RECTILINEAR FIGURES

Excerpts

• RECTILINEAR FIGURES

  A **convex polygon** is a polygon of which no side, when produced, will enter the polygon.

• RECTILINEAR FIGURES

  And, *except in the case of triangles*, two polygons may be mutually equilateral without being mutually equiangular; as, Figs. 6 and 7.

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  In general, each angle of an equiangular polygon of $n$ sides is equal to $\displaystyle \frac{2(n-2)}{n}$ right angles.

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  Two points are said to be **symmetrical** with respect to a third point, called the **centre of symmetry**, if this third point bisects the straight line which joins them.

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  A **straight line** is a line such that any part of it, however placed on any other part, will lie wholly in that part if its extremities lie in that part, as $AB$.

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  The size of an angle depends upon the *extent of opening* of its sides, and not upon the length of its sides.

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  Suppose the straight line $OC$ (Fig. 15) to move in the plane of the paper from coincidence with $OA$, about the point $O$ as a pivot, to the position $OC$; then the line $OC$ describes or generates *the angle $AOC$*, and the magnitude of the angle $AOC$ depends upon the *amount of rotation* of the line from the position $OA$ to the position $OC$.

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  The natural angular unit is one complete revolution. But this unit would require us to express the values of most angles by fractions.

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  Fold over $CFA$, on $CF$ as an axis, until it falls on the plane at the right of $CF$.

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  The beginner must not forget that in Plane Geometry all the points of a figure are in the same plane. Without this restriction in Cor. 2, an indefinite number of perpendiculars can be erected at a given point in a given line.

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  The perpendicular is the shortest line that can be drawn to a straight line from an external point.

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  A **triangle** is a portion of a plane bounded by three straight lines; as, $ABC$ (Fig. 1).

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  The sum of the three angles of a triangle is equal to two right angles.

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  All points in a plane that satisfy a single geometrical condition lie, in general, in a line or group of lines; and this line or group of lines is called the **locus** of the points that satisfy the given condition.

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  The word *locus* (pronounced lo kus) is a Latin word that signifies *place*. The plural of locus is loci (pronounced lo si).

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  In § 139 we have given two angles and the included side, in § 143 two sides and the included angle; hence, by interchanging the words *sides* and *angles*, either theorem is changed to the other. This is called the *Principle of Duality*, or the *Principle of Reciprocity*. The reciprocal of a theorem is not always true, just as the converse of a theorem is not always true.

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  The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.

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  A **polygon** is a portion of a plane bounded by straight lines.

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  A **concave polygon** is a polygon of which two or more sides, if produced, will enter the polygon.

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  Two polygons are *equal* when they can be divided by diagonals into the same number of triangles, equal each to each, and similarly placed; for if the polygons are applied to each other, the corresponding triangles will coincide, and hence the polygons will coincide and be equal.

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  If a known truth *suggests* the required proof, it is best to use the synthetic form at once. If no proof occurs to the mind, it is necessary to use the analytic method to *discover* the proof, and then the synthetic proof may be given.

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  The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.

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  A figure is symmetrical with respect to a point as a centre of symmetry, if the point bisects every straight line drawn through it and terminated by the boundary of the figure.

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  The lines joining the middle points of the sides of a square, taken in order, enclose a square.

Equations

• RECTILINEAR FIGURES

  AB = AC+CB

  The whole line AB equals the parts AC and CB added together when C lies on AB; lines are added by prolonging them.

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  AC = AB-CB

  Segment AC equals AB with CB subtracted, obtained by diminishing AB to C.

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  AC = 2AB

  A line made of two equal segments AB laid end to end is twice AB (a line multiplied by a number).

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  AD = 3AB

  Three equal segments AB laid end to end make the segment AD, equal to three times AB.

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  AE = 4AB

  Four equal segments AB laid end to end make the segment AE, equal to four times AB.

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  \angle ACB = \angle DEF

  Any two straight angles are equal, shown by superposing one on the other so that their vertices and sides coincide.

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  CE = CK

  If a perpendicular CF is folded over and the two oblique lines cut off equal segments FE and FK from the foot, the two lines CE and CK are equal.

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  \angle FCE = \angle FCK

  The angles that the equal oblique lines CE and CK make with the perpendicular CF are equal.

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  OE > OG

  Of two lines from a point in a perpendicular, the one cutting off the more remote (greater) segment from the foot is the longer.

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  \angle BHK + \angle HKD = \text{a st.\ }\angle

  Two interior angles on the same side of a transversal cut by parallel lines add to a straight angle, so they are supplementary.

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  \angle A+\angle B+\angle BCA = 2

  The three angles of a triangle together equal two right angles.

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  AB + BC > AC

  In a triangle, the sum of two sides is greater than the third side.

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  AC - BC < AB

  In a triangle, the difference of two sides is less than the third side.

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  AB > AC

  In triangle ACB, if angle C is greater than angle B, then side AB is greater than side AC (the greater angle lies opposite the greater side).

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  AO = OC

  The diagonals of a parallelogram bisect each other: the segments of diagonal AC from A and from C to the intersection point O are equal.

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  BO = OE

  The diagonals of a parallelogram bisect each other: the segments of diagonal BE from B and from E to the intersection point O are equal.

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  BC = AE

  In a parallelogram, one pair of opposite sides is equal.

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  AB = EC

  In a parallelogram, the other pair of opposite sides is equal.

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  AB = BC = CD

  If parallel lines cut off equal parts on one transversal, they cut off equal parts on every transversal: the segments AB, BC, CD on transversal AD are equal.

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  BF=FC = \frac{1}{2}BC

  The line through the midpoints of two sides of a triangle, drawn parallel to the third side, bisects the third side, so BF and FC are each half of BC.

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  DE = BF = \frac{1}{2}BC

  The segment joining the midpoints of two sides of a triangle equals half the third side.

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  \frac{1}{2} (AB + DC)

  The median of a trapezoid (the line joining the midpoints of its legs) is parallel to the bases and equal to half the sum of the bases; the book states the length as one half of AB plus DC.

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  (n-2)2

  The sum of the interior angles of a polygon of n sides equals (n-2) times two right angles.

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  \displaystyle \frac{2(n-2)}{n}

  Each angle of an equiangular polygon of n sides equals 2(n-2)/n right angles.

Problems