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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.
Standard 3 : Prove geometric theorems. (Geometry - Major Cluster)Archived
Cluster Standards
This cluster includes the following benchmarks.
Visit the specific benchmark webpage to find related instructional resources.
- MAFS.912.G-CO.3.9 : Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
- MAFS.912.G-CO.3.10 : Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; 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; the medians of a triangle meet at a point.
- MAFS.912.G-CO.3.11 : Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Cluster Information
Number:
MAFS.912.G-CO.3
Title:
Prove geometric theorems. (Geometry - Major Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
912
Domain-Subdomain
Geometry: Congruence
Cluster Access Points
This cluster includes the following Access Points.
- MAFS.912.G-CO.3.AP.9a : Measure lengths of line segments and angles to establish the facts about the angles created when parallel lines are cut by a transversal and the points on a perpendicular bisector.
- MAFS.912.G-CO.3.AP.10a : Measure the angles and sides of equilateral, isosceles, and scalene triangles to establish facts about triangles.
- MAFS.912.G-CO.3.AP.11a : Measure the angles and sides of parallelograms to establish facts about parallelograms.
Cluster Resources
Vetted resources educators can use to teach the concepts and skills in this topic.
Formative Assessments
- The Measure of an Angle of a Triangle: Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of another interior angle.
- Comparing Lengths in a Parallelogram: Students are given parallelogram ABCD along with midpoint E of diagonal AC and are asked to determine the relationship between the lengths AE + ED and BE + EC.
- Finding Angle C: Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measure of an angle opposite one of the given angles.
- Proving the Triangle Inequality Theorem: Students are asked to prove the Triangle Inequality Theorem.
- An Isosceles Trapezoid Problem: Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter.
- Frame It Up: Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.
- Proving the Corresponding Angles Theorem: Students are asked to prove that corresponding angles formed by the intersection of two parallel lines and a transversal are congruent.
- Two Congruent Triangles: Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.
- Angles of a Parallelogram: Students are given expressions that represent the measures of two angles of a parallelogram and are asked to find the measures of all four angles describing any theorems used.
- Triangles and Midpoints: Students are asked to explain why a quadrilateral formed by drawing the midsegments of a triangle is a parallelogram and to find the perimeter of the triangle formed by the midsegments.
- Interior Angles of a Polygon : Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°.
- The Third Side of a Triangle: Students are given the lengths of two sides of a triangle and asked to describe all possible lengths of the remaining side.
- Name That Triangle: Students are asked to describe a triangle whose vertices are the endpoints of a segment and a point on the perpendicular bisector of a segment.
- Locating the Missing Midpoint: Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for locating the midpoint of the remaining side using only a straight edge and pencil.
- Finding Angle Measures - 1: Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers.
- Finding Angle Measures - 4: Students are asked to find the measure of an angle in a diagram containing two parallel lines and two transversals.
- Finding Angle Measures - 3: Students are asked to find the measures of angles formed by two parallel lines and two transversals.
- Finding Angle Measures - 2: Students are asked to find the measures of angles formed by two parallel lines and a transversal.
- Proving Congruent Diagonals: Students are asked to prove that the diagonals of a rectangle are congruent.
- Proving a Rectangle Is a Parallelogram: Students are asked to prove that a rectangle is a parallelogram.
- Proving Parallelogram Angle Congruence: Students are asked to prove that opposite angles of a parallelogram are congruent.
- Proving Parallelogram Diagonals Bisect: Students are asked to prove that the diagonals of a parallelogram bisect each other.
- Proving Parallelogram Side Congruence: Students are asked to prove that opposite sides of a parallelogram are congruent.
- Proving the Alternate Interior Angles Theorem: In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent.
- Median Concurrence Proof: Students are asked to prove that the medians of a triangle are concurrent.
- Triangle Sum Proof: Students are asked prove that the measures of the interior angles of a triangle sum to 180°.
- Equidistant Points: Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.
- Proving the Vertical Angles Theorem: Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.
- Isosceles Triangle Proof: Students are asked to prove that the base angles of an isosceles triangle are congruent.
- Triangle Midsegment Proof: Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.
Image/Photograph
- Angles (Clipart ETC): This large collection of clipart contains images of angles that can be freely used in lesson plans, worksheets, and presentations.
Lesson Plans
- Engineering Design Challenge: Exploring Structures in High School Geometry : Students explore ideas on how civil engineers use triangles when constructing bridges. Students will apply knowledge of congruent triangles to build and test their own bridges for stability.
- Triangle Mid-Segment Theorem: The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.
- Keeping Triangles in Balance: Discovering Triangle Centroid is Concurrent Medians: In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the centroid is a point of concurrency for the medians of a triangle and recognize its associated usage with the center of gravity or barycenter. This set of instructional materials provides the teacher with hands-on activities using technology as well as paper-and-pencil methods.
- To Be or Not to Be a Parallelogram: Students apply parallelogram properties and theorems to solve real world problems. The acronym, P.I.E.S. is introduced to support a problem solving strategy, which involves drawing a Picture, highlighting important Information, Estimating and/or writing equation, and Solving problem.
- Parallel Thinking Debate: Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles.
- Vertical Angles: Proof and Problem-Solving: Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements.
