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. [G-CO10]

[MA2015] GEO (9-12) 16 :

16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

[MA2015] GEO (9-12) 17 :

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. [G-SRT4]

[MA2019] GEO-19 (9-12) 32 :

32. Use coordinates to prove simple geometric theorems algebraically.

[MA2019] GEO-19 (9-12) 34 :

34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

Subject: Mathematics (9 - 12), Mathematics (9 - 12) Title: Can You Solve This Pier Puzzle? | Physics Girl URL: https://aptv.pbslearningmedia.org/resource/pier-puzzle-physics-girl/pier-puzzle-physics-girl/ Description:

This math brainteaser challenges you to find a simple, elegant solution to a seemingly complex problem! Students will use geometry principles and their knowledge about triangles to solve this puzzle. Can you figure it out?

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines and the need to prove results about these quantities (G-GPE.B.5). MP.3 is highlighted in this topic as students engage in proving the criterion for perpendicularityand then extending that knowledge to reason about lines and segments. This work highlights the role of the converse of the Pythagorean theorem in the identification of perpendicular directions of motion (G-GPE.B.4). In Lesson 5, students explain the connection between the Pythagorean theorem and the criterion for perpendicularity (G-GPE.B.4). Lesson 6 extends that study by generalizing the criterion for perpendicularity to any two segments and applying this criterion to determine if segments are perpendicular.

In Lesson 7, students will recognize when a line and a normal segment intersect at the origin. Lesson 8 concludes Topic B when students recognize parallel and perpendicular lines from their slopes and create equations for parallel and perpendicular lines. The criterion for parallel and perpendicular lines and the work from this topic with the distance formula is extended in the last two topics of this module as students use these foundations to determine the perimeter and area of polygonal regions in the coordinate plane defined by systems of inequalities. Additionally, students study the proportionality of segments formed by diagonals of polygons.

Topic D concludes the work of Module 4. In Lesson 12, students find midpoints of segments and points that divide segments into more equal and proportional parts. Students also find locations on a directed line segment between two given points that partition the segment in given ratios (G-GPE.B.6). Lesson 13 requires students to connect this work to proving classical results in geometry (G-GPE.B.4). For instance, the diagonals of a parallelogram bisect one another, and the medians of a triangle meet at the point of the way from the vertex for each. Lesson 14 is an optional lesson that allows students to explore parametric equations and compare them with more familiar linear equations (G-GPE.B.6, G-MG.A.1). Parametric equations make both the variables in an equation dependent on a third variable, usually time. In this lesson, parametric equations model the robot’s horizontal and vertical motion over a period of time. Introducing parametric equations in the Geometry course prepares students for higher-level courses and also represents an opportunity to show coherence between functions, algebra, and coordinate geometry. Students extend their knowledge of parallel and perpendicular lines to lines given parametrically. Students complete the work of this module in Lesson 15 by deriving and applying the distance formula (G-GPE.B.4) and with the challenge of locating the point along a line closest to a given point, again given as a robot challenge.

6. Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

a. Given the endpoints of the diameter of a circle, use the midpoint formula to find its center and then use the Pythagorean Theorem to find its equation.

b. Derive the distance formula from the Pythagorean Theorem.

[MA2019] GEO-19 (9-12) 32 :

32. Use coordinates to prove simple geometric theorems algebraically.

Module 5, Topic D brings in coordinate geometry to establish the equation of a circle. Students solve problems to find the equations of specific tangent lines or the coordinates of specific points of contact. They also express circles via analytic equations.