Geometry Project Meagan Farber

Chapter one Basics of Geometry

1.1) Patterns and inductive reasoning Terms to know Conjecture- is an unproven statements that is based one observations Inductive reasoning- looking for patterns and making conjectures Counterexample- an example that shows a conjecture is false

1.2) Points, lines, and planes Point has no dimension. Usually represented by a small dot Line extends in one dimension. Usually represented by a straight line with two arrow heads to indicate that the line extends. Plane extends in two dimensions. It usually represented by a shape that looks like a tabletop Collinear Points lie on the same line Coplanar Points life on the same plane Two or more figures intersect if they have one or more points in common The intersection of the figures is the set of points the figures have in common

Line: Point A B Line Segment: Line A B Ray: Plane: A a d B b c Opposite Rays: A C B

1.3) segments and their measures Postulate 1: Rulers Postulate Postulates/axioms- rules that are accepted without proof Theorems- rules that are proved The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the length of AB Names of points A B x1 x2 Coordinates of points AB= |x2 – x1|

Congruent Segments- segments that have the same length Postulate 2: Segment Addition Postulate The Distance Formula If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the distance between A and B is AB= √(x2-x1)^2 + (y2-y1)^2 Congruent Segments- segments that have the same length A B C

Distance Formula Pythagorean Theorem (AB)^2 = (x2-x1)^2 + (y2-y1)^2

1.4) Angles and their Measures An angle is two different rays that have the same initial point. The rays are the sides of the angle. The initial point is the vertex of the angle. Angles that have the same measure are congruent angles Postulate 3: Protractor Postulate Consider a point A on one side of BC. The rays of the form BA can be matched one to one with real numbers from 0 to 180

Postulate 4: Angle Addition Postulate Classifying Angles

1.5) Segment and angle bisectors Midpoint- point that bisects the segments into two congruent segments. Segment bisector- segment, ray, line, or plane that intersects a segment at its midpoint Midpoint Formula- Angle bisector- is a ray that divides an angle into two adjacent that are congruent

1.6) angle pair relationships Vertical angles- if sides form two pairs of opposite rays Linear pair- if their non-common sides are opposite rays <1 and <3 are vertical angles <1 and <2 are a linear pair Complementary Adjacent Supplementary non adjacent Complementary non adjacent

1.7) intro to perimeter, circumference, and area Square- side length s P = 4s A= s^2 Triangle- side length a, b, and c, base b and height h P= a + b + c A= ½bh Rectangle- length l and width w P= 2l + 2w A=lw Circle= radius r C= 2r A=r^2

Chapter two Reasoning and Proof

2.1) Conditional Statements Conditional statement- has two parts a hypothesis and a conclusion converse- formed by switching the hypothesis and conclusion negation- writing the negative of the statement inverse-when you negate the hypothesis and conclusion Contra positive-when you negate the hypothesis and conclusion of the converse Equivalent statements- two statements are both true or both false

Point, line and Plane Postulates Postulate 5: through any two points there exists exactly one line Postulate 6: a line contains at least two points Postulate 7:if two lines intersect, then their intersection is exactly one point Postulate 8: through any 3 non collinear points there exists exactly one point Postulate 9: a plane contains at least 3 non collinear points Postulate 10: if two points lie in a plane, then the line containing them lies in the plane Postulate 11: if two planes intersect, then their intersection is a line

2.2)definitions and biconditional statements Perpendicular lines- if they intersect to form a right angle Biconditional statement- a statement that contains “ if and only if” 2.3)Deductive Reasoning Logical argument- deductive reasoning uses facts, definitions, and accepted properties in a logical order Law of Detachment- if p→q is a true conditional statement and p is true, then q is true Law of Syllogism- p→q and q→r are true conditional statements then p→r is true 2.4)Reasoning with properties from algebra Addition property- if a=b, then a+c=b+c Subtraction property- if a=b, then a-c=b-c Multiplication property- if a=b, then ac=bc Division property- if a=b and c≠0, then a/c=b/c Reflexive property-for any real number a, a=a Symmetric property- if a=b, then b=a Transitive property- if a=b and b=c, then a=c Substitution property= if a=b, then a can be substituted for b in any equation or expression

2.5)proving statements about segments Theorem 2.1- Properties of Segment Congruence Segment congruence is reflexive, symmetric and transitive Reflexive- for any segment AB, AB≈AB Symmetric- if AB≈CD, then CD≈AB transitive-if AB≈CD, and CD≈EF, then AB≈EF 2.6)Proving statements about angles Theorem 2.2- Properties of Angles Congruence Angle congruence is reflexive, symmetric and transitive Reflexive- for any angle A, <A≈<A Symmetric- If <A≈<B, then <B ≈<A Transitive- if <A≈<B and <B≈<C, then <A≈<C 2.3) Right Angle Congruence Theorem All right angles are congruent 2.4) congruent supplements theorem If two angles are supplementary to the same angle they are congruent 2.5) congruent complements theorem If two angles are complementary to the same angle then the two angles are congruent Postulate 12) Linear Pair Postulate If two angles form a linear pair, then they are supplementary Theorem 2.6) Vertical Angles Theorem Vertical angles are congruent

