Machinery's Handbook, 31st Edition
PURE GEOMETRY
61
Pure Geometry The labels that identify the parts of a figure (angle A , radius r , diameter d , and so on) are used in the formulas to indicate the measure of that feature. By definition, if any two geometric features A and B have equal measure, they are said to be congruent . So, if the measure of A equals the measure of B , then A ≅ cB . Polygons are congruent if they have the same shape and size, that is, if one can be superimposed on the other point for point. Triangles are congruent if any of the propositions for triangle congruence hold, as sum- marized below. Table 1a. Propositions of Geometry
A triangle is a three-sided polygon. It is, in fact, the polygon with the least number of sides. The sides of a triangle meet at its vertices (singular vertex ). The sum of the measures of all three angles of a triangle is 180 degrees. Hence, if the measures of any two angles are known, the third angle measure can always be found. A B C + + 180 ° = B 180 ° A C + ( ) – = A 180 ° B C + ( ) – = C 180 ° A B + ( ) – = AAS Proposition: If two angles and the non-included side of one triangle are congruent to the corresponding (similarly located) angles and sides of another triangle, the triangles are congruent. Hence, if a = a 1 , A = A 1 , and B = B 1 , the other corresponding side and angle are equal in measure, and thus the triangles are congruent. SAS Proposition: If two sides and the included angle (the angle between the sides) of one triangle are congruent (equal in measure) to the corresponding (similarly located) sides and angle of another triangle, then the triangles are congruent. Hence, in the figure, if a = a 1 , b = b 1 , and A = A 1 , then the remaining side and angles also are equal in measure, and thus the triangles are congruent. SSS Proposition: If all three sides of one triangle are congruent (equal in measure) to all three sides of another triangle, then the triangles are congruent. If the three sides in one triangle are equal in measure to the three sides of another triangle, then the angles in the two triangles are equal in measure. If a = a 1 , b = b 1 , and c = c 1 , then the corresponding angles are also equal in measure, and thus the triangles are congruent. If the three sides of a triangle are proportional to corresponding sides of another triangle, then the triangles are similar , and the angles in the one are congruent (equal in measure) to the angles in the other. Hence, if a d ⁄ b e ⁄ c f ⁄ then A D , B E , C F = = = = = Similar triangles are ones whose corresponding angles are congruent. If this is true then the corresponding sides are proportional. If the angles of one triangle are congruent (equal in measure) to the angles of another triangle, then the triangles are similar and their corresponding sides are proportional. Hence, if A D , B E , and C F then a d ⁄ b e ⁄ c f ⁄ = = = = =
A
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A 1
B 1 a 1
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b 1
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A
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B a
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