# Conjectures. Chapter 2. Chapter 3

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical angles, then they are congruent (have equal measures). (Lesson 2.5) C-3a C-3b C-3c Corresponding Angles Conjecture, or CA Conjecture If two parallel lines are cut by a transversal, then corresponding angles are congruent. (Lesson 2.6) Alternate Interior Angles Conjecture, or AIA Conjecture If two parallel lines are cut by a transversal, then alternate interior angles are congruent. (Lesson 2.6) Alternate Exterior Angles Conjecture, or AEA Conjecture If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. (Lesson 2.6) C-3 Parallel Lines Conjecture If two parallel lines are cut by a transversal, then corresponding angles are congruent, alternate interior angles are congruent, and alternate exterior angles are congruent. (Lesson 2.6) C-4 Converse of the Parallel Lines Conjecture If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior angles, or congruent alternate exterior angles, then the lines are parallel. (Lesson 2.6) Chapter 3 C-5 Perpendicular Bisector Conjecture If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints. (Lesson 3.2) C-6 Converse of the Perpendicular Bisector Conjecture If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Lesson 3.2) C-7 Shortest Distance Conjecture The shortest distance from a point to a line is measured along the perpendicular segment from the point to the line. (Lesson 3.3) C-8 Angle Bisector Conjecture If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. (Lesson 3.4) C-9 Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle are concurrent (meet at a point). (Lesson 3.7) C-10 Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of a triangle are concurrent. (Lesson 3.7) C-11 Altitude Concurrency Conjecture The three altitudes (or the lines containing the altitudes) of a triangle are concurrent. (Lesson 3.7) C-12 Circumcenter Conjecture The circumcenter of a triangle is equidistant from the vertices. (Lesson 3.7) C-13 Incenter Conjecture The incenter of a triangle is equidistant from the sides. (Lesson 3.7) C-14 Median Concurrency Conjecture The three medians of a triangle are concurrent. (Lesson 3.8) C-15 Centroid Conjecture The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side. (Lesson 3.8) 122 CONJECTURES Discovering Geometry Teaching and Worksheet Masters

2 C-16 Center of Gravity Conjecture The centroid of a triangle is the center of gravity of the triangular region. (Lesson 3.8) Chapter 4 C-17 Triangle Sum Conjecture The sum of the measures of the angles in every triangle is 180. (Lesson 4.1) C-18 Third Angle Conjecture If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle is equal in measure to the third angle in the other triangle. (Lesson 4.1) C-19 Isosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. (Lesson 4.2) C-20 Converse of the Isosceles Triangle Conjecture If a triangle has two congruent angles, then it is an isosceles triangle. (Lesson 4.2) C-21 Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (Lesson 4.3) C-22 Side-Angle Inequality Conjecture In a triangle, if one side is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. (Lesson 4.3) C-23 Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. (Lesson 4.3) C-24 SSS Congruence Conjecture If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. (Lesson 4.4) C-25 SAS Congruence Conjecture If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. (Lesson 4.4) C-26 ASA Congruence Conjecture If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (Lesson 4.5) C-27 SAA Congruence Conjecture If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. (Lesson 4.5) C-28 Vertex Angle Bisector Conjecture In an isosceles triangle, the bisector of the vertex angle is also the altitude and the median to the base. (Lesson 4.8) C-29 Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is equiangular. Conversely, every equiangular triangle is equilateral. (Lesson 4.8) Chapter 5 C-30 Quadrilateral Sum Conjecture The sum of the measures of the four angles of any quadrilateral is 360. (Lesson 5.1) C-31 Pentagon Sum Conjecture The sum of the measures of the five angles of any pentagon is 540. (Lesson 5.1) C-32 Polygon Sum Conjecture The sum of the measures of the n interior angles of an n-gon is 180 (n 2). (Lesson 5.1) Discovering Geometry Teaching and Worksheet Masters CONJECTURES 123

