5.1: Interior and Exterior Angles of a Triangle
This activity is intended to supplement Geometry, Chapter 4, Lesson 1.
ID: 8775
Time required: 40 minutes
Activity Overview
In this activity, students will measure interior and exterior angles of a triangle and make conjectures about their relationships.
Topic: Triangles & Congruence
- Use inductive reasoning to conjecture a theorem about the total measures of a triangle’s interior angles.
- Prove that the sum of the measures of the interior angles of a triangle is \begin{align*}180^\circ\end{align*}.
- Prove that the sum of the measures of the exterior angles of a triangle is \begin{align*}360^\circ\end{align*}.
Teacher Preparation
- This activity is designed to be used in a high school or middle school geometry classroom.
- 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.
- If needed, review with students the definitions of the following angles before the activity: Interior angle, exterior angle, remote interior angle, adjacent interior angle.
- 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: Interior and Exterior Angles http://www.ck12.org/flexr/chapter/9689.
- Cabri Jr. Application
Problem 1 – Interior angles of a triangle
Students should open a new Cabri Jr. file. They will construct a triangle using the Triangle tool.
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 the label, then press ENTER again to end the label.
Have students measure the three interior angles of the triangle using the Measure > Angle tool.
Note: To measure an angle, press ENTER on three points, with the vertex of the angle being the second point selected.
Students should records this data in the first row of the chart on the student worksheet.
Students should drag a vertex of the triangle to change the angle measures. Have them try to create different types of triangles (acute, obtuse, right) and record two more sets of data in the chart.
From here, they should make a conjecture about the three interior angles.
Instruct students to use the Calculate tool to find the sum of the three interior angles of the triangle.
Drag a vertex and observe the results. Ask: Do the results support your conjecture?
Have students save the file as IntAngle.
Problem 2 – One exterior angle of a triangle
Students should continue using the previous file, but instruct them to save the file as ExtAngle using the Save As option.
Have students construct a line through the two lower vertices of the triangle (\begin{align*}A\end{align*} and \begin{align*}C\end{align*}) using the Line tool.
Note: To be certain that the line passes through a vertex, be sure that the vertex point is flashing before pressing ENTER.
Next, students should create a new point on the line to the right of the triangle using the Point > Point On tool. Label it \begin{align*}D\end{align*}.
Direct students to measure the exterior angle \begin{align*}\angle{BCD}\end{align*} with the Measure > Angle tool. Record this measure and the measures of the interior angles on the chart on the student worksheet.
They can then drag a vertex of the triangle to change the angle measures, and add two more sets of data to the chart.
Students should now make some observations about the exterior angle and its relationship to other angles in the chart.
Encourage them to make calculations as needed to test their conjectures. If needed, suggest that they calculate the sums of pairs of angles in the chart.
Again, students can drag a vertex and observe the results. Ask: Do the results support your conjectures?
Before proceeding, have students save the file as ExtAngle.
Problem 3 – Three exterior angles of a triangle
Students should continue using the previous file, but instruct them to save the file as ExtAng3 using the Save As tool.
Note: Press the ALPHA button to access the numeric character “3.”
Students will select the Line tool and construct a line through vertices \begin{align*}A\end{align*} and \begin{align*}B\end{align*} of the triangle as shown below.
Next, students should measure one exterior angle at each vertex.
Note: It is not necessary to create an additional point on the line before measuring the angle.
Have them record the measures of the three exterior angles into the chart on the student worksheet.
They can then drag a vertex of the triangle and record more data into the chart.
Instruct students to make a conjecture about the three exterior angles, and have them calculate the sum of the three exterior angles.
Tell them to drag a vertex and observe the results. Ask: Do the results support your conjectures?
Solutions
- \begin{align*}\angle{B} = 43^\circ, \angle{BCD} = 143^\circ\end{align*}
- \begin{align*}\angle{BCA} = 117^\circ, \angle{B} = 32^\circ\end{align*}
- \begin{align*}\angle{BCA} = 81^\circ, \text{int}\ \angle{A} = 42^\circ, \text{ext}\ \angle{A} = 138^\circ\end{align*}
- \begin{align*}\angle{BCA} = 83^\circ, \text{ext}\ \angle{A} = 140^\circ, \angle{ABC} = 57^\circ, \text{ext}\ \angle{B} = 123^\circ\end{align*}