- Look at how many endpoints each drawing has. - A ray has one endpoint and continues in one direction. - A line continues in both directions, while a segment stops at both ends.

1. Drawing a) extends in both directions, so it is a line. 2. Drawing b) starts at one endpoint and extends in one direction, so it is a ray. 3. Drawing c) has two endpoints, so it is a line segment.

a) line b) ray c) line segment

- Find the part of the drawing that stops at both ends. - For the ray, look at which end its arrow points away from.

1. The segment stops at points \(J\) and \(K\). 2. The ray starts at \(M\) and passes through \(N\), so its endpoint is \(M\).

a) \(J\) and \(K\) b) \(M\)

- The first letter in ray notation names the endpoint. - The second letter must lie in the direction the ray travels. - Check which drawing continues past \(B\), not past \(A\).

1. The notation \(\overrightarrow{AB}\) names a ray with endpoint \(A\) that passes through \(B\). 2. Drawing b) starts at \(A\) and continues through \(B\), so b) is correct.

b)

- Look for two lines that have the same direction. - Do those two lines get closer together or cross?

1. Lines \(g\) and \(h\) both run horizontally. 2. They stay the same distance apart and never meet, so they are parallel. Line \(k\) crosses them.

\(g\) and \(h\) are parallel because they run in the same direction and stay the same distance apart.

- Follow the lines from left to right. - Compare how each line rises or falls across the squares.

1. Line \(a\) rises one grid square for each square it moves right. 2. Line \(b\) has the same direction, while line \(c\) slopes the other way.

Line \(b\) is parallel to \(a\). Both rise one square for each square to the right; \(c\) slants the other way.

- Look at the corners where the lines cross. - Which crossing makes an angle like the corner of a square?

1. Line \(g\) is horizontal and line \(h\) is vertical. 2. They form right angles where they cross, so they are perpendicular. Line \(t\) is slanted and does not meet either at a right angle.

Lines \(g\) and \(h\) are perpendicular.

- Compare the crossing to the corner of a square. - A right angle does not have to stand upright. Do these lines make one?

1. The lines meet to form four equal square corners. 2. Each angle at their intersection is a right angle, so the lines are perpendicular even though neither is horizontal.

Yes. Lines \(m\) and \(n\) meet at right angles.

- Trace a straight path to the right from \(A\). - Which marked point could lie on that path? - Think about where the arrow would be on your ray.

1. \(B\) is on the same grid row as \(A\), to its right. \(C\) is above that row. 2. The ray starts at \(A\), passes through \(B\), and continues past \(B\). Its name is \(\overrightarrow{AB}\). 3. Its arrow points right, away from \(A\).

Through \(B\); \(\overrightarrow{AB}\); it continues right past \(B\).

- Imagine joining \(P\) to each of the two upper points. - Compare the angles each possible segment would make with line \(g\).

1. The shortest path between two parallel lines is perpendicular to them. 2. The segment from \(P\) to \(Q\) goes straight across the gap and makes a right angle with both lines. 3. Joining \(P\) to \(R\) would give a slanted, longer path.

Join \(P\) to \(Q\). That segment would meet both parallel lines at right angles.

- Count the squares from one line to the other along each colored segment. - Check the corners where the segments meet the two lines. - Think about what it means for two lines to stay parallel.

1. Both segments meet the parallel lines at right angles. 2. Each segment crosses five grid squares between the two vertical lines, so both are valid distance measurements. 3. The two measurements agree: the lines are \(5\) units apart in both places.

Yes. Both segments meet the lines at right angles, and both measure the same \(5\)-unit gap.

- Do the staff lines stay the same distance apart? - Think about the angle formed where a vertical line meets a horizontal line.

1. The five staff lines stay the same distance apart and do not intersect, so they are parallel. 2. A vertical line is perpendicular to the horizontal staff lines, so right angles are formed at every intersection.

a) The five lines are parallel. b) Right angles are formed at the intersections.

- Look closely at the arrows or bar above the point names. - Recall the symbol for parallel lines. - Distinguish among lines, rays, and segments.

1. The notation \(\overleftrightarrow{AB}\) names the line through points \(A\) and \(B\). It extends forever in both directions. 2. The symbol \(\parallel\) means “is parallel to,” so line \(g\) is parallel to line \(h\). 3. The notation \(\overrightarrow{RS}\) names the ray with endpoint \(R\) that passes through \(S\), so \(k\) is that ray. 4. The notation \(\overline{PQ}\) names the segment with endpoints \(P\) and \(Q\), so \(s\) is that segment.

a) The line through points \(A\) and \(B\). b) Line \(g\) is parallel to line \(h\). c) \(k\) is the ray with endpoint \(R\) that passes through \(S\). d) \(s\) is the segment with endpoints \(P\) and \(Q\).