Chapter three Perpendicular and Parallel Lines

3.1) lines and angles Corresponding angles = 1 and 5 Parallel lines- two lines that are coplanar and do not intersect Skew lines- lines that do not intersect and are not coplanar Parallel planes- two planes that do not intersect Postulate 13) parallel postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line Postulate 14) perpendicular postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line Transversal- a line that intersects two or more coplanar lines at different points Corresponding angles = 1 and 5 Alternate exterior angles = 1 and 5 Alternate interior angles= 3 and 6 Consecutive interior angle= 3 and 5

3.2) Proof and perpendicular lines Theorem 3.1- if two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular Theorem 3.2- if two sides of two adjacent acute angles are perpendicular, then the angles are complementary Theorem 3.3- if two lines are perpendicular, then they intersect to form four right angles 3.3)Parallel lines and transversals Postulate 15) Corresponding Angles Postulates If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent Theorem 3.4- if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent Theorem 3.5- if two parallel lines are cut by a transversal, then the pair of consecutive interior angles are supplementary Theorem 3.6- if two parallel lines are cut by transversal, then the pairs of alternate exterior angles are congruent Theorem 3.7- if a transversal is perpendicular to one of two parallel lines, then its is perpendicular to the other

3.4 Proving lines are parallel Postulate 16) Corresponding angles converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel Theorem 3.8- if two liens are cut by a transversal so that alternate interior angles are congruent then the lines are parallel Theorem 3.9- if two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel Theorem 3.10- if two lines are cut by a transversal so that alternate exterior angles are congruent then the lines are parallel 3.5 using properties of parallel lines Theorem 3.11- if two lines are parallel to the same line then they are parallel to each other Theorem 3.12- in a plane, if two lines are perpendicular to the same line, then they are parallel to each other 3.6 Parallel lines in the coordinate plane Postulate 17) Slopes of parallel lines In a coordinate plane, two non vertical lines are parallel if and only if they have the same slope, any two verticall lines are parallel 3.7 Perpendicular lines in the coordinate plane Postulate 18) Slopes of perpendicular lines In a coordinate plane, two non vertical lines are perpendicular, if and only if they product of their slopes is -1. vertical and horizontal lines are perpendicular

Chapter 4 Congruent Triangles

4.1 Triangles and angles

4.2 Congruence and triangles Theorem 4.1 Triangle Sum Theorem- the sum of the measures of the interior angles of a triangle is 180° Theorem 4.2 Exterior Angle Theorem- the measure of an exterior angle of a triangle is equal to the sum of the measure of the two nonadjacent interior angles Corollary to the triangle sum theorem- the acute angles of a right triangle are complementary 4.2 Congruence and triangles Theorem 4.3 Third Angles Theorem- if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent Theorem 4.4 Properties of Congruent Triangles Reflexive property- every triangle is congruent to itself Symmetric property- if ∆abc≈∆def, then ∆def≈∆abc Transitive property- if ∆abc≈∆def and ∆def≈∆jkl, then ∆abc ≈ ∆ jkl

4.3 proving triangles are congruent: sss and sas Postulate 19- Side-side-side congruence postulate If there sides of one triangle are congruent to three sides of a second triangle then the two triangles are congruent Postulate 20- Side-angle-Side congruence postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent 4.4 Proving triangles are congruent: asa and aas Postulate 21- Angle-side-angle congruence postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent Postulate 22- Angle-angle-side congruence postulate If two angles and a non included side of one triangle are congruent to two angles and the corresponding non included side of a second corresponding non included side of a second triangle, then the two triangles are congruent

4.6 isosceles, equilateral, and right triangles Theorem 4.6- if two sides of a triangles are congruent, then the angles opposite them are congruent Theorem 4.7- if two angles of a triangle are congruent, then the sides opposite them are congruent Corollary to theorem 4.6- if a triangle is equilateral, then its is equiangular Corollary to theorem 4.7- if a triangle is equiangular, then it is equilateral Theorem 4.8- if the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle , then the two triangles are congruent

Chapter five Properties of Triangles

5.1 perpendiculars and bisectors Theorem 5.1- if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment Theorem 5.2- if a point is equidistant from the endpoints of a segment, then its on the perpendicular bisector of the segment Theorem 5.3- if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle Theorem 5.4- if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of an angle

