High School Statutory Authority:
In Activity 3, students are invited to reformulate their understanding of triangle congruence by drawing from their experiences with side-angle-side congruence in Activity 2. In side-angle-side congruence, the side opposite the angle between the two given sides will be seen to be congruent to the corresponding side of the second triangle as a consequence of the angle congruence of the two corresponding angles.
Students are asked to consider whether triangle congruence can be established by congruence if only one pair of corresponding sides compels them to develop one or more counterexamples to show the insufficiency of one pair of congruent corresponding sides.
In this case, students can visualize for themselves the increase and decrease in the length of the third side as the two congruent corresponding sides open and close in accordance with the increase and decrease of the angle between them.
Small group and partner work such as pair-share is used throughout the unit so that students can help and learn from one another. Students can help each other discover and learn mathematical ideas and specific vocabulary.
Encourage students to use this knowledge to create, explain, and solve real-world problems. Small-group activities in Lesson 1 encourage learning for students with diverse learning styles. Visual aids and spatial relationships are demonstrated within the activities.
The focus is developing an understanding of the definition of congruence and applying that information to triangles. First, students use a discovery process to develop an understanding of congruent triangles.
Then, students are developing ways to prove congruency from congruence theorems. Throughout the lesson based upon the results of formative assessmentconsider the pacing to be flexible to the needs of the students.
Information on these strategies can be found at www.
Use that information to determine when formative assessments techniques are needed e. Essential Practices for Effective Mathematics Instruction at: Students will discover that two triangles with equal perimeters do not necessarily mean that the triangles are congruent.
Divide students into groups of three or four. Each group will make triangles out of two identical lengths of string or ribbon. Construct two triangles with the same perimeter. Shift the vertex of one triangle to change the position of the vertex without changing the perimeter.
Straws with flexible joints or chenille sticks work well for this. Then draw one triangle. Draw a second triangle with the same perimeter but different side lengths on tracing paper. Overlay one triangle on the second and try to make an exact fit. Give each group two identical pieces of string or ribbon whose ends have been stapled together and several pieces of paper large enough to make triangles determined by the size of the ribbon to each small group.
Have three students each hold one vertex of a triangle created by the ribbon, and have the fourth student trace it onto paper. Have each group cut out their triangles and compare them with other groups to see if their triangles are congruent.
Have students record their discoveries about the triangles. Students should observe that even though the perimeters of the triangles are the same, the length of individual sides and measure of angles may be different.
Ask students what is significant about the perimeter being the same when the sides and angles are often different? Students should observe that even though the perimeter is the same, that does not indicate they are congruent triangles.
Then ask the class what it would indicate if the sides and angles were the same. Students should observe that the triangles with identical sides and angles are congruent. Exploring Congruence by Definition and Implying SAS The following information should be displayed on the board or an overhead projector to help students draw congruent triangles on their paper: Angle A is 90 degrees.
Side AB should be 9 cm. Side AC should be 12 cm. Then, connect B and C to create. Steps to prove congruence: Ask students to share their observations about the two triangles. Students should respond that the triangles have 3 pairs of congruent corresponding sides and 3 pairs of congruent corresponding angles.
Introduce the term corresponding to refer to and. Use the term corresponding to refer to. Write an example congruence statement for the triangles:Conventionally you use the symbol that looks like an "equals" sign but consists of three lines. It is the same symbol as is used for identities.
Conventionally you use the symbol that looks like.
Practice B Congruence LESSON Write a congruence statement for each pair of polygons. In the figure, triangle PRT triangle FJH. 5. Find a. 6. Find b. 7. Find c. 8. Find x. 9. Find y. Although congruence statements are often used to compare triangles, they are also used for lines, circles and other polygons.
For example, a congruence between two triangles, ABC and DEF, means that the three sides and the three angles of both triangles are congruent.
Similar and Congruent Polygons You classified polygons. You’ll use properties of similar Write a ratio for each pair of corresponding sides. Then substitute the Critical ThinkingTell whether the statement is true or false.
If it is false, give an example to show why. kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects).
kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only. § Implementation of Texas Essential Knowledge and Skills for Mathematics, High School, Adopted (a) The provisions of §§ of this subchapter shall be .