1.5: Relationships with Triangles
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Midsegments |
Start Perpendicular Bisectors and Angle Bisectors in Triangles Investigation 5-1 Investigation 5-2 |
Finish Perpendicular Bisectors and Angle Bisectors in Triangles Investigation 5-3 Investigation 5-4 |
Quiz 1 Start Medians and Altitudes in Triangles Investigation 5-5 |
Finish Medians and Altitudes in Triangles Investigation 5-6 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Inequalities in Triangles |
Quiz 2 \begin{align*}^\star\end{align*}Start Extension: Indirect Proof |
\begin{align*}^\star\end{align*}Finish Extension: Indirect Proof | \begin{align*}^\star\end{align*}Quiz 3 | Review Chapter 5 |
Day 11 | Day 12 | Day 13 | ||
Review Chapter 5 | Chapter 5 Test |
Finish Chapter 5 Test (if needed) Start Chapter 6 |
Midsegments
Goal
This lesson introduces students to midsegments and the properties they hold.
Vocabulary.
This lesson begins a chapter that is full of vocabulary and new types of line segments. As a new line segment is learned, have students write each one with its definition and a picture in a self-made table. By the end of this chapter, students should have: midsegment, perpendicular bisector, angle bisector, median, and altitude. You can also have students draw these line segments in acute, right and obtuse triangles. Make sure to include the appropriate labeling and congruence statements for each line segment within each triangle as well.
Notation Note
Review with students the difference between a line segment, \begin{align*}\overline{NM}\end{align*} and its distance \begin{align*}NM\end{align*}. These notations will be used frequently in this chapter.
Teaching Strategies
Stress the properties of midsegments to students and make sure they understand the definition of a midsegment before moving on to the next section. Each segment in a triangle is very similar, so students tend to get them mixed up. A midsegment is unique because it connects two midpoints.
Examples 3-5 investigate the properties of a midsegment in the coordinate plane. Give students these examples without the solutions and have them work in pairs to arrive at the Midsegment Theorem on their own. At the completion of Example 5, ask students if they notice any similarities between the slopes of \begin{align*}\overline{NM}\end{align*} and \begin{align*}\overline{QO}\end{align*} and the lengths of \begin{align*}NM\end{align*} and \begin{align*}QO\end{align*}. Explain that their findings are the Midpoint Theorem.
Discuss that a midsegment is both parallel and half the length of the third side. Stress to students that if a line is parallel to a side in a triangle that does not make it a midsegment. The parallel line must also connect the midpoints, pass through the midpoints, or cut the sides it passes through in half. Go over all of these different ways to state what a midpoint and midsegment are.
If you have access to an LCD display screen (in the classroom) or a computer lab, use the website http://www.mathopenref.com/trianglemidsegment.html (in the FlexBook) to play with midsegments within a triangle. It is a great resource to help students to better understand the Midsegment Theorem.
Additional Example: \begin{align*}B, D, F\end{align*}, and \begin{align*}H\end{align*} are midpoints of \begin{align*}\Delta ACG\end{align*} and \begin{align*}\Delta CGE\end{align*}. \begin{align*}BH = 10\end{align*} Find \begin{align*}CG\end{align*} and \begin{align*}DF\end{align*}.
Solution: \begin{align*}\overline{BH}\end{align*} is the midsegment of \begin{align*}\Delta ACG\end{align*} that is parallel to \begin{align*}\overline{CG}\end{align*}. \begin{align*}\overline{DF}\end{align*} is the midsegment of \begin{align*}\Delta CGE\end{align*} that is parallel to \begin{align*}\overline{CG}\end{align*}. Therefore, \begin{align*}BH = DF = 10\end{align*} and \begin{align*}CG = 2 \cdot 10 = 20\end{align*}.
Perpendicular Bisectors and Angle Bisectors in Triangles
Goal
Students will apply perpendicular bisectors and angle bisectors to triangles and investigate their properties.
Teaching Strategies
Review the constructions of a perpendicular bisector and angle bisector (Review Queue #1 and #2). When going over #3a ask students if the line that bisects the line segment is a perpendicular bisector (it is not, it is just a segment bisector). Explain that the markings must look like the ones in #4 to be a perpendicular bisector.