3 C-33 Exterior Angle Sum Conjecture For any polygon, the sum of the measures of a set of exterior angles is 360. (Lesson 5.2) C-34 Equiangular Polygon Conjecture You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas: n or 180 (n 2) n. (Lesson 5.2) C-35 Kite Angles Conjecture The nonvertex angles of a kite are congruent. (Lesson 5.3) C-36 Kite Diagonals Conjecture The diagonals of a kite are perpendicular. (Lesson 5.3) C-37 Kite Diagonal Bisector Conjecture The diagonal connecting the vertex angles of a kite is the perpendicular bisector of the other diagonal. (Lesson 5.3) C-38 Kite Angle Bisector Conjecture The vertex angles of a kite are bisected by a diagonal. (Lesson 5.3) C-39 Trapezoid Consecutive Angles Conjecture The consecutive angles between the bases of a trapezoid are supplementary. (Lesson 5.3) C-40 Isosceles Trapezoid Conjecture The base angles of an isosceles trapezoid are congruent. (Lesson 5.3) C-41 Isosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are congruent. (Lesson 5.3) C-42 Three Midsegments Conjecture The three midsegments of a triangle divide it into four congruent triangles. (Lesson 5.4) C-43 Triangle Midsegment Conjecture A midsegment of a triangle is parallel to the third side and half the length of the third side. (Lesson 5.4) C-44 Trapezoid Midsegment Conjecture The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the lengths of the bases. (Lesson 5.4) C-45 Parallelogram Opposite Angles Conjecture The opposite angles of a parallelogram are congruent. (Lesson 5.5) C-46 Parallelogram Consecutive Angles Conjecture The consecutive angles of a parallelogram are supplementary. (Lesson 5.5) C-47 Parallelogram Opposite Sides Conjecture The opposite sides of a parallelogram are congruent. (Lesson 5.5) C-48 Parallelogram Diagonals Conjecture The diagonals of a parallelogram bisect each other. (Lesson 5.5) C-49 Double-Edged Straightedge Conjecture If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus. (Lesson 5.6) C-50 Rhombus Diagonals Conjecture The diagonals of a rhombus are perpendicular, and they bisect each other. (Lesson 5.6) C-51 Rhombus Angles Conjecture The diagonals of a rhombus bisect the angles of the rhombus. (Lesson 5.6) C-52 Rectangle Diagonals Conjecture The diagonals of a rectangle are congruent and bisect each other. (Lesson 5.6) C-53 Square Diagonals Conjecture The diagonals of a square are congruent, perpendicular, and bisect each other. (Lesson 5.6) 124 CONJECTURES Discovering Geometry Teaching and Worksheet Masters

4 Chapter 6 C-54 Chord Central Angles Conjecture If two chords in a circle are congruent, then they determine two central angles that are congruent. (Lesson 6.1) C-55 Chord Arcs Conjecture If two chords in a circle are congruent, then their intercepted arcs are congruent. (Lesson 6.1) C-56 Perpendicular to a Chord Conjecture The perpendicular from the center of a circle to a chord is the bisector of the chord. (Lesson 6.1) C-57 Chord Distance to Center Conjecture Two congruent chords in a circle are equidistant from the center of the circle. (Lesson 6.1) C-58 Perpendicular Bisector of a Chord Conjecture The perpendicular bisector of a chord passes through the center of the circle. (Lesson 6.1) C-59 Tangent Conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. (Lesson 6.2) C-60 Tangent Segments Conjecture Tangent segments to a circle from a point outside the circle are congruent. (Lesson 6.2) C-61 Inscribed Angle Conjecture The measure of an angle inscribed in a circle is one-half the measure of the central angle. (Lesson 6.3) C-62 Inscribed Angles Intercepting Arcs Conjecture Inscribed angles that intercept the same arc are congruent. (Lesson 6.3) C-63 Angles Inscribed in a Semicircle Conjecture Angles inscribed in a semicircle are right angles. (Lesson 6.3) C-64 Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary. (Lesson 6.3) C-65 Parallel Lines Intercepted Arcs Conjecture Parallel lines intercept congruent arcs on a circle. (Lesson 6.3) C-66 Circumference Conjecture If C is the circumference and d is the diameter of a circle, then there is a number such that C d. Ifd 2r where r is the radius, then C 2 r. (Lesson 6.5) C-67 Arc Length Conjecture The length of an arc equals the circumference times the measure of the central angle divided by 360. (Lesson 6.7) Chapter 7 C-68 Reflection Line Conjecture The line of reflection is the perpendicular bisector of every segment joining a point in the original figure with its image. (Lesson 7.1) C-69 Coordinate Transformations Conjecture The ordered pair rule (x, y) ( x, y) is a reflection over the y-axis. The ordered pair rule (x, y) (x, y) is a reflection over the x-axis. The ordered pair rule (x, y) ( x, y) is a rotation about the origin. The ordered pair rule (x, y) (y, x) is a reflection over y x. (Lesson 7.2) Discovering Geometry Teaching and Worksheet Masters CONJECTURES 125