5.2 bisectors of a triangle Theorem 5.5- the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle Theorem 5.6- the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle 5.3 medians and altitudes of a triangle Theorem 5.7- the medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite sides Theorem 5.8- the lines containing the altitudes of a triangle are concurrent 5.4 midsegment theorem Theorem 5.9- the segment connecting the midpoints of two sides of a triangle is parallel to the third sides and is half as long 5.5 inequalities in one triangle Theorem 5.10- if one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side Theorem 5.11- if one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle

5.6 indirect proof and inequalities in two triangles Theorem 5.12- the measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles Theorem 5.13- the sum of the lengths of any two sides of a triangle is greater than the length of the third side 5.6 indirect proof and inequalities in two triangles Theorem 5.14- if two dies of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second Theorem 5.15- if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second

Chapter six Quadrilaterals

Polygons Number of Sides Type of Polygon 3 1.Triangle 8 6.Octagon 4 2.Quadrilateral 9 7.Nonagon 5 3.Pentagon 10 8.Decagon 6 4.Hexagon 12 9.Dodecagon 7 5.heptagon N 10.N-gon 2 1 4 3 9 6 5 8 Theorem 6.1- the sum of the measures of the interior angles of a quadrilateral is 360°

6.2 properties of parallelograms Theorem 6.2- if a quadrilateral is a parallelogram, then its opposite sides are congruent Theorem 6.3- if a quadrilateral is a parallelogram, tehn its opposites angles are congruent Theorem 6.4- if a quadrilateral is a parallelogram, tehn its consecutive angles are supplementary Theorem 6.5- if a quadrilateral is a parallelogram, then its diagonals bisect each other 6.3 proving quadrilateral are parallelograms Theorem 6.6- if both pairs of opposites sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Theorem 6.7- if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is parallelogram Theorem 6.8- if an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram Theorem 6.9- if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Theorem 6.10- if one pair of opposite sides of a quadrilateral are congruent and parallel then the quadrilateral is a parallelogram

6.4 rhombuses, rectangles, and squares Rhombus is a parallelogram with four congruent sides A rectangle is a parallelogram with four right angles A square is a parallelogram with four congruent sides and four right angles Rhombus corollary- a quadrilateral is a rhombus if and only if it has four congruent sides Rectangle corollary- a quadrilateral is a rectangle if and only if its has four right angles Square corollary- a quadrilateral is a square if and only if it’s a rhombus and a rectangle Theorem 6.11- a parallelogram is a rhombus if and only if its diagonals are perpendicular Theorem 6.12- a parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles Theorem 6.13- a parallelogram is a rectangle if and only if its diagonals are congruent

6.5 trapezoids and kites Theorem 6.14- if a trapezoid is isosceles, then each pair of base angles is congruent Theorem 6.15- if a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid Theorem 6.16- a trapezoid is isosceles if and only if its diagonals are congruent Theorem 6.17- the midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases Theorem 6.18- if a quadrilateral is a kite, then its diagonals are perpendicular Theorem 6.19- if a quadrilateral is a kite, then exactly one pair of opposite angles are congruent

Chapter seven Transformations

7.2 reflections 7.3 rotations 7.1 rigid motion in a plane image- new figure preimage- the original figure Transformation- the operation that maps or moves the preimage onto the images Isometry- a transformation that preserves lengths 7.2 reflections reflection- type of transformation uses a line that acts like a mirror , with an image reflected in the line Line of reflection- the mirror line Line of symmetry- the figure can be mapped onto itself by a reflection line 7.3 rotations Rotation- transformation in which a figure is turned about a fixed point Center of rotation- the fixed point Angle of rotation- rays drawn from the center of rotation to a point and its image form an angle Rotational symmetry- the figure can be mapped onto itself by roation of 180°

7.4 translations and vectors Theorem 7.4- a translation is isometry Theorem 7.5- if lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation. If P” is the image of p, then the following is true: PP” is perpendicular to k&m. PP”= 2d, where d is the distance between k&m. 7.5 glide reflections and compositions Glide reflection- transformation in which every point p is mapped onto a point p” by the following steps: a translation maps p onto p”. A reflection in a line k parallel to the direction of the translation maps p’ onto p” Composition- when two or more transformations are combined to produce a single transformation Theorem 7.6- the composition of two or more isometries is an isometry

