Lines, Angles, and Triangles *CISD Safety Net Standards: G.6D Title Suggested Time Frame 1 st and 2 nd Six Weeks Suggested Duration: 30 Days Geometry Big Ideas/Enduring Understandings Module 4 Parallel lines and Perpendicular Lines can be used to solve realworld problems. Module 5 Triangle congruence can be used to solve real-world problems. Module 6 Applications of Triangle Congruence can be used to solve real-world problems. Module 7 Guiding Questions Module 4 How can you find the measure of angles formed by intersecting lines? How can you prove and use theorems about angles formed by transversals that intersect parallel lines? How can you prove that two lines are parallel? What are the key ideas about perpendicular bisectors of a segment? How can you find the equation of a line that is parallel or perpendicular to a given line? How can you use parallel and perpendicular lines to solve real-world problems? Module 5 How can you show that two triangles are congruent? What does the ASA Triangle Congruence Theorem tell you about triangles? What does the SAS Triangle Congruence Theorem tell you about triangles? What does the SSS Triangle Congruence Theorem tell you about triangles? How can you use triangle congruence criteria to solve real-world problems? Module 6 How can you be sure that the result of a construction is valid? Updated April 7, 2016 Page 1 of 19
Properties of Triangles can be used to solve real-world problems. Module 8 Special segments in triangles can be used to solve realworld problems. What does the AAS Triangle Congruence Theorem tell you about two triangles? What does the HL Triangle Congruence Theorem tell you about two triangles? How can you use triangle congruence to solve real-world problems? Module 7 What can you say about the interior and exterior angles of a triangle and other polygons? What are the special relationships among angles and sides in isosceles and equilateral triangles? How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? How can you use the properties of triangles to solve real-world problems? Module 8 How can you use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle? How can you use angles bisectors to find the point that is equidistant from all the sides of a triangle? How can you find the balance point or center of gravity of a triangle? How are the segments that join the midpoints of a triangle s sides related to the triangle s sides? How can you use special segments in triangles to solve realworld problems? Vertical Alignment Expectations TEA Vertical Alignment Chart Grades 5-8, Geometry Updated April 7, 2016 Page 2 of 19
Sample Assessment Question Coming Soon... The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. District Specificity/Examples TEKS clarifying examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents. Ongoing TEKS Math Processing Skills G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the Focus is on application workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and Students should assess which tool to apply rather than trying only one or all technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; Updated April 7, 2016 Page 3 of 19
(E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication Students should evaluate the effectiveness of representations to ensure they are communicating mathematical ideas clearly Students are expected to use appropriate mathematical vocabulary and phrasing when communicating ideas Students are expected to form conjectures based on patterns or sets of examples and non-examples Precise mathematical language is expected. Knowledge and Skills with Student Expectations G.2 Coordinate and Transformational Geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and twodimensional coordinate systems to verify geometric conjectures. (C) determine an equation of a line parallel or perpendicular to a given line that passes through a given point. G.2C District Specificity/ Examples Vocabulary Suggested Resources Resources listed and categorized to indicate suggested uses. Any additional resources must be aligned with the TEKS. Prior Knowledge: Students write an equation of a line that goes through a given point that is parallel or perpendicular to another line (Algebra 2E, 2F, 2G). linear equations parallel lines perpendicular lines point-slope form slope slope-intercept form standard form y-intercept HMH Geometry Unit 2 Region XI: Livebinder NCTM: Illuminations Updated April 7, 2016 Page 4 of 19
G.5 Logical Argument and Constructions. The student uses constructions to validate conjectures about geometric figures. (A) The student is expected to investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior Teachers will still need to review the topic (point slope formula, slope intercept formula) even though it was covered in Algebra. Teachers will need to show how this will be applied to figures on a coordinate grid. Examples: (1) A triangle with endpoints is given on a coordinate grid and students need to find an equation of the altitude for one of the sides. (2) Find the equation of the tangent line to a circle given the endpoint of the radius on the coordinate grid. (3) Given Rectangle ABCD on a grid, find the equation for side AB, given the slope of side CD is ½. G.5A Teachers should: Show the relationships of parallel lines cut by a transversal & patterns it creates How patterns lead to theorems Example: Pattern in interior angle sums acute triangle adjacent angles alternate exterior angles alternate interior angles altitude angle angle bisector bisect central angle chord circle congruent congruent angles congruent segments constructions corresponding angles diagonal and exterior angles of Updated April 7, 2016 Page 5 of 19
polygons, and special segments and angles of circles choosing from a variety of tools. Students should: Investigate patterns Discover theorems based on investigation type activities Misconceptions: Angle relationships exist with any two lines at transversal. Students must understand that lines cut by a transversal must be parallel of the angles relationships to exist. Examples: Angles 8 and 4 are Alternate exterior angles that are NOT congruent. exterior angle interior angle linear pair median midsegment of a triangle obtuse triangle parallel lines perpendicular perpendicular bisectors polygon quadrilateral right triangle same side interior angles secant segment side length side-angle-side, angle-side-angle, hypotenuse-leg) supplementary tangent transversal triangle triangle congruence (side-by-side, Triangle Inequality Theorem vertex vertical angles Updated April 7, 2016 Page 6 of 19
VS. Angles 1 and 2 are Alternate exterior angles that ARE congruent BECAUSE the two lines are parallel. (B) The student is expected to construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge. G.5B Teachers should: Make formal geometric constructions with a variety of tools and methods Possible tools - compass, folding, patty paper, string, reflective mirrors, geo software Students should: Copy a segment Copy an angle Bisect an angle Construct perpendicular lines including perpendicular bisectors Construct parallel line Misconceptions: Assume one part is true (ex. 90 degrees) Updated April 7, 2016 Page 7 of 19
