Prove triangle congruence or similarity using Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg.
: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.
| Name |
Description |
| Justifying a Proof of the AA Similarity Theorem | Students are asked to justify statements of a proof of the AA Similarity Theorem. |
| Describe the AA Similarity Theorem | Students are asked to describe the AA Similarity Theorem. |
| What Is the Triangle Relationship? | Students are asked to write an informal justification of the AA Similarity Theorem. |
| 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. |
| 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. |
| 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. |
| 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. |
| Drawing Triangles AAA | Students are asked to draw a triangle with given angle measures, and explain if these conditions determine a unique triangle. |
| 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. |
| 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. |
| Name |
Description |
| 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. |
| 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. |
| What's the Problem | Students solve problems using triangle congruence postulates and theorems. |
| 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. |
| Turning to Congruence | This lesson uses rigid motions to prove the ASA and HL triangle congruence theorems. |
| 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. |
| 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. |
| How Much Proof Do We Need? | Students determine the minimum amount of information needed to prove that two triangles are similar. |
| Observing the Centroid | Students will construct the medians of a triangle then investigate the intersections of the medians. |
| 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. |
| 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. |
| 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. |
| Analyzing Congruence Proofs | 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.
|
| 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. |
| Name |
Description |
| Joining two midpoints of sides of a triangle | 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. |
| Unit Squares and Triangles | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work? | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence? | 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. |
| 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. |
| Inscribing a square in a circle | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work? | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work? | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Are the Triangles Congruent? | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Angle bisection and midpoints of line segments | 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. |
| Inscribing a circle in a triangle II | 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. |
| Is This a Rectangle? | 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. |
| Name |
Description |
| Joining two midpoints of sides of a triangle: | 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. |
| Unit Squares and Triangles: | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work?: | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence?: | 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. |
| 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. |
| Inscribing a square in a circle: | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector: | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work?: | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles: | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II: | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work?: | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Are the Triangles Congruent?: | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Angle bisection and midpoints of line segments: | 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. |
| Inscribing a circle in a triangle II: | 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. |
| Is This a Rectangle?: | 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. |
| Name |
Description |
| Joining two midpoints of sides of a triangle: | 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. |
| Unit Squares and Triangles: | This problem solving task asks students to find the area of a triangle by using unit squares and line segments. |
| Why Does ASA Work?: | This problem solving task ask students to show the reflection of one triangle maps to another triangle. |
| When Does SSA Work to Determine Triangle Congruence?: | 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. |
| 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. |
| Inscribing a square in a circle: | This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. |
| Construction of perpendicular bisector: | This problem solving task challenges students to construct a perpendicular bisector of a given segment. |
| Why Does SAS Work?: | This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points. |
| Reflected Triangles: | This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected. |
| Right triangles inscribed in circles II: | This problem solving task asks students to explain certain characteristics about a triangle. |
| Why does SSS work?: | This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation. |
| Are the Triangles Congruent?: | The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles. |
| Angle bisection and midpoints of line segments: | 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. |
| Inscribing a circle in a triangle II: | 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. |
| Is This a Rectangle?: | 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. |
