## Plane and Solid Geometry |

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**given line**at a given point in the line . PROOF If two such perpendiculars could be drawn , we should have unequal right angles , which is impos- sible ( § 101 ) . PROPOSITION II . THEOREM 103 When one straight line meets 15 RECTILINEAR ... Page 19

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**lines**. CONCLUSION . ≤ m = ≤ n , and ≤ s = Lt. PROOF Zm is the sup . of ...**line**which bisects one of two verti- cal angles bisects the other also ...**given**in Ex . 21. ] PROPOSITION IV . THEOREM 113 From a point without a ... Page 20

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**given line**at a given point in the line . " § 102 Q. E. D. 114 DEFINITION . Parallel lines are lines in the same plane which cannot meet however far produced . PK'K is an acute Exercise 23. In the figure above , show that the angle ... Page 35

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**line**ABC . " A straight**line**is the shortest**line**between two points . " .. AC < AB + BC . Ax . 15 Q. E. D. 161 ...**given**numbers may represent the sides of a triangle , is it necessary to test them by both § 160 and § 161 ? 48 ... Page 47

... line . P B D H HYPOTHESIS . P is any point without the line AB , PD is perpendicular to AB , and PH any other line ...

... line . P B D H HYPOTHESIS . P is any point without the line AB , PD is perpendicular to AB , and PH any other line ...

**given line**is perpendicular to the**given line**. EXERCISES 85 An altitude of a triangle is less than either of ...### Contents

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### Common terms and phrases

ABCD altitude angles are equal arc BC assigned quantity base bisectors bisects chord circumference circumscribed circle CONCLUSION cone construct COROLLARY cylinder diagonals diameter diedral angles divided equiangular equiangular polygon equidistant equilateral triangle exterior angle Find the area Find the locus Find the ratio frustum given circle given line given point homologous sides hypotenuse HYPOTHESIS inches inscribed intersecting isosceles trapezoid isosceles triangle lateral area legs line of centers mean proportional median mid-points number of sides parallelogram parallelopiped perimeter perpendicular polyedral angle polyedron prism PROOF Draw Prove pyramid Q. E. D. EXERCISES Q. E. D. PROPOSITION quadrilateral radii radius rectangle regular polygon rhombus right angles right triangle SCHOLIUM secant segments similar triangles slant height SOLUTION sphere spherical polygon spherical triangle straight line surface tangent THEOREM trapezoid triangle ABC triedral vertex volume

### Popular passages

Page 168 - 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 diminished by twice the product of one of those sides and the projection of the other upon that side.

Page 41 - In an isosceles triangle the angles opposite the equal sides are equal.

Page 38 - ... 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 35 - Any side of a triangle is less than the sum of the other two sides...

Page 242 - The areas of two triangles which have 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. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.

Page 174 - In any triangle, the product of two sides is equal to the product of the segments of the third side formed by the bisector of the opposite angle plus the square of the bisector.

Page 172 - If from a point without a circle a tangent and a secant are drawn, the tangent is the mean proportional between the whole secant and its external segment.

Page 171 - If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other.

Page 192 - The areas of two rectangles having equal altitudes are to each other as their bases.

Page 65 - The perpendicular bisectors of the sides of a triangle meet in a point. 12. The bisectors of the angles of a triangle meet in a point. 13. The tangents to a circle from an external point are equal. 14...