportionally. Thus, if DE be drawn parallel to the base BC, we shall have DA that is, the parts AD and AE will have to each other the same ratio as the sides AB and AC. 12. Similar triangles are those which have all the angles of the one, equal to the corresponding angles of the other, each to each. Thus, if the two triangles ABC and DEF have the angle A=D, B=E, and FC, they will be similar. The sides which lie opposite equal angles are called homolo Ᏼ Ꭰ E gous sides. Thus, AB and DE are homologous sides; also, AC and DF, and likewise CB and FE. The homologous sides of similar triangles are proportional, Thus, AB: AC:: DE: DF. QUEST.-11. If a line be drawn parallel to the base of a triangle, how I will it divide the two other sides? 12. What are similar triangles ? What are the sides called which lie opposite equal angles? Are these sides proportional? Properties of Polygons. 13. The areas of similar triangles are to each other as the squares described on their homologous sides. The similar triangles ABC, and DEF, are to each other, as the squares G and G H H, described on the homologous sides AB and DE. AB=BC=CD=DE=EA: also angle A=B=C=D=E. QUEST.-13. How are the areas of similar triangles to each other, 1. What is a regular polygon ? Properties of Polygons. 2. Similar polygons are those which have the angles of the one equal to the angles of the other, each to each, and the sides about the equal angles proportional. called homologous sides, and these sides are proportional to each other. Thus, if ABCDE, and FGHIK are two similar polygons: then Angle AF, B=G, C=H, D=I, and E=K Also AB FG: BC: GH and AB FG: CD: IH also AB: FG:: DE: IK and AB FG:: EA : KF. 3. Similar polygons are to each other as the squares QUEST.-2. What are similar polygons? Are similar polygons of like shape? May they vary in size? What are the sides called which are like situated? Are the homologous sides proportional? Make two similar hexagons-mark the equal angles and the homologous sides. Also, write down the proportional sides. 3. How are similar polygons to each other? If 3 and 4 represent the homologous sides of two similar polygons, what proportion will those polygons bear to each other? ABCDE : FGHIK :: square L: square M. 4. Any polygon may be divided by diagonals, into as many triangles less two, as the polygon has sides. Thus, if the polygon has five sides, there will be three triangles; if it has six sides, there will be four; if seven sides, five; if eight sides, six; &c. 5. Two similar polygons may be divided by diagonals, into the same number of similar triangles, Ꭰ QUEST.-4. Into how many triangles may every polygon be divided by diagonals? 5. If two similar polygons be divided by diagonals alike drawn in each, will the triangles thus formed be similar? Will they be like placed? Properties of Polygons. ABCDE, and FGHIK, the first will give the triangles ABC, ACD, and ADE; and the second, the similar triangles FHG, FHI, and FIK. 6. The sum of all the inward angles of any polygon is equal to twice as many right angles, wanting four, as the figure has sides. Thus, if the polygon has five sides, we have = E B A+B+C+D+E=10 right angles-4 right angles 6 right angles. 7. If the polygon is a quadrilateral, then the sum of the angles will be equal to four right angles. 8. When the polygon is regular, its angles will be equal to each other (Art. 1). If, then, the sum of the inward angles be divided by the number of angles, the quotient will be the value of one of the angles. We shall find the value in degrees, by simply placing 90° for the right angle. Thus, for the sum of all the angles of an equilateral triangle, we have (Art. 6). QUEST.-6. What is the sum of all the inward angles of any polygon equal to ? 7. What is the sum of the inward angles of a quadrilateral equal to? 8. If a polygon is regular, are its angles equal or unequal? When the polygon is regular, if the sum of the angles be divided by the number, what will the quotient be? What is the value of either of the angles of an equilateral triangle ? |