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# Relationships between Angles and Sides of a Triangle

Lesson 4 of 11

## Objective: SWBAT explain the relationship between angles and sides of a triangle.

#### Do Now

*5 min*

For the Do Now, I have students answer questions to review their prior knowledge about angles and measurement. Students define the terms, “acute angle” and “obtuse angle.” They then use protractors to draw an example of each type of angle. This helps me see their familiarity with using a protractor and measuring angles. The last question asks students to draw a segment, measured in centimeters. I walk around the room with a protractor to measure students’ angles.

I plan to review the definitions used in this worksheet during the Mini-Lesson, assuming my students should know them already.

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#### Mini-Lesson

*10 min*

The majority of questions in this lesson addressed are based on concepts students have explored previously. However, they may not have looked at the concepts in this way. Before students can investigate similarity, it is a good idea for teachers to see what their students know and can do in order to better inform their planning and teaching for the rest of the unit.

To begin the mini-lesson, I ask students, “What do you know about triangles?” I expect my students will answer with responses such as:

- three angles and sides
- the sum of the measures of the angles is equal to 180
^{o}

After the students share their ideas we will review terms, making sure that the following are covered if they were not introduced by the students: ** acute angles, obtuse angles, interior angles, exterior angles**.

At this time, I hand out the protractors and rulers. Often, I need to review how to measure angles with a protractor. I first ask students to decide if the angle is acute or obtuse. If the angle is acute, the students should use the smaller number on the protractor. If the angle is obtuse, the students should use the larger number on the protractor. It is also sometimes necessary to remind students how to measure with a ruler. I use centimeters for this activity because it is more precise and can be used to write measurements as decimals easily.

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#### Activity

*20 min*

In this Activity, students investigate the relationships between the sides and angles of a triangle and write explanations for what they have found out. Students work in pairs to complete questions 1 through 5 on the sheet “Relationships between Angles and Sides of a Triangle.” They work together to ensure the accuracy of their responses and to discuss their findings. We sometimes review how the term “opposite” is used in a triangle. See the answers below for expected student responses.

As they begin the activity, students first use a protractor to measure <CAB, <ABC, <BCA, <DAC, <FCB, and <CBE and a ruler to measure sides (segments) AB, CB, and AC in centimeters. They record their results in a chart. It is a good idea to measure the sides and angles of the triangles before you hand out the sheet to student to check for accuracy. I usually allow a tolerance of 2 degrees for angles and 0.1 cm for segments. The following measurements work best for the rest of the worksheet:

**Angles:**

m<CAB = 33^{o}, m<ABC= 60^{o}, m<BCA= 27^{o}, m<DAC= 147^{o}, m<FCB= 153^{o}, m<CBE= 120^{o}

**Segments**: 5.6 cm, 6.7 cm, 10.6 cm

As the students work, I circulate while they are working and question them as they work, "What information is necessary to answer questions 2 and 3?" After about 15 minutes, we go over the answers.

**Teacher's Notes about Worksheet Questions**:

- By measuring the angles and sides of the triangle, students can discover the relationship between them. Although the question only asks about the relationship between the largest angle and longest side, the teacher should question students about the relationships between the other angles and sides. When students understand the relationship between the angles and the sides, they will be able to put the sides in order given the measures of the angles and vice versa.

- Questions 4 and 5 help students better see properties of triangles and understand theorems they will use in later lessons.

**Possible Answers**:

1b. The side opposite from the largest interior angle is the longest.

2. The sum of the measure of the interior angles in a triangle is 180^{o}.

- The students may get a sum that is slightly less or slightly more than 180
^{o}. This provides a good opportunity to discuss human error and precision in measuring.

3. The sum of the measures of the exterior angles of a triangle is 360^{o}.

- This question is not a focus of the activity, however, it is included as a connection to the quadrilateral and circle units.

4b. The sum of the measures of an interior angle and it’s exterior angle is 180^{o}. Angles on a line are supplementary.

5b. The measure of an exterior angle is equal to the sum of the measures of the remote interior angles.

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#### Summary

*10 min*

**Discussion Prompt**: The triangle we looked at had three sides of different lengths. How can we order the angles if a triangle has two sides the same length? What about a triangle with three sides the same length?

There are two versions of the Exit Ticket. Some students will be able to answer the question without a copy of the drawing while some students may need a picture of the triangle to answer the questions. As I distribute the Exit Ticket I remind students to use information they are given and not just what the picture looks like.

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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