- Diagonally Half of Me!: This lesson is an exploration activity assisting students prove that diagonals of parallelograms bisect each other. It allows them to compare other quadrilaterals with parallelograms in order to make conjectures about the diagonals of parallelograms.
- Proving and Using Congruence with Corresponding Angles: Students, will prove that corresponding angles are congruent. Directions for using GeoGebra software to discover this relationship is provided.
- Proving Parallelograms Algebraically: This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram.
- Triangles: To B or not to B?: Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.
- Observing the Centroid: Students will construct the medians of a triangle then investigate the intersections of the medians.
- Determination of the Optimal Point: Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.
- The Centroid: Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.
- Parallel Lines: Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel.
- Discovering Triangle Sum: This lesson is designed to address all levels and types of learners to improve understanding of the triangle sum theorem from the simplest perspective and progress steadily by teacher led activities to a more complex level. It is intended to create a solid foundation in geometric reasoning to help students advance to higher levels in confidence.
- Geometer Sherlock: Triangle Investigations: The students will investigate and discover relationships within triangles; such as, the triangle angle sum theorem, and the triangle inequality theorem.
- Proofs of the Pythagorean Theorem: This lesson enriches the students' perspective of geometric proofs that are not in two-column form. Students will apply algebaic skills and geometric properties to prove the Pythagorean Theorem in a variety of ways. This unit is designed to help you identify and assist students who have difficulties in:
- Interpreting diagrams.
- Identifying mathematical knowledge relevant to an argument.
- Linking visual and algebraic representations.
- Producing and evaluating mathematical arguments.
- Evaluating Statements About Length and Area: This lesson unit is intended to help you assess how well students can understand the concepts of length and area, use the concept of area in proving why two areas are or are not equal and construct their own examples and counterexamples to help justify or refute conjectures.
- Intersecting Medians and the Resulting Ratios: This lesson leads students to discover empirically that the distance from each vertex to the intersection of the medians of a triangle is two-thirds of the total length of each median.
- Shape It Up: Students will derive the formula for the sum of the interior angles of a polygon by drawing diagonals and applying the Triangle Sum Theorem. The measure of each interior angle of a regular polygon is also determined.
- Right turn, Clyde!: Students will develop their knowledge of perpendicular bisectors & point of concurrency of a triangle, as well as construct perpendicular bisectors through real world problem solving with a map.
- Halfway to the Middle!: Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments.
- Location, Location, Location, Location?: Students will use their knowledge of graphing concurrent segments in triangles to locate and identify which points of concurrency are associated by location with cities and counties within the Texas Triangle Mega-region.
- Accurately Acquired Angles: Students will start the lesson by playing a game to review angle pairs formed by two lines cut by a transversal. Once students are comfortable with the angle pairs the teacher will review the relationships that are created once the pair of lines become parallel. The teacher will give an example of a proof using the angle pairs formed by two parallel lines cut by a transversal. The students are then challenged to prove their own theorem in groups of four. The class will then participate in a Stay and Stray to view the other group's proofs. The lesson is wrapped up through white board questions answered within groups and then as a whole class.
- Determination of the Optimal Point: Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.
Problem-Solving Tasks
- Finding the area of an equilateral triangle: This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.
- Points equidistant from two points in the plane: This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.
- Midpoints of the Side of a Parallelogram: This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles.
- Tangent Lines and the Radius of a Circle: This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.
- Seven Circles I: This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?
Tutorials
- Parallel lines and transversals: In this tutorial, students will find the measures of angles formed when a transversal cuts two parallel lines.
- Parallel lines, transversals and triangles: This tutorial shows students the eight angles formed when two parallel lines are cut by a transversal line. There is also a review of triangles in this video.
- Parallel lines and transversal lines: Students will see in this tutorial the eight angles formed when two parallel lines are cut by a transversal line.
- Parallel lines and transversals: In this tutorial, students will learn the angle measures when two parallel lines are cut by a transversal line.
- Sum of Exterior Angles of an Irregular Pentagon: In this video, students will learn how to use what they know about the sum of angles in a triangle to determine the sum of the exterior angles of an irregular pentagon.
- Proving vertical angles are equal: In this tutorial, students prove that vertical angles are equal. Students should have an understanding of supplementary angles before viewing this video.
- Finding the measure of vertical angles: Students will use algebra to find the measure of vertical angles, or angles opposite each other when two lines cross. Students should have an understanding of complementary and supplementary angles before viewing this video.
- Introduction to vertical angles: In this tutorial, students will use their knowledge of supplementary, adjacent, and vertical angles to solve problems involving the intersection of two lines.
- Proof: Sum of Measures of Angles in a Triangle Are 180: Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.
- Triangle Angle Example 1: Let's find the measure of an angle, using interior and exterior angle measurements.
- Using Algebra to Find Measures of Angles Formed from Transversal: We will use algebra in order to find the measure of angles formed by a transversal.
- Figuring Out Angles Between Transversal and Parallel Lines: We will be able to identify corresponding angles of parallel lines.
- Angles Formed by Parallel Lines and Transversals: We will gain an understanding of how angles formed by transversals compare to each other.
- Proof: Vertical Angles are Equal: This 5 minute video gives the proof that vertical angles are equal.