# Construction Techniques 3: Perpendicular Lines and Angle Bisectors

# Construction Techniques 3: Perpendicular Lines and Angle Bisectors

##### Problem 1

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

2) This diagram is a straightedge and compass construction. Select **all** true statements. Write each corresponding letter in the answer box and separate letters with commas.

a) Line $EF$ is the bisector of angle $BAC$. $\quad\quad$ b) Line $EF$ is the perpendicular bisector of segment $BA$.

c) Line $EF$ is the perpendicular bisector of segment $AC$. $\quad\quad$ d) Line $EF$ is the perpendicular bisector of segment $BD$.

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

##### Problem 3

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.

##### Problem 4

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

##### Problem 5

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 $BC$.

##### 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.

6) Is point $M$ closer to point $A$, closer to point $B$, or the same distance from both points?

7) Explain how you know.

##### Problem 7

8) 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.)

##### Problem 8

9) 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 $CD$.