8. Having given a point P within ABC, draw a line through P so that the segment of it lying within the angle shall be bisected at P. SUGGESTION. - If EP is to equal PF, what relation must exist between BD and DE? Hence begin by drawing PD || AB. B FA 9. Having given a point P within ▲ ABC, draw a line through P so that the segment of it lying within the angle shall be divided at P in the ratio of 1 to 2. 10. Having given a point P within ▲ ABC, draw a line through P so that the segment of it lying within the angle shall be divided at P in the ratio of m to n. 127. Similar polygons. Two polygons which have the angles of one equal respectively to the angles of the other, taken in order, are called mutually equiangular. The pairs of equal angles are called corresponding angles. The sides included between corresponding angles are called corresponding sides. Corresponding sides are also called homologous sides. Similar polygons are those which (1) are mutually equiangular and (2) have their corresponding sides proportional. Thus polygons ABCDE and MNOPQ are similar if (1) ≤ A=2M, ≤ B=2 N, ≤ C=≤ 0, ≤ D=/ P, ≤ E = < Q, (2) AB BC CD DE EA = = = = MN NO OP PQ QM Similarity of polygons is expressed by the symbol ~. Thus "ABCDE is similar to MNOPQ" is written "ABCDE~MNOPQ." It is apparent that similar polygons are of like form or shape, which is the fundamental idea of similarity. EXERCISES 1. Draw any triangle ABC. Construct a second triangle DEF with DE = 1⁄2 AB, 2 D = 2 A, and ▲ E = 2B. Why are the triangles mutually equiangular? Measure AC, BC, DF, and EF. Divide the measures thus obtained It thus appears that if any two triangles are mutually equiangular, their corresponding sides are in proportion, and conversely, if their corresponding sides are in proportion, they are mutually equiangular. That is, either condition for similarity implies the other. 2. Polygons A and B are mutually equiangular. Are their corresponding sides proportional? Polygons C and D have their sides proportional. Are they mutually equiangular? It thus appears that in the case of polygons of more than three sides neither of the conditions for similarity implies the other. NOTE. It follows from Ex. 2 that to prove any two polygons similar, both conditions for similarity must be established. In the following theorems it will be proved that in the case of triangles, as suggested in Ex. 1, either condition for similarity is sufficient to make the triangles similar. 128. Theorem.—If two triangles are mutually equiangular, they are similar. Hypothesis. In AABC and A DEF, LA=LD, LB = LE, LC LF. = Conclusion. AABC-ADEF Proof. 1. In ▲ ABC and A DEF, A = 2 D, Z B=LE, 40=ZF. Hyp. 2. ..A DEF may be placed so that F coincides with ▲ C, ▲ DEF taking the position of ▲ MNC. § 13 5. .. = how must ▲ DEF be placed upon how must ▲ DEF be placed upon 129. Theorem.-If two triangles have their corresponding sides proportional, the triangles are similar. Proof. 1. Construct EDG=ZA and ≤ DEG = L B, 10... EDF = /EDG, DEF = < DEG, F= LG. 130. Theorem. If two triangles have an angle of one equal to an angle of the other, and the including sides proportional, they are similar. Hypothesis. In ▲ ABC and A DEF, C=LF and AC BC = A DF EF 2. ..▲ DEF may be placed upon ▲ ABC, taking the position of ▲ MNC. § 13 Ax. XII MC NO 1. Two triangles are similar if they have two pairs of equal angles. 2. Two right triangles are similar if they have a pair of equal acute angles. 3. Two isosceles triangles are similar if the vertical angles are equal. CNM= B. CNM= C. B=LE. |