lane Geometry aul Yiu epartment of Mathematics Florida tlantic University Summer 2011
NTENTS 101 Theorem 1 If a straight line stands on another straight line, the sum of the adjacent angles so formed is equal to two right angles. a + b = 2rt. s. bbreviation: adj. angles on st. line. a b Theorem 2 If the sum of two adjacent angles is equal to two right angles, the exterior arms of the angles are in the same straight line. bbreviation: adj. angles supp. Theorem 3 If two straight lines intersect, the vertically opposite angles are equal. a = b and p = q. bbreviation: vert. opp. angles. a b p a b q Theorem 4 Two triangles are congruent if they have two pairs of equal sides, and the angles between the pairs of sides are equal: XY Z if X Z = XY, = XY Z, = Y Z. Y bbreviation: SS
102 NTENTS Theorem 5 Two lines are parallel if a transversal makes (i) a pair of alternate angles equal, or (ii) a pair of corresponding angles equal, or (iii) a pair of interior angles on one side of the transversal supplementary. b a d c bbreviation: (i) alt. angles equal; (ii) corr. angles equal; (iii) int. angles supp. Theorem 6 If a transversal cuts two parallel lines, then (i) alternate angles are equal, e.g., a = b, (ii) corresponding angles are equal, e.g., b = c, and (iii) interior angles on the same side of the transversal are supplementary, e.g., b + d = 2 right angles. b a d c bbreviation: (i) alt. angles between // lines; (ii) corr. angles between // lines; (iii) int. angles between // lines.
NTENTS 103 Theorem 8 (a) If one side of a triangle is extended, the exterior angle so formed is equal to the sum of the two interior opposite angles. (b) The angle sum of a triangle is equal to 2 right angles. bbreviation: (a) ext. angle of triangle; (b) angle sum of triangle. b a c 1c2 E Theorem 9 (a) The sum of the interior angles of a convex polygon of n sides is 2n 4 right angles. (b) If the sides of a convex polygon are extended in order, the sum of the exterior angles so formed are 4 right angles.
104 NTENTS Theorem 10 (a) Two triangles are congruent if they have two pairs of equal angles, and the sides adjacent to the pairs of equal angles are equal: XY Z if = Y XZ, = XY, = XY Z. bbreviation: S X Y Z (b) Two triangles are congruent if they have two pairs of equal angles, and the sides opposite to one pair if equal angles are equal: XY Z if X Z = Y XZ, = XY Z, = Y Z. Y bbreviation: S Theorem 11 If two sides of a triangle are equal, their opposite angles are equal; i.e., = = =. bbreviation base angles, isos. triangle. Theorem 12 If two angles of a triangle are equal, their opposite sides are equal; i.e., = = =. bbreviation sides opp. equal angles.
NTENTS 105 Theorem 13 (a) Two triangles are congruent if they have three pairs of equal sides: XY Z if = XY, = Y Z, = ZX. bbreviation: SSS X Y Z Theorem 14 (a) Two triangles are congruent if they have two pairs of equal sides, and the opposite angles of one pair of equal sides are right angles: XY Z if Z Y X = Y XZ = 90, = Y Z, = XY. bbreviation: RHS
106 NTENTS asic onstruction 1 isect an angle. asic onstruction 2 The perpendicular bisector of a segment. asic onstruction 3 The perpendicular to a line at a point on the line. asic onstruction 4 The perpendicular to a line from a point outside the line. asic onstruction 5 Given a segment, to construct an angle equal to a given angle. asic onstruction 6 To construct the parallel to a line through a point not on the line.
NTENTS 107 Theorem 15 (a) The opposite sides of a parallelogram are equal. (b) The opposite angles of a parallelogram are equal. (c) Each diagonal of a parallelogram bisects the area of the parallelogram. bbreviation: (a) opp. sides //gram; (b) opp. angles //gram. Theorem 16 The diagonals of a parallelogram bisect each other. bbreviation: diagonals //gram.
108 NTENTS Theorem 17 If one pair of opposite sides of a quadrilateral are equal and parallel, then the other pair of opposite sides are also equal and parallel. The quadrilateral is a parallelogram. bbreviation: 2 sides equal and //. Theorem 18 If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. bbreviation: opp. angles equal. Theorem 19 If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. bbreviation: opp. sides equal. Theorem 20 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. bbreviation: diags. bisect each other. asic onstruction 7 To construct a square on a given segment.