5 C-70 Minimal Path Conjecture If points A and B are on one side of line, then the minimal path from point A to line to point B is found by reflecting point B over line, drawing segment AB, then drawing segments AC and CB where point C is the point of intersection of segment AB and line. (Lesson 7.2) C-71 Reflections over Parallel Lines Conjecture A composition of two reflections over two parallel lines is equivalent to a single translation. In addition, the distance from any point to its second image under the two reflections is twice the distance between the parallel lines. (Lesson 7.3) C-72 Reflections over Intersecting Lines Conjecture A composition of two reflections over a pair of intersecting lines is equivalent to a single rotation. The angle of rotation is twice the acute angle between the pair of intersecting reflection lines. (Lesson 7.3) C-73 Tessellating Triangles Conjecture Any triangle will create a monohedral tessellation. (Lesson 7.5) C-74 Tessellating Quadrilaterals Conjecture Any quadrilateral will create a monohedral tessellation. (Lesson 7.5) Chapter 8 C-75 Rectangle Area Conjecture The area of a rectangle is given by the formula A bh, where A is the area, b is the length of the base, and h is the height of the rectangle. (Lesson 8.1) C-76 Parallelogram Area Conjecture The area of a parallelogram is given by the formula A bh, where A is the area, b is the length of the base, and h is the height of the parallelogram. (Lesson 8.1) C-77 Triangle Area Conjecture The area of a triangle is given by the formula A 1 bh, 2 where A is the area, b is the length of the base, and h is the height of the triangle. (Lesson 8.2) C-78 Trapezoid Area Conjecture The area of a trapezoid is given by the formula A 1 2 b 1 b 2 h, where A is the area, b 1 and b 2 are the lengths of the two bases, and h is the height of the trapezoid. (Lesson 8.2) C-79 Kite Area Conjecture The area of a kite is given by the formula A 1 2 d 1 d 2, where d 1 and d 2 are the lengths of the diagonals. (Lesson 8.2) C-80 Regular Polygon Area Conjecture The area of a regular polygon is given by the formula A 1 asn, 2 where A is the area, a is the apothem, s is the length of each side, and n is the number of sides. The length of each side times the number of sides is the perimeter P, so sn P. Thus you can also write the formula for area as A 1 ap. 2 (Lesson 8.4) C-81 Circle Area Conjecture The area of a circle is given by the formula A r 2, where A is the area and r is the radius of the circle. (Lesson 8.5) Chapter 9 C-82 The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then a 2 b 2 c 2. (Lesson 9.1) C-83 Converse of the Pythagorean Theorem If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle. (Lesson 9.2) 126 CONJECTURES Discovering Geometry Teaching and Worksheet Masters

6 C-84 Isosceles Right Triangle Conjecture In an isosceles right triangle, if the legs have length l, then the hypotenuse has length l 2. (Lesson 9.3) C Triangle Conjecture In a triangle, if the shorter leg has length a, then the longer leg has length a 3, and the hypotenuse has length 2a. (Lesson 9.3) C-86 Distance Formula The distance between points A x 1,y 1 and B x 2,y 2 is given by (AB) 2 x 2 x 1 2 y 2 y 1 2 or AB x x y 2. y 1 2 (Lesson 9.5) C-87 Equation of a Circle The equation of a circle with radius r and center (h, k) is (x h) 2 (y k) 2 r 2. (Lesson 9.5) Chapter 10 C-88a Conjecture A If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula for the volume is V BH. (Lesson 10.2) C-88b Conjecture B If B is the area of the base of a right prism (or cylinder) and H is the height of the solid, then the formula for the volume is V BH. (Lesson 10.2) C-88c Conjecture C The volume of an oblique prism (or cylinder) is the same as the volume of a right prism (or cylinder) that has the same base area and the same height. (Lesson 10.2) C-88 Prism-Cylinder Volume Conjecture The volume of a prism or a cylinder is the area of the base multiplied by the height. (Lesson 10.2) C-89 Pyramid-Cone Volume Conjecture If B is the area of the base of a pyramid or a cone and H is the height of the solid, then the formula for the volume is V = 1 BH. 3 (Lesson 10.3) C-90 Sphere Volume Conjecture The volume of a sphere with radius r is given by the formula V 4 3 r 3. (Lesson 10.6) C-91 Sphere Surface Area Conjecture The surface area, S, of a sphere with radius r is given by the formula S 4 r 2. (Lesson 10.7) Chapter 11 C-92 Dilation Similarity Conjecture If one polygon is the image of another polygon under a dilation, then the polygons are similar. (Lesson 11.1) C-93 AA Similarity Conjecture If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (Lesson 11.2) C-94 SSS Similarity Conjecture If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. (Lesson 11.2) C-95 SAS Similarity Conjecture If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. (Lesson 11.2) C-96 Proportional Parts Conjecture If two triangles are similar, then the corresponding altitudes, medians, and angle bisectors are proportional to the corresponding sides. (Lesson 11.4) C-97 Angle Bisector/Opposite Side Conjecture A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the two sides forming the angle. (Lesson 11.4) Discovering Geometry Teaching and Worksheet Masters CONJECTURES 127

7 C-98 Proportional Areas Conjecture If corresponding sides of two similar polygons or the radii of two circles compare in the ratio m n, then their areas compare in the ratio m 2 n 2 or m n 2. (Lesson 11.5) C-99 Proportional Volumes Conjecture If corresponding edges (or radii, or heights) of two similar solids compare in the ratio m n, then their volumes compare in the ratio m 3 n or 3 m n 3. (Lesson 11.5) C-100 Parallel/Proportionality Conjecture If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to the third side. (Lesson 11.6) C-101 Extended Parallel/Proportionality Conjecture If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally. (Lesson 11.6) Chapter 12 C-102 SAS Triangle Area Conjecture The area of a triangle is given by the formula A 1 ab 2 sin C, where a and b are the lengths of two sides and C is the angle between them. (Lesson 12.3) C-103 Law of Sines For a triangle with angles A, B, and C and sides of lengths a, b, and c (a opposite A, b opposite B, and c opposite C), sin A a sin B b sin C c. (Lesson 12.3) C-104 Pythagorean Identity For any angle A, (sin A) 2 (cos A) 2 1. (Lesson 12.4) C-105 Law of Cosines For any triangle with sides of lengths a, b, and c, and with C the angle opposite the side with length c, c 2 a 2 b 2 2ab cos C. (Lesson 12.4) 128 CONJECTURES Discovering Geometry Teaching and Worksheet Masters

8 Postulates and Theorems Line Postulate You can construct exactly one line through any two points. Line Intersection Postulate The intersection of two distinct lines is exactly one point. Segment Duplication Postulate You can construct a segment congruent to another segment. Angle Duplication Postulate You can construct an angle congruent to another angle. Midpoint Postulate You can construct exactly one midpoint on any line segment. Angle Bisector Postulate You can construct exactly one angle bisector in any angle. Parallel Postulate Through a point not on a given line, you can construct exactly one line parallel to the given line. Perpendicular Postulate Through a point not on a given line, you can construct exactly one line perpendicular to the given line. Segment Addition Postulate If point B is on AC and between points A and C, then AB BC AC. Angle Addition Postulate If point D lies in the interior of ABC, then m ABD m DBC m ABC. Linear Pair Postulate If two angles are a linear pair, then they are supplementary. Corresponding Angles Postulate, or CA Postulate If two parallel lines are cut by a transversal, then the corresponding angles are congruent. Conversely, if two lines are cut by a transversal forming congruent corresponding angles, then the lines are parallel. SSS Congruence Postulate If the three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. SAS Congruence Postulate If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent. ASA Congruence Postulate If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. Vertical Angles Theorem, or VA Theorem If two angles are vertical angles, then they are congruent (have equal measures). Alternate Interior Angles Theorem, or AIA Theorem If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Triangle Sum Theorem The sum of the measures of the angles of a triangle is 180. Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angle in each triangle is congruent to the third angle in the other triangle. Congruent and Supplementary Theorem If two angles are both congruent and supplementary, then each is a right angle. Supplements of Congruent Angles Theorem Supplements of congruent angles are congruent. Right Angles Are Congruent Theorem All right angles are congruent. Converse of the AIA Theorem If two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel. Alternate Exterior Angles Theorem, or AEA Theorem If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Discovering Geometry Teaching and Worksheet Masters POSTULATES AND THEOREMS 129

