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Geometry Regents By Tsun Wong (Logic not included) Fall Term Curriculum Common Postulates used Line Axiom: Given two distinct points, exactly one line contains them both. Each line contains at least 2 points. Given a line, there exists a point not on the line. The Segment Construction Postulate: A segment can be extended. Line segment Partition Postulate: If point B is between A and C, then AB + BC= AC Angle Partition Postulate: If P lies interior of Angle ABC, then ABP +PBC=ABC Trichotomy Postulate: Given two real numbers a and b, either a>b or a=b or a<b Line Segment Comparison Axiom: A whole is greater than its parts. Common Theorems used Two distinct lines have at most one point in common. Any two points have a unique midpoint. Each angle has a unique angle bisector. Through a point on the line, there exists one and only one perpendicular to the line. Through a point not on a line, there exists one and only one perpendicular to the line. Chapter 3: Congruent Triangles -SAS postulate, ASA Theorem, SSS Theorem are the three ways to prove triangles congruent. -Corresponding parts of congruent triangles are congruent. -All radii of a given circle are congruent. -A median of a triangle is a segment drawn from any vertex of the triangle to the midpoint of the opposite side. -If two sides of a triangle are congruent, then the angles opposite them are congruent. -If two angles of a triangle are congruent, then the sides opposite them are congruent. Hy-Leg Theorem: If a leg and a hypotenuse of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. Note: MUST prove that it is a right triangle. Chapter 4: Lines in a plane -All right angles are congruent. -If two points are each equidistant from the endpoint of a line segment, then the line joining them will be the perpendicular bisector of the line segment. -If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. -If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the line segment. Chapter 5: Parallel lines and related figures -The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. -The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. -If two lines are cut by a transversal and form a pair of congruent corresponding angles/alternate exterior/alternate interior angles, then the lines are parallel. -If two lines are cut by a transversal and form a pair of same-side interior/exterior angles that are supplementary, then the lines are parallel. -If two parallel lines are cut by a transversal, then each pair of alternate exterior/alternate interior/corresponding angles are congruent. -If two parallel lines are cut by a transversal, then each pair of same-side interior/exterior angles are supplementary. -If two lines are parallel, a line perpendicular to one of them is also perpendicular to the other. Parallelograms Properties of a Parallelogram 1. The opposite sides are parallel. 2. The opposite sides are congruent 3. The opposite angles are congruent. 4. The diagonals bisect each other. 5. Any pair of consecutive angles are supplementary. Properties of a rectangle 1. All the properties of a parallelogram. 2. All angles are right angles. 3. Diagonals are congruent. Properties of Kites 1. Two disjoint pairs of consecutive sides are congruent. 2. The diagonals are perpendicular. 3.One diagonal is the perpendicular bisector to other. 4. One of the diagonals bisects a pair of opposite angles. 5. One pair of opposite angles are congruent. Properties of Rhombuses 1. All the properties of a parallelogram. 2. All the properties of a kite. 3. All sides are congruent. 4. The diagonals bisect the angles. 5. The diagonals perpendicular bisect each other. 6. The diagonals divide a rhombus into 4 congruent right triangles. Properties of Squares 1. All the properties of a rectangle AND all the properties of a rhombus. 2. The diagonals form four isosceles right triangles. Properties of Isosceles Trapezoids 1. The legs are congruent. 2. The base are parallel. 3. Base angles are congruent. 4. The diagonals are congruent. 5. Lower and upper base angle are supplementary. To prove that a quadrilateral is a … Parallelogram 1. If both pairs of opposite sides of a quadrilateral are parallel/congruent, then the quadrilateral is a parallelogram. 2. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram. 3. If the diagonals of a quad bisect each other, then the quad is a parallelogram. 4. If both pairs of opposite angles of a quad are congruent, then the quad is a parallelogram. Rectangle 1. If a parallelogram contains at least one right angle, then it is a rectangle. 2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 3. If all four angles of a quadrilateral are right angles, then it is a rectangle. (No parallelogram proving required) Kite 1. If two disjoint pairs of consecutive sides of a quad are congruent, then it is a kite. 2. If one of the diagonals of a quad is the perpendicular bisector of another diagonal, then the quad is a kite. Rhombus 1. If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus. 2. If either diagonal of a parallelogram bisects two angles of the parallelogram, then ‘ ‘. 3. If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus. (No parallelogram proving required) Square If a quadrilateral is both a rectangle and a rhombus, then it is a square. A trapezoid that is isosceles 1. If the non-parallel sides of a trapezoid are congruent, then it is isosceles. 2. If the lower or the upper base angles of a trapezoid are congruent, then it is isosceles. 