5.3: Triangle Sides and Angles
ID: 8795
Time required: 40 minutes
Topic: Triangles & Congruence
- Classify triangles by angle measure.
- Prove and apply the Isosceles Triangle Theorem.
- Prove and apply the converse of the Isosceles Triangle Theorem.
- Recognize the relationship between the side lengths and angle measures of a triangle.
Activity Overview
In this activity, students will explore side and angle relationships in a triangle. First, students will discover where the longest (and shortest) side is located relative to the largest (and smallest) angle. Then, students will explore the Isosceles Triangle Theorem and its converse. Finally, students will determine the number of acute, right, or obtuse angles that can exist in any one triangle.
Teacher Preparation
- This activity is designed to be used in a high school or middle school geometry classroom.
- The screenshots on pages 1–3 demonstrate expected student results.
Classroom Management
- This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
- The student worksheet helps guide students through the activity and provides a place for students to record their answers and observations.
- Note: Measurements can display 0, 1, or 2 decimal digits. If 0 digits are displayed, the value shown will round from the actual value. To change the number of digits displayed:
- Move the cursor over the coordinate value so it is highlighted.
- Press + to display additional decimal digits or - to hide digits.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
Associated Materials
- Student Worksheet: Triangle Sides and Angles http://www.ck12.org/flexr/chapter/9689, scroll down to the third activity.
- Cabri Jr. Application
Problem 1 – Size and Location of Sides and Angles
Step 1: Opening a new Cabri Jr. file, students should first construct a triangle using the Triangle tool.
Step 2: Select the Alph-Num tool to label the vertices \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} as shown.
Note: Press ENTER to start entering the label, and then press ENTER again to end the label.
Step 3: Students should measure the three interior angles of the triangle using the Measure > Angle tool. To measure an angle, press ENTER on each of three points that define the angle, with the vertex of the angle as the second point chosen.
Step 4: Direct students to measure the three side lengths using the Measure > D. & Length tool.
Note: To measure a side of the triangle, you must click on the endpoints of the segment. If you click on the side itself, the tool will return the perimeter of the triangle.
Step 5: Now students should drag a vertex of the triangle to change the angle measures and side lengths.
Encourage them to make a conjecture about the sizes and locations of the angles and sides in a triangle.
Students can answer the question on the worksheet based on their observations, concluding that the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.
Step 6: Have students save this file as Triangle.
Problem 2 –The Isosceles Triangle Theorem
Step 1: Distribute the Cabri Jr. files Isostri1 and Isostri2 to student calculators
Have students open the file Isostri1. An isosceles triangle has been constructed and the congruent sides have side lengths displayed.
Step 2: Students should measure all three angles using the Measure > Angle tool.
Step 3: Direct students to drag a vertex of the triangle to explore what happens to the angle measures when two sides have equal lengths.
Students should conclude that a triangle with two congruent sides also has two congruent angles. This is known as the Isosceles Triangle Theorem.
Step 4: The converse of the Isosceles Triangle Theorem may be explored in the file Isostri2.
Step 5: Students should measure all three sides using the Measure > Length tool, then drag a vertex to explore.
Step 6: On their worksheets, students should make the conclusion that a triangle with two congruent angles also has two congruent sides.
Problem 3 – Types of Angles in a Triangle
Step 1: Tell students to open the file Triangle that they saved from Problem 1.
If desired, have them hide the side lengths using the Hide/Show > Object tool.
Step 2: Students should drag a vertex and notice how many angles of each type can exist in a triangle. They should conclude that a triangle:
- can have three acute angles
- cannot have three right angles
- cannot have three obtuse angles
Challenge them to observe that a triangle can have, at most, one right or one obtuse angle and that a triangle cannot have both a right and obtuse angle.