


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

1 ) Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [GCO1]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

2 ) Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

3 ) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [GCO3]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

4 ) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [GCO4]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

5 ) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [GCO5]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

6 ) Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [GCO6]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

7 ) Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [GCO7]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
1 
Lesson Plans: 
1 

8 ) Explain how the criteria for triangle congruence, anglesideangle (ASA), sideangleside (SAS), and sidesideside (SSS), follow from the definition of congruence in terms of rigid motions. [GCO8]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

9 ) Prove theorems about lines and angles. Theorems include vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [GCO9]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [GCO10]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

11 ) Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [GCO11]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
2 
Learning Activities: 
2 

12 ) Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. [GCO12]

Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

13 ) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [GCO13]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

14 ) Verify experimentally the properties of dilations given by a center and a scale factor. [GSRT1]
a. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged. [GSRT1a]
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
[GSRT1b]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

15 ) Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [GSRT2]
Example 1:
Given the two triangles above, show that they are similar.
^{4}/_{8} = ^{6}/_{12}
They are similar by SSS. The scale factor is equivalent.
Example 2:
Show that the two triangles are similar.
Two corresponding sides are proportional and the included angle is congruent. (SAS similarity)


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

16 ) Use the properties of similarity transformations to establish the angleangle (AA) criterion for two triangles to be similar. [GSRT3]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

17 ) Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [GSRT4]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

18 ) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [GSRT5]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

19 ) Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles. [GSRT6]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
1 
Lesson Plans: 
1 

20 ) Explain and use the relationship between the sine and cosine of complementary angles. [GSRT7]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

21 ) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [GSRT8]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

22 ) (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [GSRT10]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

23 ) (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).
[GSRT11]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

24 ) Prove that all circles are similar. [GC1]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

25 ) Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. [GC2]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

26 ) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [GC3]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

27 ) (+) Construct a tangent line from a point outside a given circle to the circle. [GC4]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

28 ) Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [GC5]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

29 ) Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [GGPE1]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

30 ) Use coordinates to prove simple geometric theorems algebraically. [GGPE4]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
2 
Lesson Plans: 
2 

31 ) Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). [GGPE5]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

32 ) Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [GGPE6]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

33 ) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [GGPE7]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

34 ) Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama)



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. [GGMD1]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

36 ) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [GGMD3]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

37 ) Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama)



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

38 ) Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects. [GGMD4]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

39 ) Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [GMG1]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

40 ) Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [GMG2]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

41 ) Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).* [GMG3]



Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

42 ) (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6]


Mathematics (2015) 
Grade(s): 9  12 
Geometry 
All Resources: 
0 

43 ) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [SMD7] (Alabama)
Example:
What is the probability of tossing a penny and having it land in the nonshaded region'
Geometric Probability is the NonShaded Area divided by the Total Area.