- Recall the symbol for perpendicular lines. - For a ray, the first letter names the endpoint. - Compare the notation for a line, a ray, and a segment.

1. A line through \(A\) and \(B\) is written \(\overleftrightarrow{AB}\), so \(m = \overleftrightarrow{AB}\). 2. The symbol for perpendicular lines is \(\perp\), so \(g \perp h\). 3. A ray with endpoint \(Z\) that passes through \(Y\) is written \(\overrightarrow{ZY}\), so \(r = \overrightarrow{ZY}\). 4. A segment with endpoints \(C\) and \(D\) is written \(\overline{CD}\), so \(s = \overline{CD}\).

a) \(m = \overleftrightarrow{AB}\) b) \(g \perp h\) c) \(r = \overrightarrow{ZY}\) d) \(s = \overline{CD}\)

- Decide whether each figure extends in both directions, in one direction, or not at all. - For a ray, the first letter names the endpoint. - Picture how you would draw each object with a ruler.

1. The notation \(\overleftrightarrow{AB}\) names the line through \(A\) and \(B\). It extends forever in both directions. The notation \(\overline{AB}\) names the segment with endpoints \(A\) and \(B\), so it has a fixed length. 2. Both \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are rays. The ray \(\overrightarrow{AB}\) starts at \(A\) and passes through \(B\). The ray \(\overrightarrow{BA}\) starts at \(B\) and passes through \(A\).

a) \(\overleftrightarrow{AB}\) is a line that extends forever in both directions, while \(\overline{AB}\) is a segment with endpoints \(A\) and \(B\). b) \(\overrightarrow{AB}\) starts at \(A\), while \(\overrightarrow{BA}\) starts at \(B\). The two rays point in opposite directions along the same line.

- The distance between parallel lines is not measured along just any segment joining them. - Compare the direction of each segment with the direction of the parallel lines. - Look for the segment that makes a right angle with the parallel lines.

1. The distance between two parallel lines is measured along a segment perpendicular to both lines. 2. Segment \(s_2\) meets the parallel lines at right angles. Segments \(s_1\) and \(s_3\) are slanted relative to the parallel lines. Therefore \(s_2\) represents the distance between the parallel lines.

\(s_2\), because it is perpendicular to the two parallel lines.

- Parallel lines run in the same direction and never meet. - Look for pairs that form a right angle. - Check the horizontal, vertical, and diagonal lines.

1. Lines \(a\) and \(b\) are both horizontal, so \(a \parallel b\). 2. Line \(c\) is vertical, so it is perpendicular to both horizontal lines: \(a \perp c\) and \(b \perp c\). 3. Lines \(d\) and \(e\) cross at a right angle, so \(d \perp e\).

1. \(a \parallel b\) 2. \(a \perp c\), \(b \perp c\), and \(d \perp e\)

- Start with the outside edges of the triangle. - Look for a segment inside the triangle. - A point on a longer segment creates shorter segments with new endpoint pairs.

1. The three sides of the large triangle are \(\overline{AB}\), \(\overline{AC}\), and \(\overline{BC}\). 2. Point \(D\) divides \(\overline{BC}\) into \(\overline{BD}\) and \(\overline{DC}\). 3. The interior segment is \(\overline{AD}\). 4. The six distinct segments are \(\overline{AB}\), \(\overline{AC}\), \(\overline{BC}\), \(\overline{AD}\), \(\overline{BD}\), and \(\overline{DC}\).

The six segments are \(\overline{AB}\), \(\overline{AC}\), \(\overline{BC}\), \(\overline{AD}\), \(\overline{BD}\), and \(\overline{DC}\).

- Parallel segments run in the same direction. - The angle marks at \(A\) and \(D\) show right angles.

1. Segments \(\overline{AB}\) and \(\overline{CD}\) run in the same direction, so \(\overline{AB} \parallel \overline{CD}\). 2. The right-angle marks show that \(\overline{AD}\) is perpendicular to both \(\overline{AB}\) and \(\overline{CD}\).

\(\overline{AB} \parallel \overline{CD}\). Also, \(\overline{AD} \perp \overline{AB}\) and \(\overline{AD} \perp \overline{CD}\).