Examples: Updated April 7, 2016 Page 8 of 19
(C) The student is expected to use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships; and G.5C Students: Prove theorems about lines & angles Vertical angles are congruent Alternate interior & corresponding are congruent Endpoints of a line segment that is cut by a perpendicular bisector are equidistant from the point of intersection (D) The student is expected to verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems. G.5D* Prove: The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides. Converse: A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two. Examples: 1. Which of the following could represent the lengths of the sides of a triangle? Choose one: 1, 2, 3 6, 8, 15 5, 7, 9 Updated April 7, 2016 Page 9 of 19
G.6 Proof and Congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by utilizing a variety of methods such as coordinate, transformational, axiomatic and formats such as two-column, paragraph, flow chart (A) The student is expected to verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, angles formed by parallel lines cut by a transversal, and prove equidistance between the endpoints of a segment and points on its perpendicular bisector, and apply these relationships to solve problems; G.6A Confirm theorems about angles formed by Vertical lines, parallels cut by transversal Prove equidistance b/t endpoints of a segment & points on its perpendicular bisector Apply these relationships to solve problems Students should know how to: Prove vertical angles are congruent Prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. Find midpoint and possibly use distance formula. Understand labels & symbols in order to apply the relationships. Finding missing angles and measurements. Angle-Angle-Side angles Angle-Side-Angle base angles endpoints equidistance Hypotenuse-Leg interior angles intersection isosceles triangle line line segment median mid-segment parallel lines perpendicular bisector Pythagorean Theorem Side-Angle-Side Side-Side-Side Theorem transversal Triangle vertical angles Updated April 7, 2016 Page 10 of 19
Identifying if lines are parallel Every point on a point lying on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Teachers should show: Vertical Angle Theorem Corresponding Angles Perpendicular Transversal Perpendicular Bisector Theorem Parallel Transversal (4) - Alt Int/ Ext, SS Int/Ext Examples on how to solve missing angles Misconceptions Vocabulary, proofs Vertical angles are formed by 2 straight lines, not simply rays that come together at the same point. For example: Angle RPS is not a vertical angle to Angle QPU because Q and S do not lie along the same line. Updated April 7, 2016 Page 11 of 19
Example w/ proving angle measurements Example proving theorems Updated April 7, 2016 Page 12 of 19
Example Equidistance with perpendicular bisector (B) The student is expected to prove two triangles are congruent by applying the Side-Angle- Side, Angle-Side-Angle, Side-Side-Side, Angle- Angle-Side, and Hypotenuse-Leg congruence conditions; G.6B Big Idea - Use the definition of congruence to develop and explain the triangle congruence criteria; ASA, SSS, and SAS. Teachers should: For Right Triangles: Explain postulates including Leg-Leg (SAS), Hypotenuse-Angle (AAS) **Reminder: A is not the right angle, it must be one of the acute angles Hypotenuse-Leg (SSS) - only true for right triangle bc since it has a right angle you can use the Pythagorean theorem. it only will give you one possible value for the 3rd side. Creates SSS congruence Updated April 7, 2016 Page 13 of 19
Review special segments Give examples proving congruence Students should: Be able to prove two triangles congruent using theorems. Examples Give any additional information that would be needed to prove the triangles congruent by the method given. Determine which method can be used to prove the triangles congruent from the information given. For some pairs, it may not be possible to prove the triangles congruent. For these, explain what other information would be needed to prove congruence. Updated April 7, 2016 Page 14 of 19
Misconception: Part a - Students forget that a shared side b/t two figures have the same length because it is the same side. Students mistakenly use Angle-side-side as a congruence theorem. HL - is not a version of SSA because you cannot use the 90 degree angle as your A (C) The students is expected to apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles. G.6C Big Idea - identify congruent figures and identify their corresponding parts Rigid Transformation - when the size of the figure does NOT change (translation, rotation, reflection, NOT dilation) Teachers should show: Explain what a rigid transformation is and how the size of the figure will not change Explain definition of congruence & corresponding parts Relate terminology of rigid transformations (reflection, translation, rotation) to congruent transformations CPCTC **Review how to find the length of a segment in the coordinate plane, slope of parallel/perpendicular lines Students should do: Look at two figures & identify corresponding parts Prove congruence based on congruent corresponding parts Updated April 7, 2016 Page 15 of 19
Misconceptions: Students should understand that figures can be congruent even when they are rotated or flipped. Congruence is true even if figures have different orientation. Examples: *CISD Safety Net* (D) The student is expected to verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of G.6D Big Idea - verify theorems about relationships in TRIANGLES and apply to solve problems Teaches need to show: Proof Pythagorean theorem - need to show an Algebraic proof and concrete proof Updated April 7, 2016 Page 16 of 19
isosceles triangles, midsegments, and medians and apply these relationships to solve problems Sum of interior angles - (180) Base angles of isosceles - base angles are congruent Midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle. This segment has two special properties. It is always parallel to the third side, and the length of the midsegment is half the length of the third side. Median of a triangle is a line segment that extends from one vertex of a triangle to the midpoint of the opposite side. All 3 will intersect to form centroid. The centroid splits the medium in a two to one ratio. Students need to: Understand relationships in theorems in order to apply and solve problems Find midpoint of a side, midsegment, median Find length midsegment given parallel side Apply Pythagorean formula Find missing angles knowing angle sum is 180 or find missing angle given a single base angle Example - Pythagorean proof Updated April 7, 2016 Page 17 of 19
Pythagorean Theorem Application Updated April 7, 2016 Page 18 of 19
Median Application ⅔(EM)=ET Midsegment Examples Updated April 7, 2016 Page 19 of 19