Investigation 5-1 guides students through the properties of a perpendicular bisector before placing it in a triangle. Show students that \begin{align*}\Delta ACB\end{align*} is an isosceles triangle (in the description of the Perpendicular Bisector Theorem) which reinforces the fact that \begin{align*}C\end{align*} is equidistant from the endpoints of \begin{align*}\overline{AB}\end{align*}. Explain the difference between the Perpendicular Bisector Theorem and its converse. You can also have the students put the theorems into a biconditional statement.
Investigation 5-2 places the perpendicular bisectors in a triangle. This activity should be done individually, while you show students what to do. You will need to circle around the classroom to make sure students understand step 2. Then, students should be able to do step 3 on their own. Do step 4 as a class to make sure that every student understands that the circle drawn will pass through every vertex of the triangle.
Here, two new words are introduced: circumscribe and inscribe. If students have hard time remembering their definitions use their Latin roots. Circum = around and In = inside or interior. Scribe = draw or write.
The angle bisectors are also first introduced with one angle. Investigation 5-3 explores the property of one angle bisector and its relationship to the sides of the angle. This activity should be teacher-led while students are encouraged to follow along. In step 2, the folded line does not have to be a perpendicular bisector, but just a perpendicular line through \begin{align*}D\end{align*}.
After Investigation 5-3 compare the properties of an angle bisector and a perpendicular bisector. Draw them next to each other with markings and draw a Venn diagram with the similarities and differences. If this lesson takes more than one day, this could be a good warm-up.
After analyzing the pictures, lead students towards the conclusion that \begin{align*}B\end{align*} is equidistant from the endpoints of \begin{align*}\overline{AC}\end{align*} (in picture 1) and \begin{align*}B\end{align*} is equidistant from the sides of \begin{align*}\angle ADC\end{align*}. Then, to translate a perpendicular bisector into a triangle, the point where they all intersect would be equidistant from the endpoints of all the line segments, which are the vertices of the triangle. The point of intersection of the angle bisectors would be equidistant from all the sides of the angles which are also the sides of the triangle.
Just like with Investigation 5-2, lead students through this activity. Make sure every student understands how to fold the patty paper to create an angle bisector (step 2). Once they make all the angle bisectors, do step 4 together to ensure that every student understands that the circle will pass through the sides of the circle.
Students can get the properties of perpendicular bisectors and angle bisectors within triangles confused. One way to help students remember which is which show them that the point of intersection of the perpendicular bisectors can outside the triangle so the circle would go around the outside the circle (circumscribe). The point of intersection of the angle bisectors is always inside the triangle so the circle will always be inside the triangles (inscribed).
The points of concurrency of the perpendicular bisectors, angle bisectors and altitudes were intentionally left out of the Basic Geometry FlexBook to avoid confusion and to encourage students to focus on the theorems and properties of these lines. The Enrichment Teacher’s Edition FlexBook discusses the names of these points of concurrency if you would like to include them in your curriculum.
Medians and Altitudes in Triangles
Goal
Students will be introduced to medians and their point of intersection, the centroid. They will explore the properties of a centroid as well as learn how to construct an altitude.
Teaching Strategies
The median is now the third segment that passes through at least one midpoint. Make sure students understand the difference between a median, midsegment and a perpendicular bisector. Also, students may get the angle bisector confused with a median because sometimes it “looks like” (a fatal flaw in geometry) the angle bisector will pass through the opposite side’s midpoint. Students can never assume from a picture that the angle bisector and a median are the same. Discuss the cases when they are the same, this may alleviate some confusion. When a triangle is an isosceles triangle, the line segments are all the same when drawn from the vertex. Also when a triangle is equilateral, the line segments are all the same regardless of which vertex they are drawn from. The following picture might better illustrate this point:
Points \begin{align*}F\end{align*} and \begin{align*}D\end{align*} are the midpoints of the sides they are on.
\begin{align*}\overline{FD}\end{align*} is a midsegment
\begin{align*}\overline{ED}\end{align*} is a perpendicular bisector
\begin{align*}\overline{BG}\end{align*} is an angle bisector
\begin{align*}\overline{BD}\end{align*} is a median
\begin{align*}\overline{BH}\end{align*} is an altitude
With Investigation 5-6, encourage students to construct more than one altitude on the given obtuse triangle. While we did not explore the point of intersection for the altitudes, there is one. Have students arrive at this conclusion on their own, while constructing the other altitudes in \begin{align*}\Delta ABC\end{align*}. While students are performing the constructions, encourage them to turn their paper around so that the side they are making the altitude perpendicular to is horizontal. This will make the process easier.
