# Geometry

Taking the basic distance and angle-preserving properties of rigid motions and similarity transformations as axiomatic, students establish triangle congruence and similarity criteria, then use them to prove a wide variety of theorems and solve problems involving, for example, triangles, other polygons, and circles.

Students study geometric measurement and solve problems involving length, area and volume, learning more sophisticated arguments for the circumference, area, and volume formulas that they learned in earlier grades.
They use similarity of right triangles with given angle measures to define sine, cosine, and tangent in terms of side ratios. They prove theorems and solve problems about circles, segments, angles, and arcs.

Throughout the course, students use coordinates to connect geometry with algebra, and engage in mathematical modeling using geometric principles.

## Units

#### Summary

• Know and be able to use precise definitions of geometric terms. • Make formal geometric constructions by hand and using geometry software. • Given a geometric figure and a rotation, reflection, or translation draw the transformed figure. • Develop definitions of rotation, reflection, and translation. • Represent transformations in the plane; describe transformations as functions whose inputs and outputs are points in the plane. • Describe the rotations and reflections that carry a given quadrilateral or regular polygon onto itself. • Prove that the measures of the interior angles of a triangle have sum 180°.

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#### Summary

• Specify sequences of rigid motions that will carry a figure onto another. • Understand that there can be more than one sequence of rigid motions that carries a figure onto another figure. • Use the definition of congruence in terms of rigid motions to decide if two figures are congruent. • 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 (CPCTC). • Be able to explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. • Prove theorems about lines and angles. • Prove theorems about parallelograms. • Prove base angles of isosceles triangles are congruent.

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#### Summary

• Verify experimentally properties of dilations, and use center and scale factor to describe them. • Use the definition of similarity to decide if two figures are similar. • Use the properties of similarity to establish AA criterion for two triangles to be similar. • Prove and use some theorems about triangles. • Prove and use slope criteria for parallel and perpendicular lines. • Construct points that partition a segment in a given ratio. • Explore why all circles are similar.

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#### Summary

• Using similarity, show that side ratios in right triangles are properties of the angles. • Define the trigonometric ratios for acute angles. • Explain and use the relationship between sine and cosine of complementary angles. • Use trigonometric ratios to solve a variety of modeling problems.

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#### Summary

• Use geometric shapes to describe objects and use measures of the shapes. • Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. • Give arguments that combine dissection and informal limits to yield the circumference and area formulas for a circle. • Give dissection arguments that yield the volume formula for prisms. • Use the volume formula for prisms and an informal limit argument to obtain the volume formula for cylinders. • Identify the shapes of two-dimensional cross-sections of three-dimensional objects. • Identify three-dimensional objects generated from rotations of two-dimensional shapes. • Obtain the formula for volume of a pyramid with square base via dissection. • Use Cavalieri’s Principle to obtain the formula for the volume of a pyramid from the formula for the volume of a pyramid with square base. • Use volume formulas to solve problems. • Solve volume problems involving the calculation of density.

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#### Summary

• Use the Pythagorean Theorem to derive an equation for a circle of given center and radius. • Use similarity to derive the fact that the length of the arc of a circle intercepted by an angle is proportional to the radius of the circle. • Derive a formula for the area of a sector. • Identify and describe relationships between central and inscribed angles and their arcs. • Prove that an inscribed angle that subtends a diameter is a right angle, and its converse. • Identify and describe relationships and ratios of lengths for intersecting chords. • Prove that a radius and a tangent to a circle at the same point are perpendicular. • Prove properties of angles of inscribed polygons. • Use relationships about inscribed angles to solve problems about inscribed polygons. • Use circles, cones, tangent segments, chords, and related figures, and their properties to describe objects.

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