# 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 \(C\).

### Solution

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### Problem 2

This diagram is a straightedge and compass construction.

Select **all** true statements.

Line \(EF\) is the bisector of angle \(BAC\).

Line \(EF\) is the perpendicular bisector of segment \(BA\).

Line \(EF\) is the perpendicular bisector of segment \(AC\).

Line \(EF\) is the perpendicular bisector of segment \(BD\).

Line \(EF\) is parallel to line \(CD\).

### Solution

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### 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

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(From Unit 1, Lesson 4.)### Problem 4

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

\(AB\)

\(CD\)

\(DF\)

\(CB\)

### Solution

### 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

### 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

### 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

### Problem 8

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 \(CD\).