3. If the diagonals of a trapezoid are congruent, then it is isosceles. Chapter 7: Polygons -The sum of the measure of the three angles of a triangle are 180. Midline Theorem: A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one half the length of the third side. AAA Theorem (If two angles of two triangles are congruent, then the third one must be also) AAS Theorem- Congruent triangles. Polygon formula: S=(n-2)180 Chapter 8: Similar Polygons Means Extremes Ratio Theorem- If the product of a pair of numbers is equal to the product of another pair of numbers, then they may be made the extremes, and the other pair the means, of the proportion. Means Extremes Product Theorem- In a proportion, the product of the means is equal to the product of the extremes. Arithmetic Mean VS Geometric Mean Given two distinct numbers a and b. Arithmetic Mean = a+b then divide by two. Geometric Mean= Write a proportion, use a and b as the extremes and x as both means. -AAA Theorem, AA Theorem, SSS Theorem, SAS Theorem are the four ways to prove two triangles being similar. -Corresponding sides of similar triangles are proportional. -Corresponding angles of similar triangles are congruent. Side-Splitter Theorem- If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally. Chapter 9: Pythagorean Theorem ALTITUTE ON HYPOTENUSE THEOREM! -If an altitude is drawn to the hypotenuse of a right triangle, then… a. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse it cuts through. b. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to the leg. Pythagorean Theorem: The square of the measure of the hypotenuse of a triangle is equal to the sum of the squares of the measures of the legs. ______________________________________________________________________________ Spring Term Chapter 11: Area -In a 30-60-90 right triangle, the lengths opposite the angles is the ratio of x, xroot3, 2x, respectively. -In a 45-45-90 right triangle, the lengths opposite the angles is the ratio of x, x, x root 2, respectively. - Atriangle=1/2bh -Aequilateraltriangle=root3/4 (s2) -Arectangle=lw -Asquare= s2 -Arhombus= 1/2d1d2 -Atrapezoid= 1/2(b1+b2)h -The ratio of the areas of triangles with equal heights is equal to the ratio of the corresponding bases. -The ratio of the areas of triangles with equal bases is equal to the ratio of the corresponding heights. -The ratio of the areas of two similar triangles is equal to the square of similitude. -A central triangle is a triangle formed by a central angle (an angle by two radii drawn to consecutive vertices). - Area of an n-gon = number of sides (Area of a central triangle) Chapter 13: Coordinate Geometry … EW. FUCK. EW. I HATE IT. Midpoint Formula: (x1+x2 then divide by 2, y1+y2 then divide by 2) Distance Formula: d= square root of (x2-x1)2+(y2-y1)2 Slope: y2-y1 / x2-x1 -If two non-vertical lines are perpendicular, then the product of their slopes is -1. -If the product of two lines’ slopes is -1, then they are perpendicular. -When a point divides a segment into a certain ratio. Find the coordinate of that point by: 1. Find the ratio of P1P / P2P = r1/r2 where P = (x,y) , P1= (x1,y1), and P2=(x2,y2) 2. The x value: x = r1x2+r2x1 /r1 + r2 3. The y value: same as x but change all xs with ys Point-slope form: y-yo=m(x-xo) where m is the slope and (xo,yo) are known points. -y=ax2+k, k>0, y=ax2 moves up k units. Vise versa for negatives. Axis Of Symmetry: x= -b/2a Center Radius Form of a Circle: (x-a)2+(y-b)2 = r2 where (a,b) = center and r= radius. -In a perfect square trinomial, ax2+bx+c, we can use b to find c using the equation (b/2)2=c -Also, If a>0 , then it is concave up. If a<0, then it is concave down. -The solutions (roots, zero) to the related quadratic equation ax2+bx+c=0 gives us the xintercepts for the parabola. -The c in the trinomial gives us the y intercept. Chapter 10: Circles :D (Phew, I hate coordinate geometry) Theorems 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. If 2 circles are congruent, then the lengths of their radii are the same/their radii are congruent. If 2 circles are =, then they share the same center and their radii are congruent. No circle contains 3 distinct collinear points. Through any 3 distinct non-collinear points, there is exactly one circle. In Euclidean Geometry, every triangle can be circumscribed and the center of the circumscribing circle is the point where the perpendicular bisectors of the sides meet. The three perpendicular bisectors of a triangle are concurrent. Any 2 distinct points of a circle determine two arcs. The degree measure for the circumference of a circle is 360 degrees. The linear measure for the circumference of a circle is 2(pi)r Arc Addition Postulate: If AB and BC are arcs of the same circle having an endpoint, and no other points in common, then mAB + mBC = mABC Diameter-Chord 1. The diameter of a circle is a chord with the greatest possible length. 2. If a diameter (or a radius) of a circle is perpendicular to a chord of the circle then the diameter bisects the chord. 3. If a diameter or a radius of a circle bisects a non-diameter chord of the circle, then it is perpendicular to the chord. 4. The perpendicular bisector of a chord of a circle passes through the center of the circle. 5. If two chords are equidistant from the center of a circle then they are congruent. 6. If two chords are congruent then they are equidistant from the center of the circle. Tangent 1. If a line is tangent to a circle then it is perpendicular to the radius at the point of tangency. 2. If a line is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. 