Geometry

Congruence

Definitions of angles, circles, perpendicular and parallel lines, and line segments, as well as transformations like rotations, reflections, and translations. Students learn to draw transformed figures and specify sequences of transformations. Key concepts include congruence in rigid motions, congruent triangles, and criteria for triangle congruence (ASA, SAS, SSS). Theorems about lines, angles, triangles, and parallelograms are proven, including properties of vertical angles, alternate interior angles, and corresponding angles. Geometric constructions cover copying segments and angles, bisecting, constructing perpendicular and parallel lines, and inscribing shapes in circles.

 

Similarity, Right Triangles, and Trigonometry

Students explore dilation and similarity transformations, establishing the Angle-Angle (AA) criterion for triangle similarity. They prove theorems about triangles, such as how a line parallel to one side of a triangle divides the other two sides proportionally and proving the Pythagorean Theorem using triangle similarity. The concepts of congruence and similarity criteria are applied to solve problems and prove relationships in geometric figures. Trigonometric ratios for acute angles are defined and used to solve right triangle problems, including complementary angles and special right triangles (30°-60°-90° and 45°-45°-90°). Trigonometry is applied to general triangles through the Laws of Sines and Cosines.

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Circles

Students learn to understand and apply theorems about circles, identifying relationships among inscribed angles, radii, and chords. This includes understanding the relationships between central, inscribed, and circumscribed angles, recognizing that inscribed angles on a diameter are right angles, and knowing that the radius of a circle is perpendicular to the tangent at the point of intersection. They construct inscribed and circumscribed circles of a triangle and prove angle properties for a quadrilateral inscribed in a circle. Additionally, they construct a tangent line from a point outside a circle, find arc lengths and areas of sectors, and convert between degrees and radians.

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Expressing Geometric Properties with Equations

Students derive the equation of a circle using the Pythagorean Theorem and complete the square to find a circle's center and radius from its equation. They also derive the equation of a parabola given its focus and directrix. Using coordinates, they prove simple geometric theorems algebraically, including the slope criteria for parallel and perpendicular lines. They solve geometric problems, find points that partition line segments in a given ratio, and use coordinates to compute perimeters and areas of polygons, triangles, and rectangles using the distance formula.

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Geometric Measurement and Dimension

Students provide informal arguments for formulas related to the circumference and area of a circle, and the volumes of cylinders, pyramids, and cones, using dissection arguments, Cavalieri’s principle, and informal limit arguments. They apply volume formulas to solve problems involving cylinders, pyramids, cones, and spheres. Students visualize relationships between two-dimensional and three-dimensional objects and verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side. These relationships are applied to solve real-world and mathematical problems.