Standard 1: Prove and apply geometric theorems to solve problems.

General Information
Number: MA.912.GR.1
Title: Prove and apply geometric theorems to solve problems.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning

Related Benchmarks

This cluster includes the following benchmarks.

Related Access Points

This cluster includes the following access points.

Access Points

MA.912.GR.1.AP.1
Use the relationships and theorems about lines and angles to solve mathematical or real-world problems involving postulates, relationships and theorems of lines and angles.
MA.912.GR.1.AP.2
Identify the triangle congruence or similarity criteria; Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg.
MA.912.GR.1.AP.3
Use the relationships and theorems about triangles. Solve mathematical and/or real-world problems involving postulates, relationships and theorems of triangles.
MA.912.GR.1.AP.4
Use the relationships and theorems about parallelograms. Solve mathematical and/or real-world problems involving postulates, relationships and theorems of parallelograms.
MA.912.GR.1.AP.5
Use the relationships and theorems about trapezoids. Solve mathematical and/or real-world problems involving postulates, relationships and theorems of trapezoids.
MA.912.GR.1.AP.6
Use the definitions of congruent or similar figures to solve mathematical and/or real-world problems involving two-dimensional figures.

Related 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.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Proving the Triangle Inequality Theorem:

Students are asked to prove the Triangle Inequality Theorem.

Type: Formative Assessment

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.

Type: Formative Assessment

Frame It Up:

Students are asked to explain how to determine whether a four-sided frame is a rectangle using only a tape measure.

Type: Formative Assessment

Two Congruent Triangles:

Students are asked to explain why a pair of triangles formed by the sides and diagonals of a parallelogram are congruent.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

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°.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Finding Angle Measures - 1:

Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers.

Type: Formative Assessment

Finding Angle Measures - 3:

Students are asked to find the measures of angles formed by two parallel lines and two transversals.

Type: Formative Assessment

Finding Angle Measures - 2:

Students are asked to find the measures of angles formed by two parallel lines and a transversal.

Type: Formative Assessment

Pythagorean Theorem Proof:

Students are asked to prove the Pythagorean Theorem using similar triangles.

Type: Formative Assessment

Camping Calculations:

Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes.

Type: Formative Assessment

Geometric Mean Proof:

Students are asked to prove that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

Type: Formative Assessment

Converse of the Triangle Proportionality Theorem:

Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides proportionally, then that line is parallel to the third side.

Type: Formative Assessment

Justifying a Proof of the AA Similarity Theorem:

Students are asked to justify statements of a proof of the AA Similarity Theorem.

Type: Formative Assessment

Describe the AA Similarity Theorem:

Students are asked to describe the AA Similarity Theorem.

Type: Formative Assessment

Triangle Proportionality Theorem:

Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.

Type: Formative Assessment

What Is the Triangle Relationship?:

Students are asked to write an informal justification of the AA Similarity Theorem.

Type: Formative Assessment

Same Side Interior Angles:

Students are asked to describe and justify the relationship between same side interior angles.

Type: Formative Assessment

Justifying the Triangle Sum Theorem:

Students are asked to provide an informal justification of the Triangle Sum Theorem.

Type: Formative Assessment

Justifying Angle Relationships:

Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles.

Type: Formative Assessment

Proving Congruent Diagonals:

Students are asked to prove that the diagonals of a rectangle are congruent.

Type: Formative Assessment

Proving a Rectangle Is a Parallelogram:

Students are asked to prove that a rectangle is a parallelogram.

Type: Formative Assessment

Proving Parallelogram Angle Congruence:

Students are asked to prove that opposite angles of a parallelogram are congruent.

Type: Formative Assessment

Proving Parallelogram Diagonals Bisect:

Students are asked to prove that the diagonals of a parallelogram bisect each other.

Type: Formative Assessment

Proving Parallelogram Side Congruence:

Students are asked to prove that opposite sides of a parallelogram are congruent.

Type: Formative Assessment

Drawing Triangles SSA:

Students are asked to draw a triangle given the lengths of two of its sides and the measure of a nonincluded angle and to decide if these conditions determine a unique triangle.

