Elementary Geometry: Plane |
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Page 26
Plane James McMahon. ISOSCELES TRIANGLES 59. THEOREM 8. In a triangle , if two sides are equal , then the angles opposite the equal sides are equal . Let the triangle ABC have the sides AB and AC equal . To prove that the angles ABC and ...
Plane James McMahon. ISOSCELES TRIANGLES 59. THEOREM 8. In a triangle , if two sides are equal , then the angles opposite the equal sides are equal . Let the triangle ABC have the sides AB and AC equal . To prove that the angles ABC and ...
Page 27
... triangle are not equal , then the opposite angles are not equal . [ Prove by exclusion as in 61. ] EQUALITY OF ... ABC and A'B'C ' have the sides AB , AC , and the angle BAC , respectively equal to the sides A'B ' , A'C ' , and the angle B'A' ...
... triangle are not equal , then the opposite angles are not equal . [ Prove by exclusion as in 61. ] EQUALITY OF ... ABC and A'B'C ' have the sides AB , AC , and the angle BAC , respectively equal to the sides A'B ' , A'C ' , and the angle B'A' ...
Page 29
Plane James McMahon. To prove that the triangles ABC and A'B'C ' are equal . Place the triangle ABC so that AC coincides with its equal A'C ' ; the point B falling at the same side as B ' . It is then to be proved that B falls on B ...
Plane James McMahon. To prove that the triangles ABC and A'B'C ' are equal . Place the triangle ABC so that AC coincides with its equal A'C ' ; the point B falling at the same side as B ' . It is then to be proved that B falls on B ...
Page 30
... ABC is an equilateral triangle . For AC and AB are equal , being radii of the same circle ; and BC and AB are equal , being radii of the same circle . Hence the triangle ABC has its three sides equal ; and it is therefore equilateral ...
... ABC is an equilateral triangle . For AC and AB are equal , being radii of the same circle ; and BC and AB are equal , being radii of the same circle . Hence the triangle ABC has its three sides equal ; and it is therefore equilateral ...
Page 34
... triangle O'A'B ' is trans- ferred to the position OAB . The possibility of such transference shows that there is some point ( such as B ) at which the two arcs intersect . Transference of a triangle . 78. Cor . To transfer a triangle ( ABC ) ...
... triangle O'A'B ' is trans- ferred to the position OAB . The possibility of such transference shows that there is some point ( such as B ) at which the two arcs intersect . Transference of a triangle . 78. Cor . To transfer a triangle ( ABC ) ...
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Common terms and phrases
ABCD adjacent sides altitude angle AOB angle equal angles are equal antecedents apothem base bisector bisects called central angle central line chord circumscribed circles coincide contraposite corresponding sides Definition diagonal difference divided Draw drawn equal angles equal circles equal ratios equiangular equilateral polygon equivalent figure geometry given angle given circle given line given point given polygon given ratio greater half Hence hypotenuse hypothesis inscribed interior angles intersect isosceles triangle less Let ABC line joining line-segments magnitudes measure-number mid-point mth multiple n-gon number of sides number-correspondent opposite sides parallel parallelogram perigon perimeter perpendicular PROBLEM prolonged prove quadrangle radii radius ratio compounded ratio of similitude regular polygons respectively equal rhombus right angle right triangle segments Show similar polygons similar triangles similarly solution statement straight angle straight line subtended superposable surface symmetric tangent THEOREM triangle ABC unequal vertex vertical angle
Popular passages
Page 148 - In every triangle, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle.
Page 147 - In an obtuse-angled triangle the square on the side opposite the obtuse angle is greater than the sum of the squares on the other two sides by twice the rectangle contained by either side and the projection on it of the other side.
Page 7 - LET it be granted that a straight line may be drawn from any one point to any other point.
Page 136 - If a straight line be divided into any two parts, the square on the whole line is...
Page 195 - UPON a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle.
Page 80 - 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 305 - 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 287 - ... 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 263 - If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. Let A, B, C, D be four magnitudes of the same kind, which are proportionals, viz.
Page 32 - At a given point in a given straight line, to construct an angle equal to a given angle. Let A be the given point in the straight line AB, and O the given angle.