# Lesson 5

Construction Techniques 3: Perpendicular Lines and Angle Bisectors

### Lesson Narrative

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

### Learning Goals

Teacher Facing

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

### Student Facing

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

### 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.

Building On

Building Towards

### Glossary Entries

• angle bisector

A line through the vertex of an angle that divides it into two equal angles.

• circle

A circle of radius $$r$$ with center $$O$$ is the set of all points that are a distance $$r$$ units from $$O$$

To draw a circle of radius 3 and center $$O$$, use a compass to draw all the points at a distance 3 from $$O$$.

• conjecture

A reasonable guess that you are trying to either prove or disprove.

• inscribed

We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

• line segment

A set of points on a line with two endpoints.

• parallel

Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

• perpendicular bisector

The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it.