- A line has two opposite directions from point \(S\). - Use the given perpendicular relationship between \(\overrightarrow{SC}\) and line \(g\). - Check the angles involving ray \(\overrightarrow{SD}\) separately instead of assuming every visible angle is right.

1. Because \(\overrightarrow{SC}\) is perpendicular to line \(g\), it makes a right angle with each direction of line \(g\). 2. Ray \(\overrightarrow{SA}\) and ray \(\overrightarrow{SB}\) are the two opposite directions of line \(g\). 3. Therefore \(\angle ASC\) and \(\angle CSB\) are right angles. Ray \(\overrightarrow{SD}\) splits the angle between \(\overrightarrow{SC}\) and \(\overrightarrow{SB}\), so \(\angle CSD\) and \(\angle DSB\) are not right angles.

\(\angle ASC\) and \(\angle CSB\)

- Compare the angles formed where each pair of lines intersects. - Look for the panel that resembles a rotated cross with four right angles.

1. In panel a), the smaller angle between the lines is \(90^\circ\). 2. In panels b) and c), the angles formed are not right angles. Therefore, panel a) shows perpendicular lines.

The lines are perpendicular in panel a).

- Imagine extending both lines farther in each direction. - Parallel lines keep the same distance apart. - Do not require parallel lines to be horizontal or vertical.

1. Parallel lines have the same direction and do not meet. 2. In drawing c), both lines have the same slant, so they are parallel. 3. The lines in a) and b) have different directions and would meet if extended.

c)

- First compare the directions of \(g\) and \(h\). - Look at the angle where \(t\) crosses \(g\). - Imagine following \(t\) to the other crossing. What stays the same?

1. Lines \(g\) and \(h\) have the same direction, so they are parallel. 2. Line \(t\) meets \(g\) at a right angle, so they are perpendicular. 3. Since \(h\) is parallel to \(g\), line \(t\) also meets \(h\) at a right angle.

a) parallel b) perpendicular c) perpendicular

- Check whether \(h\) passes through \(P\). - Compare the direction of \(h\) with the segment's direction. - Both conditions have to hold in the same panel.

1. In a), \(h\) passes through \(P\), but is not parallel to \(\overline{QR}\). 2. In b), \(h\) is parallel to \(\overline{QR}\), but misses \(P\). 3. In c), \(h\) passes through \(P\) and is parallel to \(\overline{QR}\), so c) meets both requirements.

c)

- Compare each street with Maple, one at a time. - Parallel streets have the same direction. - Perpendicular streets meet at a right angle.

1. Oak and Maple have the same direction, so they are parallel. 2. Pine crosses Maple at a right angle, so they are perpendicular. 3. Cedar crosses Maple at a different angle, so it is neither parallel nor perpendicular to Maple.

a) Oak b) Pine c) Cedar

- Compare the number of endpoints in the two notations. - Ask what happens after point \(B\) in each object. - Having two points in common does not make two geometric objects identical.

1. \(\overline{AB}\) is a line segment, so it has two endpoints, \(A\) and \(B\). 2. \(\overrightarrow{AB}\) is a ray, so it has endpoint \(A\), passes through \(B\), and continues beyond \(B\). 3. The two objects contain the same segment from \(A\) to \(B\), but the ray continues farther, so they are not the same object.

Ava is incorrect. \(\overline{AB}\) stops at \(A\) and \(B\), while \(\overrightarrow{AB}\) starts at \(A\), passes through \(B\), and continues beyond \(B\).

- Separate the idea of a point lying on an object from being an endpoint of it. - A line continues in both directions. - For a ray name, the first letter names the endpoint.

1. Point \(Q\) is on line \(g\), so b) is true. 2. A line continues in both directions and has no endpoints, so a) is false. 3. Ray \(\overrightarrow{RS}\) starts at \(R\) and continues through \(S\), so \(R\) is its endpoint and c) is true. 4. Point \(S\) lies on the ray but is not its endpoint, so d) is false.

b), c)

- Check the through-point condition separately from the perpendicular condition. - One choice may satisfy only one of the two requirements. - The correct line must both contain \(P\) and make a right-angle turn from \(g\).

