Geometry (Texas TEKS)

111.41(c)(1) Mathematical process standards

• 111.41(c)(1) Uses mathematical processes to acquire and demonstrate understanding through application, problem solving, tools, communication, and justification.
• 111.41(c)(1)(A) Apply mathematics to problems from everyday life, society, and the workplace.
• 111.41(c)(1)(B) Use a problem-solving model covering analysis, planning, solving, justifying, and evaluating reasonableness.
• 111.41(c)(1)(C) Select appropriate tools and techniques, such as mental math and estimation, to solve problems.
• 111.41(c)(1)(D) Communicate ideas and reasoning using multiple representations, such as symbols, diagrams, graphs, and language.
• 111.41(c)(1)(E) Create and use representations to organize, record, and communicate mathematical ideas.
• 111.41(c)(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
• 111.41(c)(1)(G) Display, explain, and justify ideas and arguments using precise mathematical language.

111.41(c)(2) Coordinate and transformational geometry

• 111.41(c)(2) Uses algebra and coordinate systems to connect algebra and geometry and verify geometric conjectures.
• 111.41(c)(2)(A) Find coordinates of a point at a given fractional distance along a segment, including the midpoint.
• 111.41(c)(2)(B) Derive and use distance, slope, and midpoint formulas to verify congruence, parallelism, and perpendicularity.
• 111.41(c)(2)(C) Determine the equation of a line through a given point that is parallel or perpendicular to a given line.

111.41(c)(3) Coordinate and transformational geometry

• 111.41(c)(3) Generates and describes rigid and non-rigid transformations and their compositions on plane figures.
• 111.41(c)(3)(A) Describe and perform transformations of figures using coordinate notation.
• 111.41(c)(3)(B) Determine pre-images or images under compositions of rigid and non-rigid transformations, including dilations centered anywhere.
• 111.41(c)(3)(C) Identify transformation sequences that carry a pre-image onto an image, on or off the coordinate plane.
• 111.41(c)(3)(D) Identify and distinguish between reflectional and rotational symmetry in plane figures.

111.41(c)(4) Logical argument and constructions

• 111.41(c)(4) Uses deductive reasoning to understand geometric relationships, including definitions and conditional statements.
• 111.41(c)(4)(A) Distinguish among undefined terms, definitions, postulates, conjectures, and theorems.
• 111.41(c)(4)(B) Determine validity of converse, inverse, and contrapositive; relate biconditionals to true conditionals with true converses.
• 111.41(c)(4)(C) Verify that a conjecture is false using a counterexample.
• 111.41(c)(4)(D) Compare Euclidean and spherical geometry relationships, such as parallel lines and triangle angle sums.

111.41(c)(5) Logical argument and constructions

• 111.41(c)(5) Uses patterns and constructions to validate conjectures about geometric figures.
• 111.41(c)(5)(A) Investigate patterns to make conjectures about angles, triangle congruence, segments, polygons, and circles.
• 111.41(c)(5)(B) Construct congruent segments and angles, bisectors, perpendiculars, and parallel lines with compass and straightedge.
• 111.41(c)(5)(C) Use constructions of congruence and bisectors to make conjectures about geometric relationships.
• 111.41(c)(5)(D) Verify the Triangle Inequality theorem using constructions and apply it to solve problems.

111.41(c)(6) Proof and congruence

• 111.41(c)(6) Uses deductive reasoning to prove and apply theorems using varied methods and proof formats.
• 111.41(c)(6)(A) Verify angle theorems and prove perpendicular bisector equidistance, applying these to solve problems.
• 111.41(c)(6)(B) Prove triangles congruent using SAS, ASA, SSS, AAS, and HL conditions.
• 111.41(c)(6)(C) Apply the definition of congruence via rigid transformations to identify corresponding sides and angles.
• 111.41(c)(6)(D) Verify triangle theorems, including the Pythagorean proof, angle sums, isosceles base angles, midsegments, and medians.
• 111.41(c)(6)(E) Prove quadrilaterals are parallelograms, rectangles, squares, or rhombuses using sides, angles, or diagonals.

111.41(c)(7) Similarity, proof, and trigonometry

• 111.41(c)(7) Applies similarity to solve problems.
• 111.41(c)(7)(A) Apply similarity as a dilation to identify similar figures, proportional sides, and congruent angles.
• 111.41(c)(7)(B) Apply Angle-Angle to verify similar triangles and use proportional sides to solve problems.

111.41(c)(8) Similarity, proof, and trigonometry

• 111.41(c)(8) Uses deductive reasoning to prove and apply theorems about similar triangles and right-triangle altitudes.
• 111.41(c)(8)(A) Prove similar triangle theorems, including Triangle Proportionality, and apply them to solve problems.
• 111.41(c)(8)(B) Apply relationships when an altitude is drawn to a right triangle's hypotenuse, including the geometric mean.

111.41(c)(9) Similarity, proof, and trigonometry

• 111.41(c)(9) Understands and applies relationships in right triangles.
• 111.41(c)(9)(A) Use sine, cosine, and tangent to find side lengths and angle measures in right triangles.
• 111.41(c)(9)(B) Apply 30-60-90 and 45-45-90 relationships, the Pythagorean theorem, and Pythagorean triples to solve problems.

111.41(c)(10) Two-dimensional and three-dimensional figures

• 111.41(c)(10) Recognizes characteristics and dimensional changes of two- and three-dimensional figures.
• 111.41(c)(10)(A) Identify cross-section shapes of solids and three-dimensional objects generated by rotating two-dimensional shapes.
• 111.41(c)(10)(B) Determine how linear dimension changes affect perimeter, area, surface area, or volume, proportionally or not.

111.41(c)(11) Two-dimensional and three-dimensional figures

• 111.41(c)(11) Applies formulas to determine measures of two- and three-dimensional figures.
• 111.41(c)(11)(A) Apply the area formula for regular polygons to solve problems using appropriate units.
• 111.41(c)(11)(B) Determine the area of composite two-dimensional figures, including sectors, to solve problems.
• 111.41(c)(11)(C) Apply total and lateral surface area formulas to three-dimensional figures and composite figures.
• 111.41(c)(11)(D) Apply volume formulas to three-dimensional figures and composite figures.

111.41(c)(12) Circles

• 111.41(c)(12) Applies theorems and equations about circles to understand geometric relationships.
• 111.41(c)(12)(A) Apply circle theorems on angles, radii, chords, tangents, and secants to solve non-contextual problems.
• 111.41(c)(12)(B) Apply the proportional relationship between arc length and circumference to solve problems.
• 111.41(c)(12)(C) Apply the proportional relationship between sector area and circle area to solve problems.
• 111.41(c)(12)(D) Describe radian measure as the ratio of intercepted arc length to the circle's radius.
• 111.41(c)(12)(E) Show the equation of a circle centered at the origin and determine equations for circles with center (h, k).

111.41(c)(13) Probability

• 111.41(c)(13) Understands probability in real-world situations, including independent and dependent events.
• 111.41(c)(13)(A) Develop strategies using permutations and combinations to solve contextual problems.
• 111.41(c)(13)(B) Determine probabilities based on area to solve contextual problems.
• 111.41(c)(13)(C) Identify independent events and compute probabilities of both occurring, with or without replacement.
• 111.41(c)(13)(D) Apply conditional probability in contextual problems.
• 111.41(c)(13)(E) Apply independence in contextual problems.