1.10: Congruent Segments, Midpoints, and Bisectors
Learning Objectives
- Be able to properly name, draw, and label angles.
- Understand and apply the Angle Addition Postulate.
- Classify angles as acute, obtuse, or straight based on their measurements.
Angle, Vertex, and Sides
An angle is formed when two rays share a common endpoint. That common endpoint is called the vertex and the two rays are called the sides of the angle:
- Two rays that share a common endpoint are called an _______________________.
- The ________________________ is the shared common endpoint of an angle.
- In an angle, the two rays are called ______________________.
In the diagram below, \begin{align*}\overrightarrow{AB}\end{align*} and \begin{align*}\overrightarrow{AT}\end{align*} form an angle, \begin{align*}\angle BAT\end{align*}, or \begin{align*}\angle{TAB}\end{align*}, or \begin{align*}\angle {A}\end{align*}. You can use one letter to name this angle since point \begin{align*}A\end{align*} is the vertex and there is only one angle at point \begin{align*}A\end{align*}. The symbol \begin{align*}\angle\end{align*} is used for naming angles.
The same basic definition for angle also holds when lines, segments, or rays intersect.
- The symbol for naming angles is ___________.
Naming Angles
If two or more angles share the same vertex, you MUST use three letters to name the angle.
The vertex letter is always in the middle of the three naming letters.
- To name an angle, use __________________ letters with the vertex letter in the _______________________.
For example, in the image below it is unclear which angle is referred to by \begin{align*}\angle L\end{align*}.
To talk about the angle with one arc, you would write \begin{align*}\angle KLJ\end{align*}.
For the angle with two arcs, you would write \begin{align*}\angle JLM\end{align*}.
Reading Check:
Name the following angle three different ways:
_____________________________________
_____________________________________
_____________________________________
Measuring Angles
We use a ruler to measure segments by their length. But how do we measure an angle?
The length of the sides does not change how wide an angle is “open.” Instead of using length, the size of an angle is measured by the amount of rotation from one side to another.
By definition, a full turn is defined as 360 degrees. Use the symbol \begin{align*}^{\circ}\end{align*} for degrees.
- One full rotation is defined as _________________ degrees.
- The symbol for degrees looks like __________.
Right Angles and Perpendicular Lines
The angle that is made by rotating through one-fourth of a full turn is very special. It measures \begin{align*}\frac{1}{4} \cdot 360^\circ = 90^\circ\end{align*} and we call this a right angle.
- A right angle measures one-quarter of \begin{align*}360^\circ\end{align*}, or ______________.
Right angles are easy to identify, as they look like the corners of most buildings, or a corner of a piece of paper.
A right angle measures exactly \begin{align*}90^\circ\end{align*}.
Right angles are usually marked with a small square. When two lines, two segments, or two rays intersect at a right angle, we say that they are perpendicular. The symbol \begin{align*}\perp\end{align*} is used for two perpendicular lines.
Below is an example of two perpendicular rays. The small square inside the vertex shows that the rays meet at a right angle:
- Right angles are marked with a small ___________________ in the vertex.
- Perpendicular lines intersect at a ___________________ angle.
- The symbol for perpendicular looks like __________.
Reading Check:
Mark the right angles in the three pictures above.
Construction: Perpendicular Bisector
The following steps show you how to construct a line that is both...
- Perpendicular to the given line segment (in other words, it intersects the segment at a \begin{align*}90^\circ\end{align*} angle) and
- A bisector (in other words, it cuts the line segment exactly in half)
Remember, for a geometric construction, you only use a compass and a straightedge. No rulers or protractors are allowed!
http://www.mathopenref.com/constbisectline.html
Reading Check:
1. True or False: A perpendicular bisector intersects the line segment at a \begin{align*}180^\circ\end{align*} angle.
2. Fill in the blank: When you bisect a line segment, you cut it exactly at its ____________________________________________.
Other Types of Angles: Acute, Obtuse, and Straight Angles
An acute angle measures between \begin{align*}0^\circ\end{align*} and \begin{align*}90^\circ\end{align*}. These are some examples of acute angles:
An obtuse angle measures between \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*}. Below are examples of obtuse angles:
- An ___________________________ angle measures between \begin{align*}0^\circ\end{align*} and \begin{align*}90^\circ\end{align*}.
- An ___________________________ angle measures between \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*}.
A straight angle measures exactly \begin{align*}180^\circ\end{align*}. These angles are easy to spot since they look like straight lines.
You can use this information to classify any angle you see.
Reading Check:
Label each angle as acute, obtuse, right, or straight:
______________ _____________ _____________ _____________
Angle Addition Postulate
The measure of any angle can be found by adding the measures of the smaller angles that comprise it.
In the diagram below, if you add \begin{align*}m\angle ABC\end{align*} and \begin{align*}m\angle CBD\end{align*}, you will get \begin{align*}m\angle ABD\end{align*}:
- The Angle Addition Postulate tells us to _______________ the measures of the smaller angles that make up a larger angle.
Use the Angle Addition Postulate just as you did the Segment Addition Postulate to identify the way different angles combine.