1. In a), line \(h\) passes through \(P\), but it is not perpendicular to \(g\). 2. In b), line \(h\) is perpendicular to \(g\), but it does not pass through \(P\). 3. In c), line \(h\) passes through \(P\) and its direction differs from \(g\) by a quarter-turn. Therefore c) satisfies both requirements.

c)

- “Do not touch” and “are parallel” are not the same condition. - Compare the directions of the two segments. - Imagine extending each segment in both directions.

1. Parallel segments must lie on lines that have the same direction and never meet. 2. \(\overline{AB}\) is horizontal, while \(\overline{CD}\) is slanted, so their directions are different. 3. If the segments were extended as lines, the two lines would meet. Therefore the segments are not parallel even though the drawn pieces do not touch.

No. The segments have different directions, and the lines containing them would meet if extended, so the segments are not parallel.

- Do both connectors take the same path across the gap? - Compare their directions with the lines they join. - Use the grid squares to check the distance you choose.

1. The distance between parallel lines is measured at right angles to the lines. 2. Segment \(AB\) meets both horizontal lines at right angles. It crosses \(3\) grid squares. 3. The slanted segment \(CD\) joins the lines but does not measure their shortest distance.

Segment \(\overline{AB}\); \(3\) units.

- Think about what happens to the gap as you move along parallel lines. - If you measure the gap in two places, what should you compare?

1. A perpendicular segment between parallel lines measures their distance. 2. Parallel lines remain the same distance apart wherever that perpendicular segment is drawn. 3. Both drawn segments are perpendicular, so they have equal lengths.

The distances are equal. Both perpendicular segments measure the same gap between parallel lines.

- Compare the direction of \(g\) with the direction from \(P\) to each other point. - A parallel line can be in a different place but must slant in the same direction.

1. Line \(g\) rises as it goes to the right. 2. From \(P\) to \(Q\), the direction matches \(g\). From \(P\) to \(R\), it slopes the opposite way. 3. Place the ruler along \(P\) and \(Q\), then extend the line both ways.

Pass the ruler through \(P\) and \(Q\), then draw a line extending in both directions.

- Imagine extending each possible line past the labeled points. - Which direction would make a square corner with \(g\)?

1. Line \(g\) slopes down to the right. 2. The direction from \(P\) to \(Q\) rises to the right at the matching steepness, so it makes a right angle with \(g\). 3. The direction from \(P\) to \(R\) runs parallel to \(g\), not perpendicular to it.

Draw the line through \(P\) and \(Q\). Extend it until it crosses \(g\), then use the square corner of a sheet of paper to check the right angle.

- Begin with the four sides of the quadrilateral. - Then identify the complete segments joining opposite vertices. - The intersection point divides each diagonal into two shorter segments.

1. The four sides are \(\overline{PQ}\), \(\overline{QR}\), \(\overline{RS}\), and \(\overline{SP}\). 2. The two complete diagonals are \(\overline{PR}\) and \(\overline{QS}\). 3. Point \(T\) divides the diagonals into \(\overline{PT}\), \(\overline{TR}\), \(\overline{QT}\), and \(\overline{TS}\). 4. Altogether, the drawing contains \(10\) such segments.

The segments are \(\overline{PQ}\), \(\overline{QR}\), \(\overline{RS}\), \(\overline{SP}\), \(\overline{PR}\), \(\overline{QS}\), \(\overline{PT}\), \(\overline{TR}\), \(\overline{QT}\), and \(\overline{TS}\).

- Start with the right angles made by the perpendicular lines \(g\) and \(h\). - Check which of those regions are split by line \(k\). - Count only the smallest angle regions shown after all three lines are drawn.

1. Lines \(g\) and \(h\) form four right angles before line \(k\) is added. 2. Line \(k\) passes through two opposite right-angle regions and splits each of them into two smaller angles. 3. The other two right-angle regions are not split. Therefore exactly \(2\) of the six smallest angle regions are right angles.

\(2\)

- Start with what parallel lines tell you about direction. - Think about the effect of a right-angle turn from that direction. - At the ray's starting point, which directions meet?

1. Parallel lines \(m\) and \(n\) have the same direction. Because \(t\) is perpendicular to \(m\), it is perpendicular to \(n\) too. 2. Ray \(r\) is parallel to \(n\) and starts on \(t\). It therefore makes a right angle with \(t\). 3. Both results follow from parallel objects sharing a line direction and perpendicular objects meeting at right angles.

a) perpendicular b) perpendicular c) Lines \(m\) and \(n\), and ray \(r\), are parallel. Line \(t\) is perpendicular to their direction, so it makes a right angle with \(n\) and with \(r\).