3. If 2 tangents are drawn to a circle from the same external point then the tangent segments are congruent. 4. If two circles are externally tangent, then the point of tangency lies on the line of centers. Arcs, Chords, central angles 1. In a given circle (or congruent circles), congruent central angles intercept congruent arcs. 2. In a given circle (or congruent circles), congruent arcs are intercepted by congruent central angles. 3. In a given circle (or congruent circles), congruent central angles intercept congruent chords. 4. In a given circle (or congruent circles) congruent chords are intercepted by congruent central angles 5. If 2 arcs of a given circle (or congruent circles) are congruent, then the corresponding chords are congruent. 6. If two chords of a given circle (or congruent circles) are congruent, then the corresponding arcs are congruent. Angles with vertex on a circle 1. 2. 3. 4. 5. The measure of an inscribed angle of a circle is one half the measure of its intercepted arc. In a given circle, inscribe dangles that intercept the same or congruent arcs are congruent. An angle inscribed in a semicircle is a right angle. If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. In a given circle, parallel chords intercept congruent arcs between them. Angles with vertex off a circle 1. The measure of a chord-chord angle is ½ of the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle. 2. The measure of a secant-secant angle, a secant-tangent angle, and a tangent-tangent angle is equal to ½ the difference of the measures of its intercepted arcs. Tangent/secant/chord 1. If 2 secants are drawn to a circle from an external point, the product of the lengths of one secant and its external segment is = to the product of the lengths of the other secant and its external segment: Secant-secant power theorem 2. If a tangent and a secant are drawn to a circle from an external point, the tangent is the mean proportional between the secant and its external segment: Secant-Tangent power theorem 3. If 2 chords intersect inside a circle, the product of the lengths of the segments of one chord is = to the product of the lengths of the segments of the other chord: Chord-chord power theorem Definitions 1. A circle is the set of all points that are equidistant from a fixed point on a plane. The common distance is the radius. The fixed point is the center. 2. Notation: C(O,r) where O is the center 3. The interior of a circle is the set of all points such that the distance from each point to the center is greater than or equal to 0 and less than the radius. 4. The exterior of a circle is the set of all points such that the distance from each point to the center is greater than or equal to 0 and greater than the radius. 5. The circumcenter of a triangle is the point of concurrency for the perpendicular bisectors. 6. A chord of a circle is a line segment whose endpoints are points of the circle. 7. A secant to a circle is a line which contains exactly 2 points of the circle. Every chord belongs to a secant and ever secant contains a chord. 8. A diameter of a circle is a chord which contains the center of the circle. Every radius belongs to a diameter and every diameter contains 2 radii. 9. A line is tangent to a circle if it intersects the circle in exactly one point. The point of intersection is called the point of tangency. 10. If 2 points of a circle are A and B and line l contains them both, then Arc AB is the set of all points of circle O on one side of line l together with points A and B. 11. Minor arcs are denoted by 2 letters. Major arcs are denoted by 3. Semicircles are named using 3 points including the endpoints of a diameter. 12. Central angle: vertex is center, rays are radii 13. Inscribed angle: vertex is on the circle, rays are chords. 14. Linear measure: length of string to “cover” the arc 15. Degree measure: measure of the central angle that intercepts the arc. 16. Tangent circles are circles that intersect each other in exactly one point. 17. 2 circles are externally tangent/internally tangent if each/one of the circles lies exterior/interior to the other. 18. A line of centers joins the centers of 2 circles. 19. A common tangent is a line tangent to two circles. Common internal and common external 20. A chord-chord angle is an angle formed by two chords that intersect inside a circle but no at the center. Chapter 14: Locus A locus is a set consisting of all the points, and only the points, that satisfy specific conditions. 1. The locus of points at a given distance from a given point is a circle whose center is the given point and the radius is the given distance. 2. The locus of points at a given distance from a given line is a pair of lines parallel to the given line at the given distance. 3. The locus of points equidistant from two parallel lines is a third line parallel to the given lines and midway between them. 4. The locus of points equidistant from two given points is the perpendicular bisector of the segment joining the two given points. 5. The locus of points equidistant from two intersecting lines is the pair of lines that bisect the angle formed by the two intersecting lines. 6. The locus of points equidistant from the sides of an angle is the angle bisector of the given angle. Note: A compound locus involve combining two or more loci. Concurrency Theorems 1. The perpendicular bisectors of the sides of a triangle are concurrent. The perpendicular bisectors of the sides of a triangle meet at the point that is equidistant from the vertices of a triangle. The point of concurrency is called the circumcenter. 