9 Converse of the AEA Theorem If two lines are cut by a transversal forming congruent alternate exterior angles, then the lines are parallel. Interior Supplements Theorem If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Converse of the Interior Supplements Theorem If two lines are cut by a transversal forming interior angles on the same side of the transversal that are supplementary, then the lines are parallel. Parallel Transitivity Theorem If two lines in the same plane are parallel to a third line, then they are parallel to each other. Perpendicular to Parallel Theorem If two lines in the same plane are perpendicular to a third line, then they are parallel to each other. SAA Congruence Theorem If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. Angle Bisector Theorem Any point on the bisector of an angle is equidistant from the sides of the angle. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equally distant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem If a point is equally distant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Isosceles Triangle Theorem If a triangle is isosceles, then its base angles are congruent. Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the triangle is isosceles. Converse of the Angle Bisector Theorem If a point is equally distant from the sides of an angle, then it is on the bisector of the angle. Perpendicular Bisector Concurrency Theorem The three perpendicular bisectors of a triangle are concurrent. Angle Bisector Concurrency Theorem The three angle bisectors of a triangle are concurrent. Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Quadrilateral Sum Theorem The sum of the measures of the four angles of any quadrilateral is 360. Medians to the Congruent Sides Theorem In an isosceles triangle, the medians to the congruent sides are congruent. Angle Bisectors to the Congruent Sides Theorem In an isosceles triangle, the angle bisectors to the congruent sides are congruent. Altitudes to the Congruent Sides Theorem In an isosceles triangle, the altitudes to the congruent sides are congruent. Isosceles Triangle Vertex Angle Theorem In an isosceles triangle, the altitude to the base, the median to the base, and the bisector of the vertex angle are all the same segment. Parallelogram Diagonal Lemma A diagonal of a parallelogram divides the parallelogram into two congruent triangles. 130 POSTULATES AND THEOREMS Discovering Geometry Teaching and Worksheet Masters

11 Intersecting Chords Theorem The measure of an angle formed by two intersecting chords is half the sum of the measures of the two intercepted arcs. Intersecting Secants Theorem The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measure of the larger intercepted arc and the measure of the smaller intercepted arc. AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. SAS Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar. SSS Similarity Theorem If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. Corresponding Altitudes Theorem If two triangles are similar, then corresponding altitudes are proportional to the corresponding sides. Corresponding Medians Theorem If two triangles are similar, then corresponding medians are proportional to the corresponding sides. Corresponding Angle Bisectors Theorem If two triangles are similar, then corresponding angle bisectors are proportional to the corresponding sides. Parallel/Proportionality Theorem If a line passes through two sides of a triangle parallel to the third side, then it divides the two sides proportionally. Converse of the Parallel/Proportionality Theorem If a line passes through two sides of a triangle dividing them proportionally, then it is parallel to the third side. Three Similar Right Triangles Theorem If you drop an altitude from the vertex of the right angle to the hypotenuse of a right triangle, then it divides the right triangle into two right triangles that are similar to each other and to the original right triangle. Altitude to the Hypotenuse Theorem The length of the altitude to the hypotenuse of a right triangle is the geometric mean between the length of the two segments on the hypotenuse. The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then a 2 b 2 c 2. Converse of the Pythagorean Theorem If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle. Hypotenuse Leg Theorem If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent. Square Diagonals Theorem The diagonals of a square are congruent, perpendicular, and bisect each other. Converse of the Parallelogram Diagonals Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Converse of the Kite Diagonal Bisector Theorem If only one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. Polygon Sum Theorem The sum of the measures of the n interior angles of an n-gon is 180 (n 2). 132 POSTULATES AND THEOREMS Discovering Geometry Teaching and Worksheet Masters

