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# Additional Properties of Triangles

Lesson 9 of 11

## Objective: SWBAT prove theorems involving the properties of triangles using Geometer’s Sketchpad

#### Do Now

*7 min*

In today's Do Now, students define the terms **midpoint of a segment** and **perpendicular bisector of a segment**. After they define the terms they construct a segment and its perpendicular bisector using a compass and a straightedge.

During this lesson we will use Geometer's Sketchpad to explore this construction. While my students are working on the Do Now, I hand out the laptops, ask students to sign on, and to open Geometer's Sketchpad.

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#### Mini-Lesson

*5 min*

When we use Geometer's Sketchpad, I often let the students play a little bit. In today's Mini-Lesson, we begin with about 3 minutes of unstructured time to explore the program. Since my students have used this program in earlier lessons, they know how to do a few things. Usually, it takes some students a few minutes to get their Sketchpad-on. I find things work better when I give students this time to explore (or to practice) before starting the activity.

As they explore, I start to make some suggestions. I display a practice sketch and I encourage students to see if they can construct the midpoint of a segment and using the midpoint to construct a perpendicular bisector for the segment. I also remind students that they should know how to measure segments and angles. All of these skills will be useful during today's Sketchpad activity.

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#### Activity

*28 min*

I like my students to work in pairs on Sketchpad activities, but I ask each student to complete end submit an Activity Sheet.

For each part of Additional Properties of Triangles Activity, students are required to create a new page in their Sketchpad document. On each page students work together to create a dynamic construction modeling a theorem or a property of triangles (**MP4, MP5**). As they explore the construction, I encourage them to discuss the theorems with their partners. For each construction, each student should write answers to questions and explain why their answers are correct (**MP3**).

As my students are working, I circulate around the room and help students who need assistance or who have a question. Some will need help finding the commands on the program. When possible, I like to have other students help each other with using Sketchpad.

Throughout the activity, I have mini discussions with students. I ask students to stop for a second and we go over certain questions. This helps me see how my students are doing, and to give them guidance with their investigation. Today, I plan to focus on Part A Question 2 and Part B Question 2 as checkpoints.

At the end of the today's activity, I will collect the activity sheet. I will not check the .gsp files, but I have asked the students to make sketches on their worksheets.

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#### Summary

*5 min*

At the end of the lesson, I have my students add the theorems that we have covered over the last few days to their theorem list in their notebook.

These theorems are:

- The mid-segments of triangles are half the length of and parallel to the third side of the triangle.
- The medians of a triangle intersect in a point.
- The median of the vertex of an isosceles triangle is the perpendicular bisector of the base and the angle bisector of the vertex.
- The base angles in an isosceles triangle are congruent.

Students will use these theorems in later lessons to write proofs.

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- UNIT 1: Preparing for the Geometry Course
- UNIT 2: Geometric Constructions
- UNIT 3: Transformational Geometry
- UNIT 4: Rigid Motions
- UNIT 5: Fall Interim Assessment: Geometry Intro, Constructions and Rigid Motion
- UNIT 6: Introduction to Geometric Proofs
- UNIT 7: Proofs about Triangles
- UNIT 8: Common Core Geometry Midcourse Assessment
- UNIT 9: Proofs about Parallelograms
- UNIT 10: Similarity in Triangles
- UNIT 11: Geometric Trigonometry
- UNIT 12: The Third Dimension
- UNIT 13: Geometric Modeling
- UNIT 14: Final Assessment

- LESSON 1: Triangle Inequality Theorem Investigation
- LESSON 2: Side-Side-Side
- LESSON 3: Angles of Triangles
- LESSON 4: Relationships between Angles and Sides of a Triangle
- LESSON 5: Applying Triangle Angle Theorems
- LESSON 6: Side-Angle-Side
- LESSON 7: Angle-Side-Angle
- LESSON 8: CPCTC
- LESSON 9: Additional Properties of Triangles
- LESSON 10: Properties of Isosceles Triangles
- LESSON 11: Properties of Isosceles Triangles Alternative Lesson with Dynamic Geometry Software