Type: Formative Assessment

Drawing Triangles SAS:

Students are asked to draw a triangle given the measures of two sides and their included angle and to explain if these conditions determine a unique triangle.

Type: Formative Assessment

Drawing Triangles ASA:

Students are asked to draw a triangle given the measures of two angles and their included side and to explain if these conditions determine a unique triangle.

Type: Formative Assessment

Drawing Triangles AAS:

Students are asked to draw a triangle given the measures of two angles and a non-included side and to explain if these conditions determine a unique triangle.

Type: Formative Assessment

Drawing Triangles AAA:

Students are asked to draw a triangle with given angle measures, and explain if these conditions determine a unique triangle.

Type: Formative Assessment

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.

Type: Formative Assessment

Proving Congruence:

Students are asked to explain congruence in terms of rigid motions.

Type: Formative Assessment

County Fair:

Students are given a diagram of a county fair and are asked to use similar triangles to determine distances from one location of the fair to another.

Type: Formative Assessment

Basketball Goal:

Students are asked to decide if a basketball goal is regulation height and are given enough information to determine this using similar triangles.

Type: Formative Assessment

Prove Rhombus Diagonals Bisect Angles:

Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.

Type: Formative Assessment

Similar Triangles - 2:

Students are asked to locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find an unknown length in the diagram.

Type: Formative Assessment

Similar Triangles - 1:

Students are asked locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find two unknown lengths in the diagram.

Type: Formative Assessment

Constructions for Parallel Lines:

Students are asked to construct a line parallel to a given line through a given point.

Type: Formative Assessment

Median Concurrence Proof:

Students are asked to prove that the medians of a triangle are concurrent.

Type: Formative Assessment

Triangle Sum Proof:

Students are asked prove that the measures of the interior angles of a triangle sum to 180°.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Isosceles Triangle Proof:

Students are asked to prove that the base angles of an isosceles triangle are congruent.

Type: Formative Assessment

Lesson Plans

Transformation and Similarity:

Using non-rigid motions (dilations), students learn how to show that two polygons are similar. Students will write coordinate proofs confirming that two figures are similar.

Type: Lesson Plan

Triangle Mid-Segment Theorem:

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

Proof of Quadrilaterals in Coordinate Plane:

This lesson is designed to instruct students on how to identify special quadrilaterals in the coordinate plane using their knowledge of distance formula and the definitions and properties of parallelograms, rectangles, rhombuses, and squares. Task cards, with and without solution-encoded QR codes, are provided for cooperative group practice. The students will need to download a free "QR Code Reader" app onto their SmartPhones if you choose to use the cards with QR codes.

Type: Lesson Plan

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.

Type: Lesson Plan

Parallel Thinking Debate:

Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles.

Type: Lesson Plan

Match That!:

Students will prove that two figures are congruent based on a rigid motion(s) and then identify the corresponding parts using paragraph proof and vice versa, prove that two figures are congruent based on corresponding parts and then identify which rigid motion(s) map the images.

Type: Lesson Plan

Airplanes in Radar's Range:

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

Type: Lesson Plan

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.

Type: Lesson Plan

What's the Problem:

Students solve problems using triangle congruence postulates and theorems.

Type: Lesson Plan

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.

Type: Lesson Plan

Turning to Congruence:

This lesson uses rigid motions to prove the ASA and HL triangle congruence theorems.

Type: Lesson Plan

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.

Type: Lesson Plan

Slip, Slide, Tip, and Turn: Corresponding Angles and Corresponding Sides:

Using the definition of congruence in terms of rigid motion, students will show that two triangles are congruent.

Type: Lesson Plan

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.

Type: Lesson Plan

How Do You Measure the Immeasurable?:

Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars.

Type: Lesson Plan

How Much Proof Do We Need?:

Students determine the minimum amount of information needed to prove that two triangles are similar.

Type: Lesson Plan

Proving quadrilaterals algebrically using slope and distance formula:

Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then justify their proofs to their classmates.