12 Hero s Formula If s is the half perimeter of a triangle with sides lengths a, b, and c, then the area A is given by the formula A s(s )(s a )(s b ). c Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and half the length of the third side. Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the bases. Rectangle Midpoints Theorem If the midpoints of the four sides of a rectangle are connected to form a quadrilateral, then the quadrilateral formed is a rhombus. Rhombus Midpoints Theorem If the midpoints of the four sides of a rhombus are connected to form a quadrilateral, then the quadrilateral formed is a rectangle. Kite Midpoints Theorem If the midpoints of the four sides of a kite are connected to form a quadrilateral, then the quadrilateral formed is a rectangle. Power of Interior Point Theorem If two chords intersect in a circle, then the product of the segment lengths on one chord is equal to the product of the segment lengths on the other chord. Discovering Geometry Teaching and Worksheet Masters POSTULATES AND THEOREMS 133

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

### Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

### HIGH SCHOOL: GEOMETRY (Page 1 of 4)

HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course

### SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### Surface Area and Volume Cylinders, Cones, and Spheres

Surface Area and Volume Cylinders, Cones, and Spheres Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable

### CHAPTER 1. LINES AND PLANES IN SPACE

CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given

### The Geometry of Piles of Salt Thinking Deeply About Simple Things

The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word

### Geometry - Semester 2. Mrs. Day-Blattner 1/20/2016

Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 3) Spot check sheet Thales Theorem - add to your response 4) Finding

### GEOMETRY OF THE CIRCLE

HTR GMTRY F TH IRL arly geometers in many parts of the world knew that, for all circles, the ratio of the circumference of a circle to its diameter was a constant. Today, we write d 5p, but early geometers

### KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007

KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### Tennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes

Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical

### Study Guide Workbook. Contents Include: 96 worksheets one for each lesson

Workbook Contents Include: 96 worksheets one for each lesson To the Teacher: Answers to each worksheet are found in Glencoe s Geometry: Concepts and Applications Chapter Resource Masters and also in the

### Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade)

Welcome to Math 7 Accelerated Courses (Preparation for Algebra in 8 th grade) Teacher: School Phone: Email: Kim Schnakenberg 402-443- 3101 kschnakenberg@esu2.org Course Descriptions: Both Concept and Application

### Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014

Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014 i May 2014 777777 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 New York

### Line Segments, Rays, and Lines

HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw

### Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### GeoGebra. 10 lessons. Gerrit Stols

GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

### 10 th Grade Math Special Education Course of Study

10 th Grade Math Special Education Course of Study Findlay City Schools 2006 Table of Contents 1. Findlay City Schools Mission Statement 2. 10 th Grade Math Curriculum Map 3. 10 th Grade Math Indicators

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY Student Name: The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 - /YJ {, ~ h (} l J/VJJJ,O 8:30 to 11:30 a.m., only The possession or use

### Functional Math II. Information CourseTitle. Types of Instruction

Functional Math II Course Outcome Summary Riverdale School District Information CourseTitle Functional Math II Credits 0 Contact Hours 135 Instructional Area Middle School Instructional Level 8th Grade

### 1.1 Identify Points, Lines, and Planes

1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean.

### Elements of Plane Geometry by LK

Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry

### GEOMETRIC FIGURES, AREAS, AND VOLUMES

HPTER GEOMETRI FIGURES, RES, N VOLUMES carpenter is building a deck on the back of a house. s he works, he follows a plan that he made in the form of a drawing or blueprint. His blueprint is a model of

### MATH 100 PRACTICE FINAL EXAM

MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number

### GRADES 7, 8, AND 9 BIG IDEAS

Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for

### EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

### Appendix A Designing High School Mathematics Courses Based on the Common Core Standards

Overview: The Common Core State Standards (CCSS) for Mathematics are organized by grade level in Grades K 8. At the high school level, the standards are organized by strand, showing a logical progression

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

### Objectives. Cabri Jr. Tools

^Åíáîáíó=NO Objectives To learn how to construct all types of triangles using the Cabri Jr. application To reinforce the difference between a construction and a drawing Cabri Jr. Tools fåíêççìåíáçå `çåëíêìåíáåö

### GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Oklahoma School Testing Program

Oklahoma School Testing Program Oklahoma Core Curriculum Tests (OCCT) End-of-Instruction ACE Geometry Parent, Student, and Teacher Guide Winter/Trimester 013 01 Oklahoma State Department of Education 70561-W

### 2009 Chicago Area All-Star Math Team Tryouts Solutions

1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total

### Unit 5 Area. What Is Area?

Trainer/Instructor Notes: Area What Is Area? Unit 5 Area What Is Area? Overview: Objective: Participants determine the area of a rectangle by counting the number of square units needed to cover the region.

