“Similarity and Congruence.” School Improvement in Maryland. Maryland Mathematics Educators
Mathematical Concept: Similarity and Congruence
Grade Levels: 7th – 8th
NC Standard Course of Study:
Competency Goal 2: The learner will understand and use measurement concepts.
2.01: Determine the effect on perimeter, area, or volume when one or more dimensions of two- and three-dimensional figures are changed.
2.02: Apply and use concepts of indirect measurement.
Competency Goal 3: The learner will understand and use properties and relationships in geometry.
3.01: Represent problem situations with geometric models.
3.02: Identify, define, and describe similar and congruent polygons with respect to angle measures, length of sides, and proportionality of sides.
3.03: Use scaling and proportional reasoning to solve problems related to similar and congruent polygons.
Scissors for each student
Worksheet: Prediction Guide
Worksheet: Comparing Sizes of Figures
Worksheets: Congruent and Similar Triangle Investigation Activity 1-4
Worksheet: Practice with Congruent and Similar Triangles
Students will know how to determine that two figures are similar or congruent by investigating figures that are similar and figures that are congruent. Upon completion of the activities, students will know how to prove that two figures are similar or congruent by using definitions, postulates, and theorems.
1. Use the prediction guide to determine what students already know.
a. Feel free to add/delete/change statements as they fit your class.
b. Accept students’ answers and justifications and ask if the class thinks they are reasonable.
2. Divide the students into small working groups of 2.
a. Have students pull out the handout titled “Comparing Sizes and Figures”
b. Ask students to measure all the segments listed using a ruler. Students should use centimeters to measure.)
c. It may be helpful to have students change their ratios to a decimal (to show that the ratios are equal)
3. An introduction for the SSS and SAS postulates of congruency and the SSS, SAS, and AA postulates of similarity are presented.
a. Activity 1:
i. Pass out the handouts for Activity 1 and scissors.
ii. Each student needs to cut out the three strips of paper.
iii. Place the strips together corner-to-corner to create a triangle
iv. Each student should compare their triangles to their partner’s and to the other triangles in the class.
v. Are they congruent? Encourage students to prove the triangles congruent using the definition of congruent triangles.
1. Measure each side with a ruler and each angle with a protractor.
2. Measuring each side may not be necessary if students reason that everyone started with the same 3 strips of paper.)
3. Have students listed the minimum amount of information used to create these 2 congruent triangles.
b. Activity 2:
i. Pass out the handouts for Activity 1 and scissors.
ii. Using the three strips of paper from Activity 1, have ONE member from each group fold and cut each strip of paper in half.
iii. Students should create a triangle using the one-half pieces of each of the original strips.
iv. Have groups to compare this new triangle to the first one and describe any similarities and differences. [The two triangles are not congruent, but are the same shape.]
v. Since the new half-pieces create triangles which are similar to the original, see if students realize that all NEW triangles are congruent to each other.
1. Have students prove this by using SSS.
2. Help students trace the new triangles and label the sides the same as Activity 1.
vi. Students will justify the similarity of the old and new triangles.
vii. Have students measure each side of the triangle and place the measure in the space requested on the worksheet. [Note: the ratios of the lengths of each pair of corresponding sides are proportional.]
viii. Have students measure each pair of corresponding angles. [Note: the measures of each pair of corresponding angles are congruent. The answers in #4 and #5 show that the triangles are similar.]
c. Activity 3:
i. Pass out Activity 3 handout and patty paper to each student.
ii. Have each student copy the two sides and included angle using patty paper.
iii. Students will draw a segment AC to create a triangle.
iv. A group discussion will take place about what parts of the triangle were given and how their measures compare with everyone else’s [SAS]
v. Students will compare their triangle to the other triangles in the group.
vi. Have students justify that the triangles are congruent, similar, or both. [By definition, they are both. Encourage them to show that the corresponding three sides of the triangles are congruent and therefore the triangles are congruent.]
vii. Ask students to locate the midpoint of AB and of BC and label these points D and E respectively.
viii. Draw segment DE. Compare this triangle to triangle ABC [the two are similar]. Use the definition of similar triangles to justify that the triangles are similar.
ix. Lead a group discussion about what parts of the triangle were given and how this is the minimal amount of information needed in order to prove two triangles are similar [SAS].
d. Activity 4:
i. Pass out Activity 4 handout.
ii. Students will now trace two angles and use these to create triangles. Since these directions may be confusing, demonstrate using the overhead and transparencies.
iii. Have the students create 2 different triangles.
iv. Lead a class discussion about why the 2 triangles are not congruent and that AA [or AAA] is not a way to prove 2 triangles are congruent.
v. The 2 triangles the students create will be similar. Give each student time to measure the sides and show that the ratios of the corresponding sides are proportional. [This will be a great revelation because the ratios of the corresponding sides will be different than the others they have seen in Activities 1-
3. Activities 1-3 ratios will be 2:1.]
vi. Have the students who that the 3 angles are congruent to each other.
vii. Lead a discussion about how the minimum amount of information needed to prove 2 triangles similar is that 2 corresponding angles must be congruent. [AA]
e. Upon completion of Activities 1-4, students will be ready to believe that the ASA theorem of congruence works. Describe that ASA works for similarity as well and can really be the same as the AA theorem.
f. As an extension assignment, hand out copies of the “Practice with Congruent and Similar Triangles” and the “Applications” worksheets.
g. Concluding this study, refer to the Prediction Guide. Allow students to make corrections to their Prediction Guides since they have worked through the entire lesson.
i. Discuss any corrections and the correct answers.
Attachment: Worksheet/Activity Student Page
• Prediction Guide
• Comparing Sizes and Figures
• Congruent and Similar Triangle Investigation: Activity 1
• Congruent and Similar Triangle Investigation: Activity 2
• Congruent and Similar Triangle Investigation: Activity 3
• Congruent and Similar Triangle Investigation: Activity 4
• Practice with Congruent and Similar Triangles
• Formula Sheet