Type: Lesson Plan

Quadrilaterals and Coordinates:

In this lesson, students will use coordinates to algebraically prove that quadrilaterals are rectangles, parallelograms, and trapezoids. A through introduction to writing coordinate proofs is provided as well as plenty of practice.

Type: Lesson Plan

Observing the Centroid:

Students will construct the medians of a triangle then investigate the intersections of the medians.

Type: Lesson Plan

Triangles on a Lattice:

In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible.

Type: Lesson Plan

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.

Type: Lesson Plan

Proving Quadrilaterals:

This lesson provides a series of assignments for students at the Getting Started, Moving Forward, and Almost There levels of understanding for the Mathematics Formative Assessment System (MFAS) Task Describe the Quadrilateral (CPALMS Resource ID#59180). The assignments are designed to "move" students from a lower level of understanding toward a complete understanding of writing a coordinate proof involving quadrilaterals.

Type: Lesson Plan

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.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

Polygon...Prove it:

While this is an introductory lesson on the standard, students will enjoy it, as they play "Speed Geo-Dating" during the Independent practice portion. Students will use algebra and coordinates to prove rectangles, rhombus, and squares. Properties of diagonals are not used in this lesson.

Type: Lesson Plan

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.

Type: Lesson Plan

Let's Prove the Pythagorean Theorem:

Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.

Type: Lesson Plan

Altitude to the Hypotenuse:

Students will discover what happens when the altitude to the hypotenuse of a right triangle is drawn. They learn that the two triangles created are similar to each other and to the original triangle. They will learn the definition of geometric mean and write, as well as solve, proportions that contain geometric means. All discovery, guided practice, and independent practice problems are based on the powerful altitude to the hypotenuse of a right triangle.

Type: Lesson Plan

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.

Type: Lesson Plan

Triangles: Finding Interior Angle Measures:

The lesson begins with a hands-on activity and then an experiment with a GeoGebra-based computer model to discover the Triangle Angle Sum Theorem. The students write and solve equations to find missing angle measures in a variety of examples.

Type: Lesson Plan

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.

Type: Lesson Plan

Halfway to the Middle!:

Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments.

Type: Lesson Plan

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.

Type: Lesson Plan

Help me Find my Relationship!:

In this lesson, students will investigate the relationship between angles when parallel lines are cut by a transversal. Students will identify angles, and find angle measures, and they will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of angle relationships.

Type: Lesson Plan

An Investigation of Angle Relationships Formed by Parallel Lines and a Transversal Using GeoGebra:

In this lesson, students will discover angle relationships formed when two parallel lines are cut by a transversal (corresponding, alternate interior, alternate exterior, same-side interior, same-side exterior). They will establish definitions and identify whether these angle pairs are supplementary or congruent.

Type: Lesson Plan

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.

Type: Lesson Plan

Concurrent Points Are Optimal:

Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.

Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.

A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.

Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.

A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.

The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students.

Type: Lesson Plan

What's the Point? Part 1:

This is a patty paper-folding activity where students measure and discover the properties of the point of concurrency of the perpendicular bisectors of the sides of a triangle.

Type: Lesson Plan

Original Student Tutorials

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

Proving Theorems About Triangles:

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally. 

Type: Original Student Tutorial

Perspectives Video: Teaching Ideas

Discovering the Triangle Inequality Theorem:

Unlock an effective teaching strategy for helping students discover the triangle inequality theorem in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Geometric Constructions- Angle Bisector:

Unlock an effective teaching strategy for making connections in geometric constructions in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Perpendicular Bisectors & Constructions:

Unlock an effective teaching strategy for teaching geometric constructions, specifically perpendicular bisectors, in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Measuring Height with Triangles and Mirrors:

Reflect for a moment on how to measure tall objects with mirrors and mathematics.

Type: Perspectives Video: Teaching Idea

Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Original Student Tutorials

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

Proving Theorems About Triangles:

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally. 

Type: Original Student Tutorial

Parent Resources

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