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2

DOE-HDBK-1014/2-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 U.S. Department of Energy Washington, D.C. 20585 FSC-6910 Distribution Statement A. Approved for public release; distribution

### Secondary Mathematics Syllabuses

Secondary Mathematics Syllabuses Copyright 006 Curriculum Planning and Development Division. This publication is not for sale. All rights reserved. No part of this publication may be reproduced without

Introduction: Summary of Goals GRADE SEVEN By the end of grade seven, students are adept at manipulating numbers and equations and understand the general principles at work. Students understand and use

### Sample Test Questions

mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about

### Mathematics Content Expectations. Content Expectations Found in Lansing Community College Math Courses High School Diploma Completion Initiative

Mathematics Content Expectations Content Expectations Found in Lansing Community College Math Courses High School Diploma Completion Initiative Based on Review of Course Syllabi and Textbooks Reviewed

### Tutorial 1: The Freehand Tools

UNC Charlotte Tutorial 1: The Freehand Tools In this tutorial you ll learn how to draw and construct geometric figures using Sketchpad s freehand construction tools. You ll also learn how to undo your

### Triangle Centers MOP 2007, Black Group

Triangle Centers MOP 2007, Black Group Zachary Abel June 21, 2007 1 A Few Useful Centers 1.1 Symmedian / Lemmoine Point The Symmedian point K is defined as the isogonal conjugate of the centroid G. Problem

### MATH CHEAT SHEET. Basic Math and Pre-Algebra Cheat Sheet

1 MATH CHEAT SHEET Basic Math and Pre-Algebra Cheat Sheet Groups of Numbers Natural or counting numbers: 1, 2, 3, 4, Whole numbers: 0, 1, 2, 3, Integers: -3, -2, -1, 0, 1, 2, 3, Rational numbers: Integers

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Solutions Manual for How to Read and Do Proofs

Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### SJO PW - Język angielski ogólnotechniczny, Poziom B2 Opracowanie: I. Zamecznik, M. Witczak, H. Maniecka, A. Hilgier,

GEOMETRY AND MEASUREMENT - Teacher s notes and key to tasks Introduction (5 minutes one week before the actual LESSON): 1. Elicit vocabulary connected with geometric figures ask the students (SS) to look

### Optical Illusions Essay Angela Wall EMAT 6690

Optical Illusions Essay Angela Wall EMAT 6690! Optical illusions are images that are visually perceived differently than how they actually appear in reality. These images can be very entertaining, but

### Essential Mathematics for Computer Graphics fast

John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made

### Chapter 5 Resource Masters

Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice,

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### COORDINATE GEOMETRY Mathematics 1 MM1G1a,b,c,d,e

Student Learning Map Unit 6 COORDINATE GEOMETRY Mathematics 1 MM1G1a,b,c,d,e Key Learning(s): Unit Essential Question(s): 1. Algebraic formulas can be used to find measures of distance on the coordinate

### About Tutorials... 1 Algebra I... 2 Geometry... 3 Algebra II... 4 English I... 5 English II... 6 English III... 7

Outlines Texas Tutorials Apex Learning Texas Tutorials provide teachers with a solution to support all students in rising to the expectations established by the Texas state standards. With content developed

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Guidelines for Mathematics Laboratory in Schools Class X

Guidelines for Mathematics Laboratory in Schools Class X Central Board of Secondary Education Preet Vihar, Delhi 110092. 1 2 1. Introduction Taking into consideration the national aspirations and expectations

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Guidelines for Mathematics Laboratory in Schools Class IX

