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Clarifications
Clarification 1: Instruction includes demonstrating that two-dimensional figures are congruent or similar based on given information.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Congruent
- Similarity
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8 students solved real-world and mathematical problems involving similar triangles, defining similarity in terms of dilations. In Geometry, students extend previous understanding of congruence and similarity to solve mathematical and real-world problems involving congruent or similar polygons. In later courses, students will use similar triangles to develop trigonometry related to the unit circle.- Instruction includes discussing the definitions of congruent polygons and similar polygons based on corresponding parts. Students should understand that if a problem involves polygons, they will have to use the definitions of congruent and similar (there are no congruence or similarity criteria for polygons) to show the polygons are congruent or similar and use this information to solve the task. (MTR.4.1)
- When two polygons are congruent, corresponding sides and corresponding angles are congruent.
- When two polygons are similar, corresponding angles are congruent and corresponding sides are in proportion.
- Problem types includes using congruence and similarity criteria to determine whether two triangles are congruent or similar, and using the definition of congruence and similarity to find missing angle measures and side lengths.
- Instruction includes the understanding of the geometric mean in right triangles and its connection to similarity criteria.
- Geometric Mean Altitude Theorem
- In a right triangle, the altitude from the right angle to the hypotenuse separates the hypotenuse into two segments. The length of the altitude, in triangle ABC, is the geometric mean between the lengths of the two line segments the altitude creates on the hypotenuse. Therefore, by applying the fact that ΔABC~ΔHBA~ΔHAC , students can conclude that = , which is equivalent to HC ⋅ HB = (HA)2 , which is equivalent to = HA.
- Geometric Mean Leg Theorem
- In a right triangle, the altitude from the right angle to the hypotenuse separates the hypotenuse into two segments. The length of one of the legs, in triangle ABC, is the geometric mean between the length of the hypotenuse and the line segment of the hypotenuse adjacent to that leg. Therefore, by applying the fact that ΔABC~ΔHBA~ΔHAC, students can conclude that = , which is equivalent to HC ⋅ BC = (BA)2, which is equivalent to = BA.
- In a right triangle, the altitude from the right angle to the hypotenuse separates the hypotenuse into two segments. The length of one of the legs, in triangle ABC, is the geometric mean between the length of the hypotenuse and the line segment of the hypotenuse adjacent to that leg. Therefore, by applying the fact that ΔABC~ΔHBA~ΔHAC, students can conclude that = , which is equivalent to HC ⋅ BC = (BA)2, which is equivalent to = BA.
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Instruction includes the connection between triangle similarity and the Triangle Proportionality Theorem, or Side-Splitter Theorem. Students can explore and conclude that if a line is parallel to one side of a triangle intersecting the other two sides of the triangle, then the line divides these two sides proportionally.
- For example, given triangle ABC and PQ || AB, students can begin the proof that = by first proving ΔABC~ΔPQC using the Angle-Angle (AA) criterion. Since corresponding sides of similar triangles are in proportion, students can determine the relationship =. Students should be able to
realize that CA = CP + PA and CB = CQ + QB using the segment addition postulate. Therefore, = can be written as = . Students can use their algebraic reasoning to rewrite this relationship equivalently as + = + which is equivalent to 1+ = 1+ , which is equivalent to = .
- For example, given triangle ABC and PQ || AB, students can begin the proof that = by first proving ΔABC~ΔPQC using the Angle-Angle (AA) criterion. Since corresponding sides of similar triangles are in proportion, students can determine the relationship =. Students should be able to
realize that CA = CP + PA and CB = CQ + QB using the segment addition postulate. Therefore, = can be written as = . Students can use their algebraic reasoning to rewrite this relationship equivalently as + = + which is equivalent to 1+ = 1+ , which is equivalent to = .
Common Misconceptions or Errors
- Students may have difficulty separating overlapping similar or congruent triangles. To help address this misconception, have students draw the two triangles separately with their corresponding known measures and lengths.
- Students may expect that there are congruence or similarity criteria for non-triangular polygons that are like the criteria for triangles. To help address this misconception, have students create examples in which two quadrilaterals are not congruent even though they have corresponding congruent sides to see that there is no Side-Side-Side-Side congruence criterion for quadrilaterals.
Instructional Tasks
Instructional Task 1 (MTR.7.1)- An artist rendering for the Hapbee Honey Company logo is on a 24” x 36” canvas. The company wants to use the logo on a postcard and is determining the size of the logo based on the different mailing costs. According to the United States Postal Service, mailing costs are determined using the following information.
- Part A. What is the maximum length of the postcard if it is similar to the original rendering and falls within the First-Class Mail® Postcards dimensions?
- Part B. What is the maximum width of the postcard if it is similar to the original rendering and falls within the First-Class Mail® Stamped Large Postcards dimensions?
- Polygons ABCDE and A'B'C'D'E' are similar and shown.
- Part A. If ∠A′ = 103°, what is the measure of angle A?
- Part B. If ∠D = 97°, what other angle has a measure of 97°?
- Part C. Find the value of if DC = 2 + 1.5, D′C′ = 5.1 and = .
- Figure ABCDEFG is similar to Figure LKJIHFM with a scale factor of 0.5. Assume that the measure of angle B within Figure ABCDEFG is 90°.
- Part A. If point F is located at the origin and line segments EF and CD are vertical and line segments GF and DE are horizontal, determine possible coordinates of each of the points, except points A and B, on Figure ABCDEFG.
- Part B. What is the perimeter of Figure ABCDEFG?
- Part C. What is the length of segment DE?
- Part D. What is the perimeter of triangle AGC?
Instructional Items
Instructional Item 1- Triangles ABC and DEF are shown where ∠A ≅ ∠D, ∠C ≅ ∠F and AC ≅ DF,
- Part A. Determine whether the triangles are congruent.
- Part B. If the triangles are congruent, find EF, in units
- If ΔADE and ΔABC are similar, what is the length of AC, in units?
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Perspectives Video: Teaching Idea
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to decide if a basketball goal is regulation height and are given enough information to determine this using similar triangles.
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.
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.
Student Resources
Problem-Solving Tasks
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 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
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 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 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 problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).
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
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
Parent Resources
Problem-Solving Tasks
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 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
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 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 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 problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).
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
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
