Download MAFS.912.G-CO.3 - Prove geometric theorems | CPALMS.org

Standard 3 : Prove geometric theorems. (Geometry Major Cluster)
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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.
Number: MAFS.912.G-CO.3
Title: Prove geometric theorems. (Geometry - Major
Cluster)
Type: Cluster
Subject: Mathematics
Grade: 912
Domain: Geometry: Congruence
Related Standards
Code
MAFS.912.G-CO.3.9
MAFS.912.G-CO.3.10
MAFS.912.G-CO.3.11
Description
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.
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.
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.
Related Access Points
Access Point
Access Point Number
MAFS.912.G-CO.3.AP.9a:
MAFS.912.G-CO.3.AP.10a:
MAFS.912.G-CO.3.AP.11a:
Access Point Title
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.
Measure the angles and sides of equilateral, isosceles, and scalene triangles to establish facts about triangles.
Measure the angles and sides of parallelograms to establish facts about parallelograms.
Related Resources
Lesson Plan
Name
Accurately Acquired Angles:
Detemination of the Optimal
Point:
Description
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.
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.
page 1 of 6 Students will use dynamic geometry software to determine the optimal location for a facility under a variety of
Determination of the Optimal scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in
Point (formerly where to build some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the
a house):
conjectures.
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 some quadrilaterals with parallelograms in order to make conjectures about the diagonals of
parallelograms.
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 lead 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.
Engineering Design Challenge: Students explore ideas on how civil engineers use triangles when constructing bridges. Students will apply knowledge of
Exploring Structures in High
congruent triangles to build and test their own bridges for stability.
School Geometry :
This lesson unit is intended to help you assess how well students can understand the concepts of length and area, use
Evaluating Statements About
the concept of area in proving why two areas are or are not equal and construct their own examples and
Length and Area:
counterexamples to help justify or refute conjectures.
Geometer Sherlock: Triangle The students will investigate and discover relationships within triangles; such as, the triangle angle sum theorem, and
Investigations:
triangle inequalities.
Halfway to the middle!:
Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments.
Intersecting Medians and the 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.
Resulting Ratios:
In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the
Keeping Triangles in Balance: centroid is the point of concurrency for the medians of a triangle and recognize its associated usage with the center
Discovering Triangle Centroid of gravity or barycenter. This set of instructional materials provides the teacher with hands-on activities using
is Concurrent Medians:
technology as well as paper-and-pencil methods.
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.
Observing the Centroid:
Students will construct the medians of a triangle then investigate the intersections of the medians.
Parallel Lines:
Students will be able to prove that alternate interior angles are congruent and corresponding angles are congruent
given two parallel lines and a traversal line. Students will use GeoGebra to explore real-world images to see if they can
prove that their line segments are parallel.
Parallel Thinking Debate:
Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles.
Proofs of the Pythagorean
Theorem:
This lesson is intended to help you assess how well students are able to produce and evaluate geometrical proofs. In
particular, this unit is intended 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.
Proving and Using Congruence Students, with the aid of GeoGebra software, will prove and use congruence of corresponding angles produced by
parallel lines intersected by a traversal.
with Corresponding Angles:
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.
Right turn, Clyde!:
Students will develop their knowledge of perpendicular bisectors & point of concurrency of a triangle, as well as
construct perpendicular biesctors through real world problem solving with a map.
Shape It Up:
Students will draw diagonals for different polygons, separating the polygons into triangles. Using the fact that the sum
of the measures of the interior angles of a triangle is 180 degrees, and the fact the angles of the triangles are used
to form the angles of the polygons, students will derive the formula for finding the sum of the measures of the angles
of a polygon with n sides. Students will also learn to use this formula, along with the fact that all angles of a regular
polygon are congruent, to find the measures of the angles of a regular polygon.
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.
