THEOREM XI. From a given point without a straight line one perpendicular can be drawn to the line, and but one. The perpendicular is the shortest line that can be drawn from a point to a straight line. THEOREM XII. If two oblique straight lines drawn from a point to a straight line meet the line at equal distances from the foot of the perpendicular drawn from the point to the line, they are equal. If they meet the line at unequal distances from the foot of the perpendicular, the more remote is the greater. THEOREM XIII. If two right triangles have the hypotenuse and a side of one respectively equal to the hypotenuse and a side of the other, the triangles are equal. THEOREM XIV. If a perpendicular is erected at the middle of a straight line, then every point in the perpendicular is equally distant from the extremities of the line, and every point not in the perpendicular is unequally distant from the extremities of the line; that is, the locus of points equidistant from the extremities of a line is a line bisecting that line at right angles. THEOREM XV. Every point in the bisector of an angle is equally distant from the sides of the angle; and every point not in the bisector is unequally distant from the sides of the angle; that is, the bisector of an angle is the locus of the points within the angle and equally distant from its sides. PARALLELS AND PARALLELOGRAMS. THEOREM XVI. Two straight lines perpendicular to the same straight line are parallel. THEOREM XVII. When two straight lines are cut by a third, if the alternate interior angles are equal, the two straight lines are parallel. Corollary I. When two straight lines are cut by a third, if a pair of corresponding angles are equal, the lines are parallel. Corollary II. When two straight lines are cut by a third, if the sum of two interior angles on the same side of the secant line is equal to two right angles, the two lines are parallel. Corollary III. Two straight lines parallel to the same straight line are parallel to each other. THEOREM XVIII. If two parallel lines are cut by a third straight line, the alternate interior angles are equal. Corollary I. If two parallel lines are cut by a third straight line, any two corresponding angles are equal. Corollary II. If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles. Corollary III. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other. THEOREM XIX. The sum of the three angles of any triangle is equal to two right angles. Corollary. If one side of a triangle is extended, the exterior angle is equal to the sum of the two interior opposite angles. THEOREM XX. The sum of the angles of a polygon of n sides is 2n right angles. THEOREM XXI. If the sides of one angle are perpendicular respectively to the sides of another, the angles are either equal or supplementary. THEOREM XXIII. The diagonals of a parallelogram bisect each other. THEOREM XXII. The opposite sides of a parallelogram are equal, and the opposite angles are equal. THEOREM XXIV. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. THEOREM XXV. If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram. BOOK II. THE CIRCLE AND THE MEASURE OF ANGLES. THEOREM I. The diameter of a circle is greater than any other chord; and it bisects the circle and its circumference. THEOREM II. A straight line can intersect a circumference in only two points. THEOREM III. Through three points not lying in a straight line one circumference, and only one, can be drawn. THEOREM IV. Two circumferences can intersect each other in only two points. THEOREM V. In the same circle or in equal circles, equal angles at the centre intercept equal arcs on the circumference. Conversely, in the same circle or in equal circles, equal arcs subtend equal angles at the centre. THEOREM VI. If, in the same circle or in equal circles, two arcs are equal, the chords subtending them are equal. Conversely, if in the same circle or in equal circles, two chords are equal, the arcs subtended by them are equal. THEOREM VII. In the same circle or in equal circles, the greater of two unequal arcs, neither of which exceeds a semi-circumference, is subtended by the greater chord. Conversely, in the same circle or in equal circles, the greater of two unequal chords subtends the greater arc, if neither arc exceeds a semi-circumference. THEOREM VIII. The diameter perpendicular to a chord bisects the chord and the arcs which the chord subtends. Corollary I. A line bisecting a chord at right angles passes through the centre of the circle. Corollary II. When two circumferences intersect each other, the straight line joining their centres bisects at right angles their common chord. THEOREM IX. In the same circle or in equal circles, equal chords are equally distant from the centre; and of two unequal chords the less is at the greater distance from the centre. Conversely, in the same circle or in equal circles, chords equally distant from the centre are equal; and of two chords unequally distant from the centre, that is the greater whose distance from the centre is the less. THEOREM X. A straight line tangent to a circle is perpendicular to the radius drawn to the point of contact. Corollary. A perpendicular to a tangent at the point of contact passes through the centre of the circle. THEOREM XI. When two tangents to the same circle intersect each other, the distances from their point of intersection to their points of contact are equal. |