Example 1
What is \begin{align*}m \angle QRT\end{align*} in the diagram below?
You can see that \begin{align*}m\angle QRS\end{align*} is \begin{align*}15^\circ\end{align*}. You can also see that \begin{align*}m\angle SRT\end{align*} is \begin{align*}30^\circ\end{align*}. Using the Angle Addition Postulate, you can add these values together to find the total measure of \begin{align*}\angle QRT\end{align*}:
\begin{align*}m \angle QRS = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*} and \begin{align*}m \angle SRT = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
So \begin{align*}m \angle QRS + m\angle SRT = 15^\circ + 30^\circ = 45^\circ\end{align*}
Therefore, \begin{align*}m\angle QRT\end{align*} is \begin{align*}45^\circ\end{align*}.
Example 2
What is \begin{align*}m \angle LMN\end{align*} in the diagram below given \begin{align*}m \angle LMO = 85^\circ\end{align*} and \begin{align*}m \angle NMO = 53^\circ\end{align*}?
To find \begin{align*}m \angle LMN\end{align*}, you must subtract \begin{align*}m \angle NMO\end{align*} from \begin{align*}m \angle LMO\end{align*}:
If \begin{align*}m\angle LMO = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*} and \begin{align*}m \angle NMO = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*},
Then \begin{align*}m \angle LMO - m \angle NMO = 85^\circ - 53^\circ = 32^\circ\end{align*}
So \begin{align*}m \angle LMN = 32^\circ\end{align*}.
Reading Check:
1. Find the measure of angle \begin{align*}\angle WER\end{align*} in the diagram below:
2. Samantha is trying to find the measure of angle \begin{align*}\angle ABD\end{align*} in the figure below. Can you identify Samantha's mistake?
Congruent Angles
You already know that congruent line segments have exactly the same length. You can also apply the concept of congruence to other geometric figures: when angles are congruent, they have exactly the same measure. They may point in different directions, have different side lengths, have different names or other attributes, but their measures will be equal.
- Congruent angles have the _____________________ measure.
When writing that two angles are congruent, we use the congruent symbol:
\begin{align*}\angle ABC \cong \angle XYZ\end{align*}
Alternatively, the symbol \begin{align*}m \angle ABC\end{align*} refers to the measure of \begin{align*}\angle ABC\end{align*}, so we could write \begin{align*}m \angle ABC = m \angle XYZ\end{align*} and that has the same meaning as \begin{align*}\angle ABC \cong \angle XYZ\end{align*}.
You may notice then, that numbers (such as measurements) are equal while objects (such as angles and segments) are congruent.
- Numbers are equal while objects are _____________________________.
When drawing congruent angles, you use an arc in the middle of the angle to show that two angles are congruent. If two different pairs of angles are congruent, use one set of arcs for one pair, then two for the next pair and so on:
\begin{align*}\angle ABC \cong \angle XYZ\end{align*} (one arc)
\begin{align*}\angle K \cong \angle L\end{align*} (one arc)
\begin{align*}\angle M \cong \angle N\end{align*} (two arcs)
Example 3
The two angles shown below are congruent:
What is the measure of each angle?
Start by setting the angles equal to each other:
\begin{align*}5x+7 &= 3x+23\\ 5x-3x &= 23-7\\ 2x &= 16\\ x &= 8\end{align*}
So, the value of \begin{align*}x\end{align*} is 8. You are not done, however, because the question asks you for the measure of each angle. Use the value of \begin{align*}x\end{align*} to find the measure of one of the angles in the problem (you can choose either angle because we know the angles are congruent):
\begin{align*}m\angle ABC &= 5x+7\\ &= 5(8)+ 7\\ &= 40 + 7\\ &= 47\end{align*}
Since we know \begin{align*}m\angle ABC = m\angle XYZ\end{align*}, both angles measure \begin{align*}47^\circ\end{align*}.
Note: If you wanted to check your work, try substituting the value of \begin{align*}x\end{align*} into the other angle:
\begin{align*}m\angle XYZ &= 3x+23\\ &= 3(\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;})+ 23\\ &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}+ 23\\ &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
Did you get the same angle measure?
Construction: Copying A Congruent Angle
http://www.mathopenref.com/constcopyangle.html
Reading Check:
1. Circle the correct answer choice: When two angles are congruent, they have the same (direction / degree / measure).
2. Fill in the blank: When you are constructing a congruent angle, you use the _____________ to measure how open the angle is.
Angle Bisectors
An angle bisector divides an angle into two congruent angles, each having a measure of exactly half of the original angle.
Angle Bisector Postulate
Every angle has exactly one bisector.
Do you remember what a bisector is?
A bisector cuts something in half.
So an angle bisector cuts an angle in half into two congruent angles.
Reading Check:
1. How many angle bisectors can one angle have?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. In your own words, describe the angles that are formed by an angle bisector.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Construction: Angle Bisector
http://www.mathopenref.com/constbisectangle.html
Reading Check:
True or false: The first mark you make when bisecting an angle is an arc that crosses both rays of the angle.
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