To Be or Not to Be a
Parallelogram:
A lesson where students apply parallelogram properties and theorems to solve real world problems. The teacher
models a problem solving strategy, which involves drawing a picture, highlighting important information, estimating
and/or writing equation, and solving problem (P.I.E.S.).
Triangle Mid-Segment
Theorem:
This lesson provides a straightforward way to show the steps and the thought process of a proof involving the
Triangle Mid-Segment Theorem and algebra.
Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will
Triangles: To B or not to B?: determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a
triangle.
Vertical Angles: Proof and
Problem-Solving:
In this lesson, students learn about the relationship between vertical angles by making inferences and proving the
Vertical Angle Theorem. They then use this relationship to prove other angle relationships and to find angle
measurements by using vertical angles.
Formative Assessment
page 2 of 6 Name
An Isosceles Trapezoid
Problem:
Description
Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its
perimeter.
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.
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.
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.
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.
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 - 2:
Students are asked to find the measures of angles formed by two parallel lines and a transversal.
Finding Angle Measures - 3:
Students are asked to find the measures of angles formed by two parallel lines and two transversals.
Finding Angle Measures - 4:
Students are asked to find the measure of an angle in a diagram containing two parallel lines and two transversals.
Frame It Up:
Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.
Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the
Interior Angles of a Polygon : formula (n – 2)180°.
Isosceles Triangle Proof:
Students are asked to prove that the base angles of an isosceles triangle are congruent.
Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for
Locating the Missing Midpoint: locating the midpoint of the remaining side using only a straight edge and pencil.
Median Concurrence Proof:
Students are asked to prove that the medians of a triangle are concurrent.
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.
Proving a Rectangle Is a
Parallelogram:
Students are asked to prove that a rectangle is a parallelogram.
Proving Congruent Diagonals: Students are asked to prove that the diagonals of a rectangle are congruent.
Proving Parallelogram Angle
Congruence:
Proving Parallelogram
Diagonals Bisect:
Proving Parallelogram Side
Congruence:
Students are asked to prove that opposite angles of a parallelogram are congruent.
Students are asked to prove that the diagonals of a parallelogram bisect each other.
Students are asked to prove that opposite sides of a parallelogram are congruent.
Proving the Alternate Interior 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.
Angles Theorem:
Proving the Triangle Inequality Students are asked to prove the Triangle Inequality Theorem.
Theorem:
Proving the Vertical Angles
Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.
Theorem:
The Measure of an Angle of a Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of
another interior angle.
Triangle:
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.
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.
Triangle Sum Proof:
Students are asked prove that the measures of the interior angles of a triangle sum to 180°.
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.
Two Congruent Triangles:
Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are
congruent.
Image/Photograph
Name
Angles (Clipart ETC):
Description
This large collection of clipart contains images of angles that can be freely used in lesson plans, worksheets, and
presentations.
Tutorial
Name
Description
page 3 of 6 Angles Formed by Parallel
We will gain an understanding of how angles formed by transversals compare to each other.
Lines and Transversals:
Figuring Out Angles Between We will be able to identify corresponding angles of parallel lines.
Transversal and Parallel Lines:
Finding the measure of vertical 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.
angles:
In this tutorial, students will use their knowledge of supplementary, adjacent, and vertical angles to solve problems
Introduction to vertical angles: involving the intersection of two lines.
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.
Parallel lines and transversals: In this tutorial, students will find the measures of angles formed when a transversal cuts two parallel lines.
This tutorial shows students the eight angles formed when two parallel lines are cut by a transversal line. There is
Parallel lines, transversals and also a review of triangles in this video.
triangles:
Proof: Sum of Measures of
Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.
Angles in a Triangle Are 180:
Proof: Vertical Angles are
This 5 minute video gives the proof that vertical angles are equal.
Equal:
In this tutorial, students prove that vertical angles are equal. Students should have an understanding of
Proving vertical angles are
supplementary angles before viewing this video.
equal:
Sum of Exterior Angles of an 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.
Irregular Pentagon:
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.
