The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and Exercises |
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Page 13
... opposite and vertical angles may be shown by means of the equation . 40. When an angle adjacent to two opposite angles is 92 ° , 120 ° , 112 ° , or 146 ° , what is each of the opposite angles ? III . INTERSECTED PARALLELS . 1. When are ...
... opposite and vertical angles may be shown by means of the equation . 40. When an angle adjacent to two opposite angles is 92 ° , 120 ° , 112 ° , or 146 ° , what is each of the opposite angles ? III . INTERSECTED PARALLELS . 1. When are ...
Page 16
... angles B and C in degrees. For word problems where you are not given a diagram, a rough sketch may help you work out ... opposite angles is equal. The diagonals bisect the opposite angles. A parallelogram has opposite sides equal and ...
... angles B and C in degrees. For word problems where you are not given a diagram, a rough sketch may help you work out ... opposite angles is equal. The diagonals bisect the opposite angles. A parallelogram has opposite sides equal and ...
Page 13
... angle is equal to the two interior and opposite angles ; and the three angles of every triangle are together equal to two right angles ( Euc . I. 32 ) . Hence the angles of an equilateral triangle are each equal to one - third of two right ...
... angle is equal to the two interior and opposite angles ; and the three angles of every triangle are together equal to two right angles ( Euc . I. 32 ) . Hence the angles of an equilateral triangle are each equal to one - third of two right ...
Page 21
... ADJACENT ANGLES are those which lie on the same side of one line , and on opposite sides of the other ; thus , ACE and ECB , or ACE and ACD , are adjacent angles . A E 2o . OPPOSITE , or VERTICAL ANGLES , are those which lie on opposite ...
... ADJACENT ANGLES are those which lie on the same side of one line , and on opposite sides of the other ; thus , ACE and ECB , or ACE and ACD , are adjacent angles . A E 2o . OPPOSITE , or VERTICAL ANGLES , are those which lie on opposite ...
Page 33
... opposite angles being mutually equal , each pair of adja- cent angles must be equal to two right angles ; therefore , the opposite sides must be equal and parallel . PROP . XXXIV . THEOR . The opposite sides and angles of a ...
... opposite angles being mutually equal , each pair of adja- cent angles must be equal to two right angles ; therefore , the opposite sides must be equal and parallel . PROP . XXXIV . THEOR . The opposite sides and angles of a ...
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Common terms and phrases
ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral radius rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Popular passages
Page 286 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 30 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Page 34 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles...
Page 37 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Page 12 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Page 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Page 220 - If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 354 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.
Page 104 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Page 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.