NTENTS 109 Theorem 21 If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. bbreviation: greater side opp. angle. greater Theorem 22 If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. bbreviation: greater angle opp. greater side. Theorem 23 mong all lines drawn from a point to a line (not containing it), the perpendicular is the shortest. Q Theorem 24 ny two sides of a triangle are together greater than the third side. bbreviation: triangle ineq.
110 NTENTS Theorem 25 The line joining the midpoints of two sides of a triangle is parallel to the third side, and is equal to one half of its length. bbreviation: Midpoint theorem. Theorem 26 The line through the midpoint of a side of a triangle, parallel to a second side, passes through the midpoint of the third side. F E bbreviation: Intercept theorem. Theorem 27 If three or more parallel lines make equal intercepts on one traversal, then they make equal intercepts on any other transversal. bbreviation: Intercept theorem. asic onstruction 8 To divide a given segment into a given number of equal parts.
NTENTS 111 Theorem 28 (a) The three medians of a triangle are concurrent (at the centroid of the triangle). (b) The centroid of a triangle divides each median in the ratio 2 : 1. F G H E bbreviation: centroid theorem. Theorem 29 (a) ny point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. (b) ny point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. bbreviation: perp. bisector locus. Theorem 30 The perpendicular bisectors of the three sides of a triangle are concurrent (at the circumcenter of the triangle). bbreviation: circumcenter theorem. Theorem 31 The three altitudes of a triangle are concurrent (at the orthocenter of the triangle). bbreviation: orthocenter theorem. Z X H Y
112 NTENTS Theorem 32 (a) ny point on a bisector of an angle is equidistant from the lines containing the angle. (b) ny point equidistant from two intersecting lines lies on the bisector of an angle between the two lines. K H bbreviation: angle bisector locus. Theorem 33 The internal bisectors of the three angles of a triangle are concurrent (at the incenter of the triangle). bbreviation: incenter theorem. I
NTENTS 113 Theorem 35 The area of a rectangle is measured by the product of the measures of two adjacent sides. Theorem 36 The area of a triangle is equal to one half of the area of a rectangle on the same base and between the same parallels. Theorem 37 Triangles on the same base and between the same parallels are equal in area. Theorem 38 Triangles of equal area on the same base and on the same side of the base are between the same parallels. Theorem 39 If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangles is one half of that of the parallelogram. asic onstruction 9 To construct a triangle equal in area to a given quadrilateral. E
114 NTENTS Theorem 40 In a right angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the remaining two sides. bbreviation: yth. theorem. K Theorem 41 If the area of the square on one side of a triangle is equal to the sum of the areas of the squares on the remaining two sides, then these two remaining sides contain a right angle. F G L E H bbreviation: converse yth. theorem. Y a c a b Z b X
NTENTS 115 Theorem 42 The line joining the center of a circle to the midpoint of a chord (which is not a diameter) is perpendicular to the chord. Theorem 43 The perpendicular from the center of a circle to a chord bisects the chord. Theorem 44 Equal chords of a circle are equidistant from the center. M Theorem 45 hords of a circle equidistant from the center are equal in length. N M Theorem 46 Given three points not on the same line, there is a unique circle passing through them. bbreviation: circumcircle theorem.
116 NTENTS Theorem 47 The angle which a minor arc of a circle subtends at the center is twice of the angle it subtends at any point on the complementary major arc. Theorem 48 ngles in the same segment of a circle are equal. Q Q Theorem 49 The angle in a semicircle is a right angle. Theorem 50 (a) The opposite angles of a cyclic quadrilateral are supplementary. (b) If one side of a cyclic quadrilateral is extended, the exterior angle so formed is equal to the interior opposite angle. Q Q Theorem 51 The circle described on the hypotenuse of a right-angled triangle as diameter passes through the vertex of the right angle.
NTENTS 117 Theorem 52 If a segment subtends equal angles at two points on the same side, the four points are concyclic. Q Q Theorem 53 If a pair of opposite angles of a quadrilateral are supplementary, the vertices are concyclic. Theorem 54 (a) In equal circles (or the same circle), equal angles at the centers (or centter) stand on equal arcs. (b) In equal circles (or the same circle), equal angles at the circumferences (or circumference) stand on equal arcs. Theorem 55 (a) In equal circles (or the same circle), equal arcs subtend equal angles at the centers (or centter). (b) In equal circles (or the same circle), equal arcs subtend equal angles at the circumferences (or circumference). Theorem 56 In equal circles (or the same circle), equal chords cut off equal arcs. Theorem 57 In equal circles (or the same circle), chords of equal arcs are equal.