Virtual Manipulative
Name
Congruent Triangles:
Description
This manipulative is a virtual realization of the kind of physical experience that might be available to students given
three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles
(SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle,
students can experience congruence by actually moving one on the top of the other.
Problem-Solving Task
Name
Finding the area of an
equilateral triangle:
Description
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.
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.
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?
This problem solving task asks students to find the area of an equilateral triangle.
Tangent Lines and the Radius This problem solving task challenges students to find the perpendicular meeting point of a segment from the center
of a circle and a tangent.
of a Circle:
Educational Game
Name
Opposite Angles:
Description
Play a game to discover the relationship between opposite angles and identify names of angles by their measures.
Students may select Teach Me to learn about these angle relationships prior to beginning play. Hints and feedback
are provided to players.
Assessment
Name
Sample 1 - High School
Geometry State Interim
Assessment:
Sample 2 - High School
Geometry State Interim
Assessment:
Description
This is a State Interim Assessment for 9th-12th grade.
This is a State Interim Assessment for 9th-12th grade.
page 4 of 6 Sample 3 - High School
Geometry State Interim
Assessment:
Sample 4 - High School
Geometry State Interim
Assessment:
This is a State Interim Assessment for 9th-12th grade.
This is a State Interim Assessment for 9th-12th grades.
3D Modeling
Name
The 3D Pantograph:
Description
A pantograph is a device that is used to enlarge or reduce a two-dimensional figure or object. In this activity,
students will manipulate a 3D-printed pantograph to help them visualize and understand parallelograms and similarity.
Student Resources
Title
Angles Formed by Parallel
Lines and Transversals:
Congruent Triangles:
Description
We will gain an understanding of how angles formed by transversals compare to each other.
This manipulative is a virtual realization of the kind of physical experience that might be available to students given
three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles
(SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle,
students can experience congruence by actually moving one on the top of the other.
Figuring Out Angles Between We will be able to identify corresponding angles of parallel lines.
Transversal and Parallel Lines:
Finding the area of an
This problem solving task asks students to find the area of an equilateral triangle.
equilateral triangle:
Finding the measure of vertical 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.
angles:
In this tutorial, students will use their knowledge of supplementary, adjacent, and vertical angles to solve problems
Introduction to vertical angles: involving the intersection of two lines.
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.
Opposite Angles:
Play a game to discover the relationship between opposite angles and identify names of angles by their measures.
Students may select Teach Me to learn about these angle relationships prior to beginning play. Hints and feedback
are provided to players.
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.
Parallel lines and transversals: In this tutorial, students will find the measures of angles formed when a transversal cuts two parallel lines.
This tutorial shows students the eight angles formed when two parallel lines are cut by a transversal line. There is
Parallel lines, transversals and also a review of triangles in this video.
triangles:
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.
Proof: Sum of Measures of
Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.
Angles in a Triangle Are 180:
Proof: Vertical Angles are
This 5 minute video gives the proof that vertical angles are equal.
Equal:
In this tutorial, students prove that vertical angles are equal. Students should have an understanding of
Proving vertical angles are
supplementary angles before viewing this video.
equal:
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?
Sum of Exterior Angles of an 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.
Irregular Pentagon:
Tangent Lines and the Radius This problem solving task challenges students to find the perpendicular meeting point of a segment from the center
of a circle and a tangent.
of a Circle:
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.
Parent Resources
page 5 of 6 Title
Congruent Triangles:
Description
This manipulative is a virtual realization of the kind of physical experience that might be available to students given
three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles
(SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle,
students can experience congruence by actually moving one on the top of the other.
Finding the area of an
equilateral triangle:
This problem solving task asks students to find the area of an equilateral triangle.
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.
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.
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?
Tangent Lines and the Radius This problem solving task challenges students to find the perpendicular meeting point of a segment from the center
of a circle and a tangent.
of a Circle:
page 6 of 6