Chapter eight Similarity

8.2 problems solving in geometry with proportions 8.1 ratio and proportion Properties of proportions Cross products property- the product of the extremes equals the product of the means Reciprocal property- if two ratios are equal, then their reciprocals are also equal 8.2 problems solving in geometry with proportions Additional properties of proportions If a/b=c/d then a/c=b/d If a/b=c/d then a+b/b=c+d/d 8.3 similar polygons Theorem 8.1- if two polygons are similar, then the ratio of their perimeter is equal to the ratios of their corresponding side lengths 8.4 similar triangles Postulate 25- if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar

8.5 proving triangles are similar` ` Theorem 8.2- if the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar Theorem 8.3- if an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional then the triangles are similar 8.6 proportions and similar triangles Theorem 8.4- if a line parallel to one side of triangle intersects the other two sides, the it divides the two sides proportionally Theorem 8.5- if a line divides two sides of a triangle proportionally then it is parallel to the third side Theorem 8.6- if three parallel lines intersect two transversals, then they divide the transversals proportionally Theorem 8.7- if a ray bisects an angle of a triangle then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides 8.7 dilations Dilation- with center c and scale factor k is a transformatin that maps ever point p in the plane to point p’ so that the following properties are true: if p is not the center point c, then the image point p’ lies on CP, the scale factor k is a postivie number such that k=cp’/cp, and k≠1. if p is the center point c, then p=p’

Chapter nine Right triangles and trigonometry

9.1 similar right triangles Theorem 9.1- if the altitude is drawn to the hypotenuse of a right triangle., then the two triangles formed are similar to the original triangle and to each other Theorem 9.2- in a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments Theorem 9.3- in a right triangle the altitude from the right angle to the hypotenuse divides the hypotenuse into two segmetns. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg 9.4 special righ triangles Theorem 9.8- in a 45-45-90 degree triangle the hypotenuse is √2 times as long as each leg Theorem 9.9- in a 30-60-90 degree triangle the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg

9.5 trigonometric ratios 9.7 vectors Adding vectors: sum of two vectors- the sum of u=(a1,b1) and v=(a2,b2) is u+v= (a1+a2, b1+b2)

Chapter 10 Circles

10.1 tangents to circles Theorem 10.1- if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency Theorem 10.2- in a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then hte line is tangent to the circle Theorem 10.3- if two segments from the same exterior point are tangent to a circle, then they are congruent

10.2 Arcs and chords Postulate 26- the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs Theorem 10.4- if the same circle, or in a congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent Theorem 10.5- if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc Theorem 10.6- if one chord is a perpendicular bisector of an other chord, then the first chord is the diameter Theorem 10.7- in the same circle, or in two congruent circles, two chords are congruent if and only if they are equidistant from the center

10.3 inscribed angles Theorem 10.8- if an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc Theorem 10.9- if two angles of a circle intercept the same arc, then the angles are congruent Theorem 10.10- if a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle Theorem 10.11- a quadrilateral can be inscribed in a circle if and only if opposites = 180° Theorem 10.12- if a tangent and a chord intersect at a point on a circle the measure of each angle formed is one half the measure of its intercepted arc Theorem 10.13- if two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle Theorem 10.14- if a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measures of the angle formed is one half the difference of the measure of the intercepted arcs

10.5 segment lengths in circles Theorem 10.15- if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord Theorem 10.16- if two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment Theorem 10.17- if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment

Chapter 11 Area of polygons and circles

11.1 angle measures in polygons Theorem 11.1- the sum of the measures of the interior angles of a convex n-gon is (n-2)*180° Corollary to theorem 11.1- the measure of each interior angle of a regular n-gon is Theorem 11.2- the sum of the measure of the exterior angles of a convex polygon, one angle at each vertex is 360 Corollary to theorem 11.2- the measure of each exterior angle of a regular n-gon is 360°/n

11.2 areas of regular polygons Theorem 11.3- the area of an equilateral triangle is one fourth the square of the length of the side time √3 Theorem 11.4- the area of a regular n-gon with side length s is half the product of the apothem a and the perimeter p, so A= ½ap or A=12a*ns 11.3 perimeters and areas of similar figures Theorem 11.5- if two polygons are similar with the lengths of corresponding sides in the ratio a:b, then the areas will be a^2:b^2 11.4 circumference and arc length Theorem 11.6- the circumference c of a circle is c=d or c=2r, where d is diameter and r is the radius Arc length corollary- in a circle the ratio of the length of a given arc to circumference is equal to the ratio of the measure of the arc to 360°

11.5 areas of circles and sectors Theorem 11.7- the areas of a circle is  times the square of the radius. A= r^2 Theorem11.8- the ratio to the area a of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360° 11.6 geometric probability Probability and length- let AB be segment that contains the segment CD. If a point K on AB is chosen at random, then the probability that it is on CD is as follows: P(point k is on CD) = length of CD/ length of AB