Question

1. the diagram below shows the construction of the perpendicular bisector of \\( \overline{ae} \\)

which statement is not true?

1. \\( ac = ce \\)
2. \\( ce = \frac{1}{2} ae \\)
3. \\( ac = 2 ae \\)
4. \\( ac + ce = ae \\)

1. the diagram below illustrates the construction of \\( \overleftrightarrow{pb} \\) parallel to \\( \overleftrightarrow{rq} \\) through point \\( p \\)

which statement justifies this construction?

1. \\( m\angle 1 = m\angle 2 \\)
2. \\( m\angle 1 = m\angle 3 \\)
3. \\( \overline{pr} \cong \overline{rq} \\)
4. \\( \overline{pb} \cong \overline{rq} \\)

Explanation:

Problem 37 (Perpendicular Bisector)

Step1: Recall Perpendicular Bisector Properties

A perpendicular bisector of a segment divides it into two equal parts, so \( AC = CB \) (since the line is the perpendicular bisector of \( \overline{AB} \)), and \( CB=\frac{1}{2}AB \) (because \( AC = CB \), so \( AB = AC + CB = 2CB \)). Also, \( AC + CB = AB \) (segment addition postulate).

Step2: Analyze Each Option

1. \( AC = CB \): True (perpendicular bisector divides segment into two equal parts).
2. \( CB=\frac{1}{2}AB \): True (since \( AB = AC + CB = 2CB \), so \( CB=\frac{AB}{2} \)).
3. \( AC = 2AB \): False (since \( AC=\frac{AB}{2} \), so \( AC \) is half of \( AB \), not twice).
4. \( AC + CB = AB \): True (segment addition postulate: the sum of two parts of a segment equals the whole segment).

Problem 38 (Parallel Line Construction)

Step1: Recall Parallel Line Construction (Corresponding Angles)

To construct a line parallel to another, we use corresponding angles (if corresponding angles are equal, lines are parallel). Here, \( \angle 1 \) and \( \angle 2 \) are vertical angles? No, wait—actually, the construction uses equal angles (like corresponding angles) to ensure parallelism. Let's analyze each option:

1. \( m\angle 1 = m\angle 2 \): \( \angle 1 \) and \( \angle 2 \) are vertical angles? Wait, no—actually, in the diagram, \( \angle 1 \) and \( \angle 2 \) are constructed to be equal (maybe alternate interior or corresponding). Wait, no—let's check the options:

1. \( m\angle 1 = m\angle 3 \): \( \angle 1 \) and \( \angle 3 \) are corresponding angles? If \( \angle 1 = \angle 3 \), then \( \overline{PR} \parallel \overline{MQ} \) (corresponding angles postulate). Wait, no—the line constructed is \( \overline{PR} \) parallel to \( \overline{MQ} \)? Wait, the question is "Which statement justifies this construction?"

Wait, the construction is of \( \overline{PR} \) parallel to \( \overline{MQ} \) through point \( P \). Let's re-express:

• Option 1: \( m\angle 1 = m\angle 2 \): \( \angle 1 \) and \( \angle 2 \) are vertical angles? No, maybe alternate interior. Wait, no—let's think again. The correct justification for constructing parallel lines is using equal corresponding angles (or alternate interior angles). If \( m\angle 1 = m\angle 3 \), that would be corresponding angles? Wait, no—\( \angle 1 \) and \( \angle 3 \): if \( \angle 1 = \angle 3 \), then lines are parallel (corresponding angles). Wait, but let's check the options:

Wait, the options are:

1. \( m\angle 1 = m\angle 2 \)
2. \( m\angle 1 = m\angle 3 \)
3. \( \overline{PR} \cong \overline{MQ} \) (length doesn't determine parallelism)
4. \( \overline{PR} \cong \overline{RQ} \) (length of segments doesn't determine parallelism)

So, for parallel lines, we need angle equality (corresponding/alternate interior). \( m\angle 1 = m\angle 3 \) would mean corresponding angles are equal, so lines are parallel. Wait, but let's confirm:

• Option 1: \( m\angle 1 = m\angle 2 \): \( \angle 1 \) and \( \angle 2 \) are vertical angles? No, vertical angles are equal, but that's not the reason for parallelism.
• Option 2: \( m\angle 1 = m\angle 3 \): If \( \angle 1 = \angle 3 \), then by corresponding angles postulate, \( \overline{PR} \parallel \overline{MQ} \).
• Options 3 and 4: Length of segments (\( \cong \)) doesn't imply parallelism (parallelism is about direction, not length).

So the correct justification is \( m\angle 1 = m\angle 3 \) (option 2)? Wait, no—wait the question is "Which statement justifies this construction?" Let's re-examine:

Wait, the construction is of \( \overline{PQ} \) parallel to \( \overline{RQ} \)? No, the diagram says "construction of \( \overline{PQ} \) parallel to \( \overline{RQ} \) through point \( P \)"? Wait, no—the original problem says "construction of \( \overline{PQ} \) parallel to \( \overline{RQ} \) through point \( P \)"? Wait, maybe a typo. Wait, the options include \( \overline{PR} \parallel \overline{MQ} \) or \( \overline{PQ} \parallel \overline{RQ} \). Wait, no—let's check the options again:

1. \( \overline{PR} \cong \overline{MQ} \) (length congruence, not parallelism)
2. \( \overline{PQ} \cong \overline{RQ} \) (length congruence, not parallelism)

So the key is angle equality. \( m\angle 1 = m\angle 3 \) (option 2) would mean corresponding angles are equal, so lines are parallel. Wait, but \( \angle 1 \) and \( \angle 3 \): if they are equal, then \( \ov…

Answer:

(Problem 37):

1. \( AC = 2AB \)