At the end of this lesson, compare all of the line segments within triangles. Use the table below. The answers are filled in, but draw on the board without answers and generate the answers as a class. You can also use the picture above to help students determine the properties of the line segments.
Pass through the midpoint(s)? | Pass through a vertex? | Perpendicular? | Properties of the point of intersection | |
---|---|---|---|---|
Midsegment | Yes, two. | No | No | No point of intersection |
Angle Bisector | No | Yes | No | Equidistant from the sides of a triangle; inscribed circle. |
Perpendicular Bisector | Yes | No | Yes | Equidistant from the vertices of a triangle; circumscribed circle. |
Median | Yes | Yes | No | The centroid; the “center of gravity” of a triangle. Also it splits the medians into \begin{align*}\frac{2}{3} - \frac{1}{3}\end{align*} pieces. |
Altitude | No | Yes | Yes | None |
Check and recheck that students understand these five line segments before moving on. It is very common for students to get the definitions and properties confused.
Inequalities in Triangles
Goal
The purpose of this lesson is to familiarize students with the angle inequality theorems and the Triangle Inequality Theorem and the SAS and SSS Inequality Theorems.
Teaching Strategies
Students have probably figured out the Triangle Inequality Theorem but not actually put it into words. Ask the class if they can make a triangle out of the lengths 3 in, 5 in, and 9 in. You can give each student a few pieces of dry spaghetti and have them break the pieces so that they are the lengths above and then attempt to make a physical model. They will discover that it is impossible. Then, tell the class to break off 1 in of the 9 in piece and try again. Again, this will not work. Finally, tell them to break off another \begin{align*}\frac{1}{2}\end{align*}-inch and try a third time. This time it will work. Analyze each set of numbers. You could also have students do this a fourth time and make the longest piece either 6 or 7 inches. They will still be able to make a triangle.
3, 5, 9 \begin{align*}\rightarrow\end{align*} No triangle
3, 5, 8 \begin{align*}\rightarrow\end{align*} No triangle
3, 5, 7.5 \begin{align*}\rightarrow\end{align*} Yes!
3, 5, 6 \begin{align*}\rightarrow\end{align*} Yes!
Guide students towards the Triangle Inequality Theorem. Example 4 explores the possible range of the third side, given two sides. Explain to students that this third side can be the shortest side, the longest side or somewhere in-between. We have no idea, so we have to propose a range of lengths that the third side could be. Have students shout out possible lengths of the third side and place them in a table. Then, show them the way to write the lengths as a compound inequality.
Example 4 leads students into the SAS Inequality Theorem, which compares two triangles where two sides are the same length and the included angles are different measurements. We know, from Chapter 4, that if the included angles are congruent, then the triangles would be congruent, but in this case, we know that one is bigger than the other. Logically, it follows that the triangle with the bigger included angle will have the longer opposite side. This is a very wordy theorem; it might help to explain using the symbols and picture in the text. This theorem is also called the Hinge Theorem.
The SSS Inequality Theorem is the converse of the SAS Inequality Theorem. Now we know that two sides are congruent and the third sides are not. It follows that the angle opposite the longer side is going to be larger than the same angle in the other triangle. Example 6 is a good example of how this theorem works. You can also reverse the question and ask: If \begin{align*}m \angle 1 > m \angle 2\end{align*}, what can we say about \begin{align*}XY\end{align*} and \begin{align*}XZ\end{align*}?
Extension: Indirect Proof
Goal
Students should be able to understand how an indirect proof is organized and executed.
Teaching Strategy
An indirect proof is a powerful reasoning tool that students might find useful outside of mathematics. Ask students what professions they think would use indirect proofs (also called proof by contradiction). Examples are lawyers (disproving innocence/guilt), doctors (disproving diagnosis), crime scene investigators (collecting evidence and trying to prove or disprove).
Additional Example: Prove \begin{align*}\sqrt{15} \neq 4\end{align*}.
Solution: Assume \begin{align*}\sqrt{15} = 4\end{align*}
Squaring both sides, we get \begin{align*}15 = 16\end{align*}.
But \begin{align*}15 \neq 16\end{align*}, therefore, \begin{align*}\sqrt{15} \neq 4\end{align*}.