118 NTENTS Theorem 58 The line perpendicular to a radius of a circle at its extremity is tangent to the circle. Theorem 59 tangent to a circle is perpendicular to the radius through the point of tangency. Theorem 60 If two tangents are drawn to a circle from an external points, (a) the tangents are equal, (b) the line joining the point to the center bisects the angle between the two tangents. Theorem 61 The angle between a tangent to a circle and a chord of the circle through the point of tangency is equal to the angle subtended by the chord at any point on the opposite arc of the circle. Theorem 62 line through an endpoint of a chord of a circle making the same angle with the chord as the angle subtended at any point on the opposite arc of the circle is tangent to the circle. Theorem 63 If two circles are tangent to each other, the line joining their centers passes through the point of tangency. l l
NTENTS 119 asic onstruction 10 To construct the tangent to a circle at a point on the circumference. asic onstruction 11 To construct the tangents to a circle from a point outside the circumference. asic onstruction 12 To construct the exterior common tangents of two circles. asic onstruction 13 To construct the interior common tangents of two circles. asic onstruction 14 To construct the incircle of a triangle. asic onstruction 15 To construct an excircle of a triangle. asic onstruction 16 n a given line segment, to construct a segment of a circle containing an angle equal to a given angle.
120 NTENTS Theorem 64 In an obtuse angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing it, plus twice the rectangle contained by one of these sides and the projection of the other side on it. c b c b a X X a Theorem 65 In any triangle, the square on the opposite side of an acute angle is equal to the sum of the squares on the sides containing it, minus twice the rectangle contained by one of these sides and the projection of the other side on it. Theorem 66 (pollonius Theorem) In any triangle, the sum of the squares on two sides is equal to twice the square on half of the third side, plus twice the square of the median which bisects the third side.
NTENTS 121 Theorem 67 (Intersecting chords theorem) If two chords and of a circle intersect at a point inside the circle, then area of the rectangle formed by and is equal to that of the rectangle formed by and. T Theorem 68 (Intersecting chords theorem) If two chords and of a circle intersect at a point outside the circle, then the area of the rectangle formed by and is equal to that of the rectangle formed by and, and also equal to the square of the tangent from to the circle. Theorem 69 If two segments and are divided, both internally or both externally, at a point such that the rectangle formed by and has the same area as that formed by and, then the four points,,, are concyclic. asic onstruction 17 To construct a square equal in area to a given rectangle. asic onstruction 18 To construct a circle passing through two given points and tangent to a given line (not containing the given points).
122 NTENTS Theorem 70 If two triangle have equal altitudes, then the ratio of their areas is equal to the ratio of their bases. Theorem 71 straight line parallel to a side of a triangle divides the remaining two sides, extended if necessary, in the same ratio. Theorem 72 If two sides of a triangle are divided in the same ratio, both internally or both externally, the line joining the points of division is parallel to the third side. Theorem 73 (ngle bisector theorem) The bisectors of an angle of a triangle divide its opposite side in the ratio of the remaining sides. If X and X respectively the internal and external bisectors of angle, then X : X = c : b and X : X = c : b. Z c Z b X X Theorem 74 (ngle bisector theorem) (a) If X is a point on the side such that X : X = :, then the line X is the (internal) bisector of angle. (b) If X is a point on the extension of such that X : X = :, then the line X bisects angle externally. asic onstruction 19 To divide a given line segment in a given ratio, internally and externally. asic onstruction 20 To construct the fourth proportional of three given line segments.
NTENTS 123 Theorem 75 If two triangles are equiangular, their corresponding sides are proportional. Theorem 76 If the three sides of one triangle are proportional to the three sides of a second triangle, then the two triangles are equiangular. Theorem 77 If two triangle have one pair of equal angles, and if the sides containing these angles are proportional, then the two triangles are equiangular. Theorem 78 If a perpendicular is drawn from the right angle of a right-angled triangle to the hypotenuse, the triangles on each side of the perpendicular is similar to the whole triangle, and to one another. Theorem 79 The ratio of the areas of two similar triangles is equal to the ratio of the squares of corresponding sides. Theorem 80 If a segment is divided externally at X, and if is a point (not on the line ) such that X X = X 2, then the circle is tangent to X at. X asic onstruction 21 To construct the mean proportional of two given line segments. asic onstruction 22 To divide a segment in the golden ratio, e.g., to construct a point such that = 2. asic onstruction 23 To construct a triangle with = = 2.