149. THEOREM. The median of a trapezoid is parallel to the bases and equal to half their sum. [This is another form of stating the theorem of 144.] 150. THEOREM. The perpendiculars from the vertices of a triangle to the opposite sides meet in a point. Z Given: A ABC, AX 1 to BC, BY 1 to AC, and CZ L to AB. To Prove These three s meet in a point. Proof Through A suppose RS drawn | to BC; through B, TS to AC; through C, RT to AB, forming ▲ RST. The figure ABCR is a □ (Const.) and ABTC is a □ (?). .. RC AB and CT = AB (?) (130). .. RC = CT (Ax. 1). Now Cz is to RT (?) (95). That is, cz is Similarly AX is And BY is to to RT at its midpoint. to RS at its midpoint. TS at its midpoint. Therefore AX, BY, CZ meet at a point (?) (85). Q.E.D. Ex. 1. Draw the three altitudes of an obtuse triangle and prolong them until they meet. Ex. 2. Prove that each of the three outer triangles in the figure of 150 is equal to ▲ ABC. Ex. 3. Prove that any altitude of A RST is double the parallel altitude of ▲ ABC. [Use 143 and 130.] 151. THEOREM. The point at which two medians of a triangle intersect is two thirds the distance from either vertex of the triangle to the midpoint of the opposite side. Given: ▲ ABC, BD and CE two medians intersecting at O. To Prove: BO = 3 BD and CO = =CE. Proof: Suppose H is the midpoint of BO and I is the midpoint of co. Draw ED, DI, IH, HE. In ▲ ABC, DE is to BC and = BC (?) (142). In A OBC, HI is to BC and = BC (?). .. ED = HI (Ax. 1), and ED is to HI (?) (94). .. EDIH is a □ (?) (135). 8 .. HO = OD and 10=0E (?) (137). .'. BH= HO = OD and CI=10= 0E (Ax. 1). = BD, and CO = 2. EO = 3 CE. Q. E.d. 152. THEOREM. The three medians of a triangle meet in a point which is two thirds the distance from any vertex to the midpoint of the opposite side. : Proof Suppose AX is the third median of ▲ ABC and meets BD at o'. Then BO' BD and BO= BD (?) (151). = •. BO' =BO (?). That is o' coincides with o and the three medians meet at o which is the distance, etc. Q.E.D. Ex. 1. In the figure of 151, prove EH = 1⁄2 AO = DI, by 142. Ex. 2. In the figure of 151, if BC>AC, prove the angle BEC obtuse. [Use 87.] Ex. 3. If one angle of a rhombus is 30°, find all the angles of the four triangles formed by drawing the diagonals. Ex. 4. Show that any trapezoid can be divided into a parallelogram and a triangle by drawing one line. Ex. 5. Prove that every right triangle can be divided into two isos celes triangles by drawing one line. [Use 148.] POLYGONS 153. A polygon is a portion of a plane bounded by straight lines. The lines are called the sides. of intersection of the sides are the vertices. of a polygon are the angles at the vertices. The points The angles 154. The number of sides of a polygon is the same as the number of its vertices or the number of its angles. An exterior angle of a polygon is an angle without the polygon, between one side of the polygon and another side prolonged. 155. An equilateral polygon has all of its sides equal to one another. An equiangular polygon has all of its angles equal to one another. 156. A convex polygon is a polygon no side of which if produced will enter the surface bounded by the sides of the polygon. A concave polygon is a polygon two sides of which if produced will enter the polygon. EQUILATERAL EQUIANGULAR CONVEX POLYGONS CONCAVE, OR NOTE. A polygon may be equilateral and not be equiangular; or it may be equiangular and not be equilateral. The word "polygon" is usually employed to signify convex figures. 157. Two polygons are mutually equiangular if for every angle of the one there is an equal angle in the other and similarly placed. Two polygons are mutually equilateral, if for every side of the one there is an equal side in the other, and similarly placed. 158. Homologous angles in two mutually equiangular polygons are the pairs of equal angles. Homologcas sides in two polygons are the sides between two pairs of homologous angles. 159. Two polygons are equal if they are mutually equiangular and their homologous sides are equal; or if they are composed of triangles, equal each to each and similarly placed. (Because in either case the polygons can be made to coincide.) 160. Two polygons may be mutually equiangular without being mutually equilateral; also, they may be mutually equilateral without being mutually equiangular- except in the case of triangles. The first two figures are mutually equilateral but not mutually equiangular. The last two figures are mutually equiangular but not mutually equilateral. 161. A 3-sided polygon is a triangle. A 4-sided polygon is a quadrilateral. Ex. Draw a pentagon and all the possible diagonals from one vertex. How many triangles are formed? Draw a decagon and the diagonals from one vertex. How many triangles are thus formed? Construct a 20gon and the diagonals from one vertex. How many triangles are formed? 162. THEOREM. The sum of the interior angles of an n-gon is equal to (n-2) times 180°. Proof: By drawing all possible diagonals from any vertex it is evident that there will be formed (n - 2) triangles. The sum of the of one ▲ = 180° (?) (110). ... the sum of the 4 of (n-2) A = (n− 2) 180° (Ax. 3). But the sum of the 4 of the triangles = the sum of the of the polygon (Ax. 4). ... sum of 4 of the polygon = (n−2) 180° (Ax. 1). Q.E.D. 163. COR. The sum of the interior angles of an n-gon is equal to 180°n - 360°. 164. COR. Each angle of an equiangular n-gon = (n-2) 180° n 165. COR. The sum of the angles of any quadrilateral is equal to four right angles. 166. COR. If three angles of a quadrilateral are right angles, the figure is a rectangle. Ex. 1. How many degrees are there in each angle of an equiangular pentagon? of an equiangular pentedecagon? of a 30-gon? Ex. 2. If two angles of a quadrilateral are right angles, what is true of the other two? Ex. 3. How many sides has that polygon the sum of whose interior angles is equal to 20 rt. ? Ex. 4. How many sides has that equiangular polygon each of whose angles contains 160°? Ex. 5. If in the figure of 105, ≤a = 65°, how many degrees are there in each of the other angles of the figure? |