# Lesson 6

Construction Techniques 4: Parallel and Perpendicular Lines

• Let’s use tools to draw parallel and perpendicular lines precisely.

### 6.1: Math Talk: Transformations

Each pair of shapes is congruent. Mentally identify a transformation or sequence of transformations that could take one shape to the other.

### 6.2: Standing on the Shoulders of Giants

Here is a line $$m$$ and a point $$C$$ not on the line. Use straightedge and compass tools to construct a line perpendicular to line $$m$$ that goes through point $$C$$. Be prepared to share your reasoning.

1. The line segment $$AB$$ has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment $$AB$$. How far is this new point from $$A$$?

2. We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment $$AB$$. How far is this new point from $$A$$?

3. If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?

### 6.3: Parallel Constructions Challenge

Here is a line $$m$$ and a point $$C$$ not on the line. Use straightedge and compass moves to construct a line parallel to line $$m$$ that goes through point $$C$$

### Summary

When we write the instructions for a construction, we can use a previous construction as one of the steps. We now know 2 new constructions that are made up of a sequence of moves.

• Perpendicular lines are lines that meet at a 90 degree angle.
• Parallel lines are lines that don’t intersect. One way to make parallel lines is to draw 2 lines perpendicular to the same line.

### 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.