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

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

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

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

Students are asked to prove the Triangle Inequality Theorem.

Type: Formative Assessment

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

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

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

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

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

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

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

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

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

Type: Formative Assessment

Students are asked to describe the AA Similarity Theorem.

Type: Formative Assessment

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

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

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

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

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

Type: Formative Assessment

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

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

Type: Formative Assessment

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

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

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

Type: Formative Assessment

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

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

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Type: Formative Assessment

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

Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.

Type: Formative Assessment

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

Type: Formative Assessment

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

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

Type: Lesson Plan

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

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

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

Type: Lesson Plan

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

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

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

Students solve problems using triangle congruence postulates and theorems.

Type: Lesson Plan

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

Students will use formulas they know (distance, midpoint, and slope) to classify quadrilaterals.

Type: Lesson Plan

This lesson will have students exploring different types of triangles and their medians. Students will construct mid-points and medians to determine that the medians meet at a point.

Type: Lesson Plan

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

Type: Lesson Plan

Students, will prove that corresponding angles are congruent. Directions for using GeoGebra software to discover this relationship is provided.

Type: Lesson Plan

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

Type: Lesson Plan

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

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

Type: Lesson Plan

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

Type: Lesson Plan

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

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

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

Type: Lesson Plan

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

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

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

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

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

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

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

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

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

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

Students work on the concept of congruency whilst developing their understanding of proof in a geometric context. This lesson is intended to help students learn to:

• Work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles.
• Identify and understand the significance of a counterexample.
• Prove, and evaluate proofs in a geometric context.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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

Students will act as mathematicians and scientists as they use models, observations and space science concepts to perform calculations and draw inferences regarding a fictional solar system with three planets in circular orbits around a sun. Among the calculations are estimates of the size of the home planet (using a method more than 2000 years old) and the relative distances of the planets from their sun.

Type: Lesson Plan

This activity allows students to go outdoors to measure the height of objects indirectly. Similar right triangles are formed when mirrors are placed on the ground between the object that needs to be measured and the student observing the object in the mirror. Students work in teams to measure distances and solve proportions.

This activity can be used as a review or summative assessment after teaching similar triangles.

Type: Lesson Plan

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

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

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

Type: Lesson Plan

This lesson uses a discovery approach to identify the special angles formed when a set of parallel lines is cut by a transversal. During this lesson, students identify the angle pair and the relationship between the angles. Students use this relationship and special angle pairs to make conjectures about which angle pairs are considered special angles.

Type: Lesson Plan

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

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

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

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

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

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

Through a hands-on-activity and guided practice, students will explore parallel lines intersected by a transversal and the measurements and relationships of the angles created. They will solve for missing measurements when given a single angle's measurement. They will also use the relationships between angles to set up equations and solve for a variable.

Type: Lesson Plan

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

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

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

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

Type: Perspectives Video: Teaching Idea

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

Type: Perspectives Video: Teaching Idea

<p>Reflect for a moment on how to measure tall objects with mirrors and mathematics.</p>

Type: Perspectives Video: Teaching Idea

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Type: Problem-Solving Task

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Type: Problem-Solving Task

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Type: Problem-Solving Task

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

Type: Problem-Solving Task

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Type: Problem-Solving Task

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Type: Problem-Solving Task

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Type: Problem-Solving Task

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Type: Problem-Solving Task

This problem solving task asks students to explain certain characteristics about a triangle.

Type: Problem-Solving Task

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Type: Problem-Solving Task

This problem solving task challenges students to bisect a given angle.

Type: Problem-Solving Task

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

Type: Problem-Solving Task

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Type: Problem-Solving Task

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

Type: Problem-Solving Task

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

Type: Problem-Solving Task

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.

Type: Problem-Solving Task

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Type: Problem-Solving Task