 Lesson plan

# Lesson 5: Construction Techniques 3: Perpendicular Lines and Angle Bisectors

teaches Alabama State Standards Geo-30.
teaches Alabama State Standards Geo-29.b.
teaches Alabama State Standards Geo-29.a.
teaches Alabama State Standards Geo-29.
teaches Arizona State Standards G.G-CO.D.12
teaches Arizona State Standards G.G-CO.A.1
teaches Common Core State Standards MP3 http://corestandards.org/Math/Practice/MP3
teaches Common Core State Standards MP5 http://corestandards.org/Math/Practice/MP5
teaches Common Core State Standards HSG-CO.D.12 http://corestandards.org/Math/Content/HSG/CO/D/12
teaches Common Core State Standards MP1 http://corestandards.org/Math/Practice/MP1
teaches Common Core State Standards MP7 http://corestandards.org/Math/Practice/MP7
teaches Georgia State Standards MGSE9-12.G.CO.12.
teaches Georgia State Standards MGSE9-12.G.CO.1.
teaches Kansas State Standards G.CO.11.
teaches Minnesota State Standards 9.3.2.5.
teaches Minnesota State Standards 9.3.2.1.
teaches Ohio State Standards G.CO.12.
teaches Pennsylvania State Standards CC.2.3.HS.A.4.

# Lesson 5: Construction Techniques 3: Perpendicular Lines and Angle Bisectors

In this lesson, students learn two constructions:

• a line perpendicular to a given line through a point on the line
• an angle bisector

For the perpendicular line construction, students rely on their experience with the perpendicular bisector construction. The angle bisector construction is then connected to the perpendicular line construction with the observation that constructing a perpendicular line is the same as bisecting a straight angle. Students make use of structure when they decide how to apply what they already know about constructions to construct perpendicular lines and angle bisectors (MP7). Students are likely to struggle to do so; this is an opportunity to encourage them to persevere in solving problems (MP1).

There is a significant connection between the angle bisector and the perpendicular bisector in triangles that is made in this lesson and built on in the next unit. For isosceles triangles, in particular, the angle bisector of the vertex between the congruent sides is the same as the perpendicular bisector of the side opposite that vertex. This connection is essential for proving that the perpendicular bisector and the set of points equidistant to 2 given points are the same set.

If students have ready access to digital materials in class, they can choose to perform all construction activities with the GeoGebra Construction tool accessible in the Math Tools or available at https://www.geogebra.org/m/VQ57WNyR.

Lesson overview

• 5.1 Warm-up: Two Circles (5 minutes)
• 5.2 Activity: Make It Right (10 minutes)
• Digital applet in this activity
• 5.3 Activity: Bisect This (20 minutes)
• Digital applet in this activity
• Includes "Are you Ready for More?" extension problem
• Lesson Synthesis
• 5.4 Cool-down: Bisect That (5 minutes)

Learning goals:

• Construct a line that’s perpendicular to a given line through a given point on the line.
• Construct an angle bisector.

Learning goals (student facing):

• Let’s use tools to solve some construction challenges.

Learning targets (student facing):

• I can construct a line that is perpendicular to a given line through a point on the line.
• I can construct an angle bisector.

Required materials:

• Geometry toolkits

Glossary:

• angle bisector - A line through the vertex of an angle that divides it into two equal angles. • Access the complete Geometry Course glossary.

Standards:

• This lesson builds on the standard: CCSS.HSG-CO.A.1MS.G-CO.1MO.G.CO.A.1
• This lesson builds towards the standards: CCSS.HSG-CO.C.9MS.G-CO.9CCSS.HSG-CO.D.12MS.G-CO.12CCSS.HSG-CO.D.13MS.G-CO.13MO.G.CO.C.8MO.G.CO.D.11

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