Guidelines for Mathematics Laboratory in Schools Class IX Central Board of Secondary Education Preet Vihar, Delhi 110092. 1 1. Introduction 1.1 Rationale Mathematics is erroneously regarded as a difficult

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### Exact Values of the Sine and Cosine Functions in Increments of 3 degrees

Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms

### 8th Grade Texas Mathematics: Unpacked Content

8th Grade Texas Mathematics: Unpacked Content What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards mean a student must

Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of

### Support Materials for Core Content for Assessment. Mathematics

Support Materials for Core Content for Assessment Version 4.1 Mathematics August 2007 Kentucky Department of Education Introduction to Depth of Knowledge (DOK) - Based on Norman Webb s Model (Karin Hess,

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and

### MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model

MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### How to Enroll in an American School Online Course

The American School now offers these online courses: Economics Geography Earth Science English IV Pre-Algebra Geometry Calculus* Each of these courses is worth 1 unit of credit except Economics, which

### Mathematics Notes for Class 12 chapter 10. Vector Algebra

1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

### Resource Guide to the Arkansas Curriculum Framework for Students with Disabilities for Ninth Grade Mathematics. Summer 2005

Resource Guide to the Arkansas Curriculum Framework for Students with Disabilities for Ninth Grade Mathematics Summer 2005 Purpose and Process The Individuals with Disabilities Education Act and No Child

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### AMC 10 Solutions Pamphlet TUESDAY, FEBRUARY 13, 2001 Sponsored by Mathematical Association of America University of Nebraska

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO AMERICAN MATHEMATICS COMPETITIONS nd Annual Mathematics Contest 10 AMC 10 Solutions Pamphlet TUESDAY, FEBRUARY 1, 001

### Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

### 04 Mathematics CO-SG-FLD004-03. Program for Licensing Assessments for Colorado Educators

04 Mathematics CO-SG-FLD004-03 Program for Licensing Assessments for Colorado Educators Readers should be advised that this study guide, including many of the excerpts used herein, is protected by federal

### http://www.xtremepapers.net

http://www.xtremepapers.net An ARCO Book ARCO is a registered trademark of Thomson Learning, Inc., and is used herein under license by Peterson s. About The Thomson Corporation and Peterson s With revenues

### Graphing and Solving Nonlinear Inequalities

APPENDIX LESSON 1 Graphing and Solving Nonlinear Inequalities New Concepts A quadratic inequality in two variables can be written in four different forms y < a + b + c y a + b + c y > a + b + c y a + b

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### Chapter 3. Inversion and Applications to Ptolemy and Euler

Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent

### Further Steps: Geometry Beyond High School. Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu

Further Steps: Geometry Beyond High School Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu Geometry the study of shapes, their properties, and the spaces containing

### Name Summer Assignment for College Credit Math Courses 2015-2016

Name Summer Assignment for College Credit Math Courses 015-016 To: All students enrolled in Pre-AP PreCalculus and Dual Credit PreCalculus at El Campo High School College math classes utilizes skills and

### College Prep. Geometry Course Syllabus

College Prep. Geometry Course Syllabus Mr. Chris Noll Turner Ashby High School - Room 211 Email: cnoll@rockingham.k12.va.us Website: http://blogs.rockingham.k12.va.us/cnoll/ School Phone: 828-2008 Text:

### GeoGebra Workshop for the Initial Teacher Training in Primary Education

GeoGebra Workshop for the Initial Teacher Training in Primary Education N. Ruiz natalia.ruiz@uam.es Facultad de Formación de Profesorado y Educación (Faculty of Education) Universidad Autónoma de Madrid

### Finding diagonal distances on the coordinate plane. Linear Inequalities. Permutation & Combinations. Standard Deviation

Introduction of Surface Area Measurement Unit Conversion Volume of Right Rectangular Prisms Statistical Variability (MAD and IQR) Multiplication of Fractions Scientific Notation Introduction of Transformations

### David Bressoud Macalester College, St. Paul, MN. NCTM Annual Mee,ng Washington, DC April 23, 2009

David Bressoud Macalester College, St. Paul, MN These slides are available at www.macalester.edu/~bressoud/talks NCTM Annual Mee,ng Washington, DC April 23, 2009 The task of the educator is to make the

### http://school-maths.com Gerrit Stols

For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

### Gymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz

SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October