2.2: Vertical and Adjacent Angles
This activity is intended to supplement Geometry, Chapter 1, Lesson 5.
ID: 10893
Time required: 15 minutes
Activity Overview
In this introductory or review activity, students will explore vertical and adjacent angles. They will define and identify pairs of angles. Then they will change the intersecting lines of a geometric model to make conjectures about the relationships of the pairs of angles.
Topic: Points, Lines & Planes
- Congruency of vertical angles
- Adjacent angles formed by two intersecting lines are supplementary.
Teacher Preparation and Notes
- This activity was written to be explored with Cabri Jr.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=10893 and select VERTICAL.8xv.
Associated Materials
- Student Worksheet: Vertical and Adjacent Angles http://www.ck12.org/flexr/chapter/9686, scroll down to the second activity.
- Cabri Jr. Application
- VERTICAL.8xv
Problem 1 – Exploring Vertical Angles
Students should first define the term vertical angles using their textbook or other source. After opening Cabri Jr., student should press \begin{align*}Y=\end{align*} and select Open and then VERTICAL to view the file. Introduce the geometric model (two intersecting lines).
Students should name the two pairs of vertical angles of the model.
Direct students to explore the model independently, by grabbing and dragging points \begin{align*}B\end{align*} and/or \begin{align*}C\end{align*}. To grab a point, move the cursor over the point and then press a. The cursor will change to a closed fist. Press a again to release the point.
Then they need to answer the questions on the worksheet.
Problem 2 – Exploring Adjacent Angles
Students are to repeat the steps from Problem 1 with adjacent angles. They will need to first define the term adjacent angles using a textbook or other source.
Students are to use the same geometric model of intersecting lines from Problem 1. They need to name the four pairs of adjacent angles.
Have them explore the model independently and make a conjecture about adjacent angles.
They can also use the Calculate tool from the \begin{align*}F5\end{align*} menu to add the pairs of angles confirming that adjacent angles are supplementary. To do this, select one angle measurement, press +, and then select the adjacent angle measurement.
Solutions
1. Two angles whose sides are opposite rays.
2. \begin{align*}\angle{AOB}\end{align*} and \begin{align*}\angle{COD}; \ \angle{BOC}\end{align*} and \begin{align*}\angle{AOD}\end{align*}
3. Sample answers.
Location | \begin{align*}1^{st}\end{align*} | \begin{align*}2^{nd}\end{align*} | \begin{align*}3^{rd}\end{align*} | \begin{align*}4^{th}\end{align*} |
---|---|---|---|---|
\begin{align*}m\angle{AOB}\end{align*} | 130.6 | 118.5 | 90.4 | 79.4 |
\begin{align*}m\angle{BOC}\end{align*} | 49.4 | 61.5 | 89.6 | 110.6 |
\begin{align*}m \angle{COD}\end{align*} | 130.6 | 118.5 | 90.4 | 79.4 |
\begin{align*}m \angle{AOD}\end{align*} | 49.4 | 61.5 | 89.6 | 100.6 |
4. If \begin{align*}\angle{AOD}\end{align*} and \begin{align*}\angle{BOC}\end{align*} are vertical angles, then the \begin{align*}m\angle{AOD} = m\angle{BOC}\end{align*}.
5. If \begin{align*}\angle{AOB}\end{align*} and \begin{align*}\angle{COD}\end{align*} are vertical angles, then the \begin{align*}m\angle{AOB} = m\angle{COD}\end{align*}.
6. Vertical angles are congruent.
7. Adjacent angles are two coplanar angles that have a common side and a common vertex but no common interior points.
8. \begin{align*}\angle{AOB}\end{align*} and \begin{align*}\angle{BOC}\end{align*}; \begin{align*}\angle{BOC}\end{align*} and \begin{align*}\angle{COD}; \angle{AOD}\end{align*} and \begin{align*}\angle{COD}; \angle{AOD}\end{align*} and \begin{align*}\angle{AOB}\end{align*}
9. Adjacent angles formed by two intersecting lines are supplementary.
10. If \begin{align*}\angle{AOB}\end{align*} and\begin{align*}\angle{BOC}\end{align*} are adjacent angles formed by two intersecting lines, then \begin{align*}\angle{AOB}\end{align*} and \begin{align*}\angle{BOC}\end{align*} are supplementary.
11.
\begin{align*}3x & =75 \\ x & = 25 \\ y & = 180-75 = 105\end{align*}
12.
\begin{align*}x+10 & = 4x-35 \\ 3x & = 45 \\ x & = 15\end{align*}
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