# MA.912.GR.1.6

Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures.

### Clarifications

Clarification 1: Instruction includes demonstrating that two-dimensional figures are congruent or similar based on given information.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• 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 ΔABCHBAHAC , students can conclude that $\frac{\text{HC}}{\text{HA}}$ = $\frac{\text{HA}}{\text{HB}}$, which is equivalent to HC ⋅ HB = (HA)2 , which is equivalent to $\sqrt{HC · HB}$ = 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 ΔABCHBAHAC, students can conclude that $\frac{\text{HB}}{\text{BA}}$ = $\frac{\text{BA}}{\text{BC}}$, which is equivalent to HC ⋅ BC = (BA)2, which is equivalent to $\sqrt{HB · BC}$ = BA.
• 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 $\frac{\text{PA}}{\text{CP}}$ = $\frac{\text{QB}}{\text{CQ}}$ by first proving ΔABCPQC using the Angle-Angle (AA) criterion. Since corresponding sides of similar triangles are in proportion, students can determine the relationship $\frac{\text{CA}}{\text{CP}}$=$\frac{\text{CB}}{\text{CQ}}$. Students should be able to realize that CA = CP + PA and CB = CQ + QB using the segment addition postulate. Therefore, $\frac{\text{CA}}{\text{CP}}$ = $\frac{\text{CB}}{\text{CQ}}$ can be written as $\frac{\text{CP+PA}}{\text{CP}}$ = $\frac{\text{CQ+QB}}{\text{CQ}}$. Students can use their algebraic reasoning to rewrite this relationship equivalently as $\frac{\text{CP}}{\text{CP}}$+$\frac{\text{PA}}{\text{CP}}$ = $\frac{\text{CQ}}{\text{CQ}}$+$\frac{\text{QB}}{\text{CQ}}$ which is equivalent to 1+ $\frac{\text{PA}}{\text{CP}}$ = 1+ $\frac{\text{QB}}{\text{CQ}}$, which is equivalent to $\frac{\text{PA}}{\text{CP}}$ = $\frac{\text{QB}}{\text{CQ}}$.

### 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.

• 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 $m$A′ = 103°, what is the measure of angle A
• Part B. If $m$D = 97°, what other angle has a measure of 97°?
• Part C. Find the value of $x$ if DC = 2$x$ + 1.5, DC′ = 5.1 and $\frac{\text{BC}}{\text{B'C'}}$ = $\frac{\text{1}}{\text{3}}$
• 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
Instructional Item 2
• If ΔADE and ΔABC are similar, what is the length of AC, in units?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
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 benchmark.

## Formative Assessments

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

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

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

## 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

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

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

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

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

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

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

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

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

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

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

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.

Type: Lesson Plan

Mirror, Mirror on the ... Ground?:

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

## Perspectives Video: Teaching Idea

Measuring Height with Triangles and Mirrors:

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

Type: Perspectives Video: Teaching Idea

Bank Shot:

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.

Extensions, Bisections and Dissections in a Rectangle:

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

Solar Eclipse:

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.

Mt. Whitney to Death Valley:

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

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.

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.

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.

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:

Locating Warehouse:

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

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.

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.

## 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.

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.

Proving Congruence:

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

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.

## Student Resources

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

Bank Shot:

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.

Extensions, Bisections and Dissections in a Rectangle:

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

Solar Eclipse:

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.

Mt. Whitney to Death Valley:

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

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.

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.

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.

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:

Locating Warehouse:

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

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.

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.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Bank Shot:

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.

Extensions, Bisections and Dissections in a Rectangle:

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

Solar Eclipse:

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.

Mt. Whitney to Death Valley:

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

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.

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.

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.

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:

Locating Warehouse:

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

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.

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.