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2.2: Vertical and Adjacent Angles

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Geometry, Chapter 1, Lesson 5.

Problem 1 – Exploring Vertical Angles

1. Define Vertical (or Opposite) Angles.

2. Open the Cabri Jr. file VERTICAL. \begin{align*}\overleftrightarrow{AC}\end{align*} intersects \begin{align*}\overleftrightarrow{BD}\end{align*} at point \begin{align*}O\end{align*}.

Name two pairs of vertical angles.

3. Move point \begin{align*}B\end{align*} or point \begin{align*}C\end{align*} to four different locations where the angles have different measures. Record \begin{align*}m\angle{AOB}\end{align*}, \begin{align*}m\angle{BOC}\end{align*}, \begin{align*}m \angle{COD}\end{align*}, and \begin{align*}m\angle{AOD}\end{align*} for each of your four locations.

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*}
\begin{align*}m\angle{BOC}\end{align*}
\begin{align*}m\angle{COD}\end{align*}
\begin{align*}m\angle{AOD}\end{align*}

What patterns do you notice?

4. If \begin{align*}\angle{AOD}\end{align*} and ______ are vertical angles, then the \begin{align*}m\angle{AOD}\end{align*} ______.

5. If \begin{align*}\angle{AOB}\end{align*} and ______ are vertical angles, then the \begin{align*}m\angle{AOB}\end{align*} ______.

6. Based on your data from Question 3, make a conjecture about vertical angles in general.

Problem 2 – Exploring Adjacent Angles

7. Define Adjacent Angles.

8. Use the file VERTICAL from Problem 1. Identify all four pair of adjacent angles.

9. Move point \begin{align*}B\end{align*} or point \begin{align*}C\end{align*} and make a conjecture about adjacent angles formed by two intersecting lines. Hint: You may have to do a calculation.

10. If \begin{align*}\angle{AOB}\end{align*} and ___________ are adjacent angles formed by two intersecting lines, then the \begin{align*}m\angle{AOB}\end{align*} and ______________ are ______________.

Complete the following problems.

11. Find the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

12. Find the value of \begin{align*}x\end{align*}.

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