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5. If two parallel lines are cut by a transversal, the corresponding angles are equal.

6. If two parallel lines are cut by a transversal, the two interior angles on the same side of the transversal are supplementary.

7. If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also.

This proposition may be considered a special case of number 6, above.

8. When two lines in the same plane are cut by a transversal, if the alternate angles are equal, the two lines are parallel.

9. Two lines in the same plane perpendicular to the same line are parallel.

10. Two lines in the same plane parallel to a third line are parallel to each other.

11. When two lines in the same plane are cut by a transversal, if two corresponding angles are equal or if two interior angles on the same side of the transversal are supplementary, the lines are parallel.

12. If two angles have their arms respectively parallel, and if both pairs of parallels extend either in the same direction or in opposite directions from the vertices, the angles are equal.

13. Segments of parallel lines cut off by parallel lines are equal.

14. Two parallel lines are everywhere equally distant from each other.

15. If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal.

16. If a line parallel to one side of a triangle bisects another side, it bisects the third side also.

17. The line which joins the midpoints of two sides of a triangle is parallel to the third side and is equal to half the third side.

10. Parallelograms. 1. A quadrilateral is a rectilinear figure of four sides.

2. A parallelogram is a quadrilateral whose opposite sides are parallel.

If all the angles of a parallelogram are right angles, it is a rectangle; if all the angles are right angles and the sides are equal, it is a square; if the sides are equal, it is a rhombus.

3. The opposite sides of a parallelogram are equal and the opposite angles are also equal.

4. A diagonal divides a parallelogram into two congruent triangles.

5. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.

6. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

7. If both pairs of opposite angles of a quadrilateral are equal, the figure is a parallelogram.

8. The diagonals of a parallelogram bisect each other. 9. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

11. Trapezoids. 1. A trapezoid is a quadrilateral which has two of its sides parallel.

2. If a line parallel to the base of a trapezoid bisects one of the other sides, it bisects the opposite side and is equal to half the sum of the bases.

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12. Angles of a Polygon. 1. A polygon is a rectilinear figure of three or more sides.

A polygon is said to be equilateral if all its sides are equal; equiangular if all its angles are equal; and regular if it is both equilateral and equiangular.

Two polygons are said to be mutually equiangular if the angles of one are equal to the angles of the other, taken in the same order; mutually equilateral if the sides of one are equal to the sides of the other taken in the same order.

2. The sum of the interior angles of a polygon is as many straight angles less two as the figure has sides. 3. Each angle of a regular polygon of n sides is equal n 2

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straight angles.

4. The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is two straight angles.

13. Fundamental Constructions. 1. Through a given point on a given straight line construct a perpendicular to the line.

While there are many other constructions of plane geometry, those given here are sufficient for the needs of solid geometry, for we do not construct the solid figures, and make but relatively few constructions in plane sections of these figures.

2. Through a given external point construct a perpendicular to a given line.

3. Through a given external point construct a line parallel to a given line.

4. Bisect a given line segment.

5. Bisect a given angle.

6. From a given point on a given line construct a line which shall make with the given line an angle equal to a given angle.

14. Perpendiculars and Obliques. 1. The sum of two line segments from a given external point to the ends of a given line segment is greater than the sum of two other line segments similarly drawn but included by them.

2. If two line segments, drawn from a point on a perpendicular to a given line, cut off on the given line equal segments from the foot of the perpendicular, the line segments are equal and make equal angles with the perpendicular.

3. If two line segments, drawn from a point on a perpendicular to a given line, cut off on the given line unequal segments from the foot of the perpendicular, the line segment more remote is the greater.

4. Equal obliques from a point to a line cut off equal segments from the foot of the perpendicular from the point to the line.

5. If two unequal line segments are drawn from a point to a line, the greater cuts off the greater segment from the foot of the perpendicular from the point to the line.

6. The perpendicular is the shortest line segment that can be constructed to a given line from a given external point.

7. The shortest line segment to a given line from an external point is the perpendicular from the point to the line.

15. Inequalities in Relation to Triangles. 1. If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater.

2. If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater.

3. If two sides of one triangle are equal respectively to two sides of another, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first is greater than the third side of the second.

4. If two sides of one triangle are equal respectively to two sides of another, but the third side of the first triangle is greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second.

16. General Properties of Circles. 1. A circle is a closed curve lying in a plane and such that all its points are equally distant from a fixed point in the plane.

2. In a plane one and only one circle can be drawn with a given point as center and a given segment as radius.

3. Through any three given points not lying in a straight line one and only one circle can pass.

4. All radii of the same circle or of equal circles are equal. 5. All circles with equal radii are equal.

6. All diameters of the same circle or of equal circles are equal.

7. A point is within, on, or outside a circle according as its distance from the center is less than, equal to, or greater than a radius.

8. A diameter bisects the circle and the surface inclosed, and conversely.

9. If two central angles of the same circle or of equal circles are equal, the angles have equal arcs; and if two central angles are unequal, the greater angle has the greater arc,

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