2. The medians of a triangle meet at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side. The point of concurrency for the medians is called the centroid. 3. The angle bisectors of a triangle meet at a point that is equidistant from the sides of a triangle. The point of intersection is called the incenter. 4. The altitudes of a triangle are concurrent. The point of concurrency is called the orthocenter. NOTE. 1. The circumcenter and orthocenter of an obtuse triangle lie outside of the triangle. 2. The orthocenter of a right triangle is the vertex of its right angle. 3. The circumcenter of a right triangle is the midpoint of its hypotenuse. 4. The incenter and centroid of a triangle ALWAYS lie in its interior. Not Included in Textbook: Transformation Geometry Unit Vocabularies: 1. Transformation is a one-to-one mapping of points in the plane to points in the plane. 2. Original point is called preimage and point it is mapped to is called the image. 3. Definition of a line reflection: A reflection in line l is a transformation of the plane such that: 1. If a point P is not on line l, then its image is the point p1 where l is the perpendicular bisector of pp1. 2. If a point P is on line l, then P is its own image. 4. Geometric properties that are preserved by a line reflection: Distance, Area, Angle measure, Parallelism, Collinearity, Midpoint. 5. A translation is a transformation that moves each point in the plane the same number of units in the same direction. (Preserves six geometric properties) 6. A composition of transformation is an operation where the preimage for the second transformation is the image from the first transformation. NOTE: Compositions for transformations is not (necessarily) commutative. 7. A glide reflection is a composition in either order of a line reflection and a translation parallel to the line of reflection. 8. A reflection in point P is a transformation of the plane such that: a. the image of point P is itself. b. the image of any other point A is the point A1 such that P is the midpoint of AA1 NOTE: A point reflection is equivalent to a rotation of 180 degrees. Also preserves geometry properties. And remember, use Midpoint Formula. 9. A rotation about point P through an angle of theta degrees is a transformation of the plane such that: a. the image of point P is itself. b. the image of any other point A is the point A1 where: mAPA1= Theta and also A1P= AP NOTE: Counterclockwise rotations are positive and clockwise rotations are negative. Also note that, (x,y) r90-> (-y,x) (x,y) r180-> (-x,-y) (x,y) r270-> (y,-x) Last but not least, 6 basic geometric properties are preserved under rotations. 10. A dilation is the …? NOTE: In coordinate geometry, the center of dilation is (0,0). 11. An isometry is a transformation that preserves distance. There are 2 types of isometries: a. Direct Isometry- preserves orientation (ex. Point reflection, Rotation, Translation) b. Opposite Isometry- Doesn’t preserve orientation (ex, Line reflection, Glide reflection) Chapter 6: Planes Line and Plane relationships 1. If a line passes through 2 points that lie in a plane, then the line lies entirely in the plane. 2. If a line not contained in a plane intersects a plane, then they intersect at most 1 point. Plane determination Axioms and Theorems Conditions: Under which “one and only one” plane exists. 1. (Axiom) Three non-collinear points determine a plane. 2. (Theorem) A line and a point not on a line determine a plane. 3. (Theorem) Two intersecting lines determine a plane (Starts with point of intersection) 4. (Theorem) Two parallel lines determine a plane. Perpendicularity of a line and a plane Definition: If a line is perpendicular to a plane, then it is perpendicular to every line in the plane that passes through its foot. Theorem: If a line is perpendicular to two distinct lines in a plane that pass through its foot, then the line is perpendicular to the plane. Parallel Planes A. Definition 1. Two lines are parallel if they are coplanar and they do not intersect. 2. Skew lines are lines that are non-coplanar. 3. A line and a plane are parallel if they do not intersect. 4. Two lines are parallel if they do not intersect. B. Theorems 1. If a plane intersects two parallel planes, then the lines of intersection are parallel. 2. If two planes are perpendicular to the same line, then they are parallel to each other. 3. If two lines are perpendicular to the same plane, then they are parallel to each other. 4. If a line is perpendicular to one of two parallel planes, it is also perpendicular to the other. 5. If a plane is perpendicular to one of two parallel lines, it is also perpendicular to the other. 6. If two planes are parallel to the same plane, then they are parallel to each other. C. Notes 1. Planes are either intersecting or parallel. 2. Lines in parallel planes are either skew or parallel. 3. If two lines in parallel panes are coplanar, then they are parallel. Chapter 12: Surface Area There will be a reference sheet on the regents. Don’t panic. (: Chapter 14 ; Special: Constructions: You have to know how the seven basic constructions. Which are to: -Copy a line segment. -Copy an angle. -Bisect a line segment. -Bisect an angle. -Construct a perpendicular to a line through a point on a line. -Construct a perpendicular to a line through a point NOT on a line. -Construct parallel lines. I cant draw diagrams here. Sorry. But you get the point . Now we are done ^_^.