# Lesson 6Justifying ConstructionsPractice Understanding

## Jump Start

We examined this diagram in a previous Jump Start and wondered if we could claim if the two segments, and , were congruent.

How can you use triangle congruence criteria, such as ASA, SAS, or SSS, to prove that these two segments are congruent? (If you need help, your teacher has some hints to get you started.)

## Learning Focus

Justify construction strategies.

How can I explain why particular constructions (made only with lines and circular arcs) work?

How might I draw upon triangle congruence criteria and the definitions of rigid transformations in my explanations?

## Open Up the Math: Launch, Explore, Discuss

Compass and straightedge constructions can be justified using such tools as:

• definitions and properties of the rigid transformations

• identifications of the corresponding parts of congruent triangles

• conjectures about the attributes of sides, angles, and diagonals of special types of quadrilaterals

### 1.

Study the steps of the following procedure for constructing an angle bisector, and complete the illustration based on the descriptions of the steps.

Steps

Illustration

Using a compass, draw an arc (portion of a circle) that intersects each ray of the angle to be bisected, with the center of the arc located at the vertex of the angle.

Without changing the span of the compass, draw two arcs in the interior of the angle, with the center of the arcs located at the two points where the first arc intersected the rays of the angle.

With the straightedge, draw a ray from the vertex of the angle through the point where the last two arcs intersect.

Explain in detail why this construction works. It may be helpful to identify some congruent triangles or a familiar quadrilateral in the final illustration. You may also want to use definitions or properties of the rigid-motion transformations in your explanation. Be prepared to share your explanation with your peers.

### 2.

Study the steps of the following procedure for constructing a line perpendicular to a given line through a given point, and complete the illustration based on the descriptions of the steps.

Steps

Illustration

Using a compass, draw an arc (portion of a circle) that intersects the given line at two points, with the center of the arc located at the given point.

Without changing the span of the compass, locate a second point on the other side of the given line by drawing two arcs on the same side of the line, with the center of the arcs located at the two points where the first arc intersected the line.

With the straightedge, draw a line through the given point and the point where the last two arcs intersect.

Explain in detail why this construction works. It may be helpful to identify some congruent triangles or a familiar quadrilateral in the final illustration. You may also want to use definitions or properties of the rigid-motion transformations in your explanation. Be prepared to share your explanation with your peers.

### 3.

Study the steps of the following procedure for constructing a line parallel to a given line through a given point, and complete the illustration based on the descriptions of the steps.

Steps

Illustration

Using a straightedge, draw a line through the given point to form an arbitrary angle with the given line.

Using a compass, draw an arc (portion of a circle) that intersects both rays of the angle formed, with the center of the arc located at the point where the drawn line intersects the given line.

Without changing the span of the compass, draw a second arc on the same side of the drawn line, centered at the given point. The second arc should be as long or longer than the first arc and should intersect the drawn line.

Set the span of the compass to match the distance between the two points where the first arc crosses the two lines. Without changing the span of the compass, draw a third arc that intersects the second arc, centered at the point where the second arc intersects the drawn line.

With the straightedge, draw a line through the given point and the point where the last two arcs intersect.

Explain in detail why this construction works. It may be helpful to identify some congruent triangles or a familiar quadrilateral in the final illustration. You may also want to use definitions or properties of the rigid-motion transformations in your explanation. Be prepared to share your explanation with your peers.

Draw a line segment and a point not on the line. Using the line segment as one side of a parallelogram and the point as one of the vertices of the parallelogram, construct the parallelogram with a segment representing the height of the parallelogram constructed from one vertex perpendicular to the opposite side.

## Takeaways

Strategies I might use when justifying constructions:

These strategies will also be useful in future geometry work.

## Lesson Summary

In this lesson, we examined some standard constructions, such as bisecting an angle, constructing a line perpendicular to a given line through a given point, and constructing a line parallel to a given line through a given point. We found that we could explain why these constructions work based on properties of quadrilaterals, using corresponding parts of congruent triangles, or relying on the defining features and properties of rigid transformations.

## Retrieval

### 1.

This regular octagon is to be used as a decorative window in the front of a house. The carpenters framed in an opening for it. How many degrees would they need to rotate it to make the window fit into the opening eight different ways?

### 2.

Given:

Find the distance between point and point . Then find the slope of the line that passes through them.