## Modern geometry [ed.] with an appendix by W.B. Jack1876 |

### From inside the book

Results 1-5 of 55

Page iv

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**opposite**angle . 7. Proof is given of the proposition that two**sides**of a triangle are together greater than the third . 8. The distinction between commensurables and incommensurables is fully exhibited . 9. The nature of ratio and ... Page vi

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**Sides**45 The shortest distance be- SECT . I 79 tween two Points . Distinction between THEOREMS XII - XV . ... FOR ...**Opposite**-The Contra - Positive -Method of Proof . EXERCISES 68 71 ་ ་ 12 72 the SECT . II . - Loci ... Page 16

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**opposite sides**of the common vertex , E ; such that the sides of one are the continuation of the sides of the other . Two such angles are called vertically - opposite angles . Vertically - opposite angles are equal ( Fig . 34 ) . The ... Page 18

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**opposite sides**of a common side . 18. When one straight line standing on another straight line makes the adjacent angles equal to one another , each of these angles is called a right angle ; and the straight line which stands on the ... Page 20

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**sides**are in the same straight line . Let the adjacent angles AC D , BCD be supplementary ; that is , let them be ...**opposite**angles are equal . Let A B and C D intersect in E , then shall △ AEC = 4 BED , and AED 4 BEC . = AEC ...### Other editions - View all

### Common terms and phrases

A B and C D A B C D adjacent angles alternate angles angles equal angular points base bisector centre chord circumference circumscribing circle Construct a triangle contra-positive converse decagon describe a circle diameter distance divided draw a straight extremities figure fixed point fulfils the given given angle given circle given condition given straight line gonals greater Hence hypothenuse inches inscribed inscribed angle inscribed circle interior angles isosceles triangle less Let A B C line A B locus of points magnitudes middle points number of sides opposite angles opposite sides parallel parallelogram perpendicular point of contact point of intersection PROBLEM propositions proved quadrilateral ratio rect rectangle contained regular polygon respectively equal rhombus right angles right-angled triangle segment semicircle side opposite square on A B tangent termed Theo THEOREMS ON CHAPTER triangles are equal vertical angle

### Popular passages

Page 242 - 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 240 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 240 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.

Page 242 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

Page 242 - If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

Page 241 - If the first has to the second the same ratio which the third has to the fourth...

Page 240 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 242 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM 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 18 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.

Page 239 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.