# Lesson 5

Construction Techniques 3: Perpendicular Lines and Angle Bisectors

### Problem 1

This diagram is a straightedge and compass construction of a line perpendicular to line $$AB$$ passing through point $$C$$. Explain why it was helpful to construct points $$D$$ and $$A$$ to be the same distance from $$C$$.

### Problem 2

This diagram is a straightedge and compass construction.

Select all true statements.

A:

Line $$EF$$ is the bisector of angle $$BAC$$.

B:

Line $$EF$$ is the perpendicular bisector of segment $$BA$$.

C:

Line $$EF$$ is the perpendicular bisector of segment $$AC$$.

D:

Line $$EF$$ is the perpendicular bisector of segment $$BD$$.

E:

Line $$EF$$ is parallel to line $$CD$$.

### Problem 3

This diagram is a straightedge and compass construction. $$A$$ is the center of one circle, and $$B$$ is the center of the other. A rhombus is a quadrilateral with 4 congruent sides. Explain why quadrilateral $$ACBD$$ is a rhombus.

### Solution

(From Unit 1, Lesson 4.)

### Problem 4

$$A$$, $$B$$, and $$C$$ are the centers of the three circles. Which line segment is congruent to $$HF$$?

A:

$$AB$$

B:

$$CD$$

C:

$$DF$$

D:

$$CB$$

### Solution

(From Unit 1, Lesson 4.)

### Problem 5

In the construction, $$A$$ is the center of one circle, and $$B$$ is the center of the other. Explain why segment $$EA$$ is the same length as segment $$BC$$.

### Solution

(From Unit 1, Lesson 2.)

### Problem 6

In this diagram, line segment $$CD$$ is the perpendicular bisector of line segment $$AB$$. Assume the conjecture that the set of points equidistant from $$A$$ and $$B$$ is the perpendicular bisector of $$AB$$ is true. Is point $$M$$ closer to point $$A$$, closer to point $$B$$, or the same distance from both points? Explain how you know.

### Solution

(From Unit 1, Lesson 3.)

### Problem 7

A sheet of paper with points $$A$$ and $$B$$ is folded so that $$A$$ and $$B$$ match up with each other.

Explain why the crease in the sheet of paper is the perpendicular bisector of segment $$AB$$. (Assume the conjecture that the set of points equidistant from $$A$$ and $$B$$ is the perpendicular bisector of segment $$AB$$ is true.)

### Solution

Here is a diagram of a straightedge and compass construction. $$C$$ is the center of one circle, and $$B$$ is the center of the other. Explain why the length of segment $$CB$$ is the same as the length of segment $$CD$$.