## Plane Geometry |

### From inside the book

Results 1-5 of 42

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**ratios**in connection with similar polygons . ( 5 ) Elimination from elementary geometry of proofs of incom- mensurable cases of theorems by use of the theory of limits . ( 6 ) A grading of geometry , through a rearrangement of the ... Page 94

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**ratio**of AB to CD A is 3 divided by 5 , or . C B If one of two angles contains 40 ° and the other one 60 ° , the**ratio**of the first to the second is 8 . In any**ratio**the dividend is called the antecedent and the divisor is called the ... Page 97

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**ratios**are equal , let each**ratio**equal r , that is , r , = r , = r , f etc. Clearing fractions , a = rb , c = rd , e = rf , etc. Adding , a + c + e + etc. = rb + rd + rf + etc. Factoring , Dividing , Hence , by Ax . I , a + c + e + etc ... Page 100

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**ratio**Ax . I - NOTE . The proof above is incomplete theoretically , since it assumes that DC and AD have a common unit of measure , which need not be true . There are line - segments , such as the side and diagonal of a square , which ... Page 101

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**ratios**AB BC and DE ' EF '**ratios**AB BC AC and DE ' EF ' DF What is observed about the AC DF ? It thus appears that if any two triangles are mutually equiangular , their corresponding sides are in proportion , and conversely , if their ...### Other editions - View all

### Common terms and phrases

AABC ABCD AC and BD acute angle altitude angle equal angles are equal apothem base bisects central angle chord circle with center circumscribed compasses and straightedge Conclusion congruent Construct a triangle Corollary corresponding sides decagon diagonals diameter distance Divide a given drawn equal angles equal arcs equal circles equidistant equilateral triangle EXERCISES exterior angles follows formed geometry given angle given circle given line given line-segment given point given triangle Hence hypotenuse Hypothesis inscribed angle intercepted internally tangent intersect isosceles triangle locus measure medians middle points number of sides parallel lines parallelogram perimeter perpendicular bisector point of contact proof in full proof is left quadrilateral radii radius ratio rectangle regular polygon rhombus right angle right triangle secant segment Show similar polygons square straight angle straight line student SUGGESTION tangent trapezoid vertex Write the proof

### Popular passages

Page 130 - If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second.

Page 76 - The straight line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of it 46 INTERCEPTS BY PARALLEL LINES.

Page 223 - ... they have an angle of one equal to an angle of the other and the including sides are proportional; (c) their sides are respectively proportional.

Page 4 - PERIPHERY of a circle is its entire bounding line ; or it is a curved line, all points of which are equally distant from a point within called the center.

Page 21 - If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third.

Page 222 - Two triangles having an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.

Page 131 - ... if two triangles have two sides of one equal, respectively, to two sides of the other...

Page 72 - There are three important theorems in geometry stating the conditions under which two triangles are congruent: 1. Two triangles are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other.

Page 258 - S' denote the areas of two circles, R and R' their radii, and D and D' their diameters. Then, I . 5*1 = =»!. That is, the areas of two circles are to each other as the squares of their radii, or as the squares of their diameters.

Page 197 - In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the product of one of these sides and the projection of the other side upon it.