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Geometry — 8.G ELG.MA.8.G.1 Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Circles – G-C ELG.MA.HS.G.9: Understand and apply theorems about circles. G-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. Geometry: Circles — G-C ELG.MA.HS.G.9: Understand and apply theorems about circles. G-C.A.1 Prove that all circles are similar. G-C.A.2 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. G-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Students will demonstrate command of the ELG by: Constructing a tangent line from a point outside a given circle to the circle. Vocabulary: circle construct tangent line Sample Assessment Questions: Standard(s): G-C.A.4 Source: https://www.illustrativemathematics.org/content-standards/HSG/C/A/4/tasks/1096 Item Prompt: Suppose C is a circle with center O and P is a point outside of C. Let M be the midpoint of ̅̅̅̅ 𝑂𝑃 and let D be the circle with center M passing through O. Let A and B be the two points of intersection of C and D, pictured below along with several line segments of interest: a. Show that angles OAP and OBP are right angles. b. Show that ⃡𝑃𝐴 and ⃡𝑃𝐵 are tangent lines from P to the circle C. Correct Answer(s): Below is a picture of the different points and triangles used in the solution of the problem: a. Segment ̅̅̅̅ 𝑂𝑃 is a diameter of circle D since the center of D is the midpoint M of this segment. The points A and B are also both on D since they are the points of intersection of C and D. The angles OAP and OBP are both right angles because segment ̅̅̅̅ 𝑂𝑃 is a diameter of circle D, and A and B are points on D: this means that triangles OAP and OBP are inscribed in circle D and so the angle opposite the diameter must be a right angle. b. Since angles OAP and OBP are right angles it follows that ⃡𝑃𝐴 meets the radius ̅̅̅̅ 𝑂𝐴 in a right angle and similarly ⃡𝑃𝐵 meets radius ̅̅̅̅ 𝑂𝐵 in a right angle. This means that ⃡ is tangent to C at A and ⃡𝑃𝐵 is tangent to C at B. 𝑃𝐴 Below is a picture with the two tangent lines constructed above: