# Geometry Critical Areas

CRITICAL AREA #1: Establish criteria for congruence of triangles based on rigid motions
Students have prior experience with drawing triangles based on given measurements and performing rigid motions including translations, reflections, and rotations. They have used these to develop notions about what it means for two objects to be congruent. In this course, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats including deductive and inductive reasoning and proof by contradiction—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.

The Domain and Clusters below relate to this Critical Area:

Congruence

• Experiment with transformations in the plane.
• Understand congruence in terms of rigid motions.
• Prove geometric theorems.
• Make geometric constructions.

CRITICAL AREA #2: Establish criteria for similarity of triangles based on dilations and proportional reasoning
Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students derive the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on their work with quadratic equations done in Model Algebra I. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles.

The Domain and Clusters below relate to this Critical Area:

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity in terms of similarity transformations.
• Prove theorems involving similarity.
• Define trigonometric ratios and solve problems involving right triangles.
• Apply trigonometry to general triangles.

CRITICAL AREA #3: Informally develop explanations of circumference, area, and volume formulas
Students’ experience with three-dimensional objects is extended to include informal explanations of circumference, area, and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.

The Domain and Clusters below relate to this Critical Area:

Geometric Measurement and Dimension

• Explain volume formulas and use them to solve problems.
• Visualize relationships between two-dimensional and three-dimensional objects.

Modeling with Geometry

• Apply geometric concepts in modeling situations.

CRITICAL AREA #4: Apply the Pythagorean Theorem to the coordinate plan
Building on their work with the Pythagorean Theorem in eighth grade to find distances, students use the rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals, and slopes of parallel and perpendicular lines, which relates back to work done in the Model Algebra I course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.

The Domain and Clusters below relate to this Critical Area:

Expressing Geometric Properties with Equations

• Translate between the geometric description and the equation for a conic section.
• Use coordinates to prove simple geometric theorems algebraically.

CRITICAL AREA #5: Prove basic geometric theorems
Students prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations—which relates back to work done in the Model Algebra I course—to determine intersections between lines and circles or parabolas and between two circles.

The Domain and Clusters below relate to this Critical Area:

Circles

• Understand and apply theorems about circles.
• Find arc lengths and area of sectors of circles.

CRITICAL AREA #6: Extend work with probability
Building on probability concepts that began in the middle grades, students use the language of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.

The Domain and Clusters below relate to this Critical Area:

Conditional Probability and the Rules of Probability

• Understand independence and conditional probability and use them to interpret data.
• Use the rules of probability to compute probabilities of compound events in a uniform probability model.

DOMAIN AND CLUSTERS TO BE INCORPORATED INTO ALL CRITICAL AREAS
All standards listed within the domain and clusters below should be integrated throughout all critical areas of focus.

Quantities

• Reason quantitatively and use units to solve problems.

DOMAIN AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS
All standards listed within the domain and clusters below are beyond the College and Career Readiness requirements of the 9-12 sequence.

Using Probability to Make Decisions

• Use probability to evaluate outcomes of decisions.