Free math worksheets

- Identify the vertex. - In a three-letter angle name, the vertex is the middle letter. - The order of the points on the two sides can be reversed.

1. The vertex is \(K\), so the angle can be named \(\angle K\) because only one angle at \(K\) is shown. 2. When three points are used, the vertex must be the middle letter. The angle can be named \(\angle MKN\) or \(\angle NKM\).

\(\angle MKN\), \(\angle NKM\), or \(\angle K\)

- Recall the names of the parts of an angle. - The two rays meet at one point.

1. The common endpoint \(S\) is the vertex of the angle. 2. Rays \(g\) and \(h\) are the sides of the angle.

a) The vertex b) The sides of the angle

- 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

- A segment stops at both ends. - A ray has exactly one endpoint. - In ray notation, the first letter names the endpoint.

a) A line segment has two endpoints. The endpoints are \(J\) and \(K\). b) A ray starts at its first named point. The 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)

- 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 \(\overline{AD} \perp \overline{CD}\).

- Count the two adjacent parts. - Then count the whole angle that contains both parts.

1. The two smaller adjacent angles are \(\angle XSY\) and \(\angle YSZ\). 2. The entire angle is \(\angle XSZ\). 3. Therefore, \(3\) angles are visible.

\(3\) angles: \(\angle XSY\), \(\angle YSZ\), and \(\angle XSZ\)

- 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, so they would meet if extended.

c)

- Perpendicular lines make a square-corner turn where they meet. - The lines do not have to be horizontal and vertical. - Compare the opening made by each pair to a right angle.

1. Perpendicular lines meet to form right angles. 2. In drawing b), the directions differ by \(90^\circ\), so the lines are perpendicular. 3. The pairs in a) and c) do not meet at right angles.

b)

- First compare the directions of \(g\) and \(h\). - Perpendicular lines meet to form right angles. - A line perpendicular to one of two parallel lines will meet the other at the same kind of angle.

a) Lines \(g\) and \(h\) have the same direction and never meet, so they are parallel. b) Line \(t\) meets \(g\) at a right angle, so they are perpendicular. c) Because \(h\) has the same direction as \(g\), line \(t\) also meets \(h\) at a right angle. They are perpendicular.

a) parallel b) perpendicular c) perpendicular

- Check the “passes through \(P\)” requirement and the “parallel to \(\overline{QR}\)” requirement separately. - A line can satisfy one requirement without satisfying the other. - The correct drawing must satisfy both conditions at the same time.

1. In a), line \(h\) passes through \(P\), but it has a different direction from \(\overline{QR}\), so it is not parallel. 2. In b), line \(h\) is parallel to \(\overline{QR}\), but it does not pass through \(P\). 3. In c), line \(h\) passes through \(P\) and has the same direction as \(\overline{QR}\). Therefore c) satisfies 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.

a) Oak has the same direction as Maple, so Oak is parallel to Maple. b) Pine crosses Maple at a right angle, so Pine is perpendicular to Maple. c) Cedar is slanted at a different non-right angle, so it is neither parallel nor perpendicular to Maple.

a) Oak b) Pine c) Cedar

- 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)

- List every pair of rays systematically. - Do not count the same pair twice in reverse order.

1. List each pair once: \((a,b)\), \((a,c)\), \((a,d)\), \((b,c)\), \((b,d)\), and \((c,d)\). 2. There are \(6\) different pairs, so there are \(6\) different smaller angles.

\(6\) angles

- 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.

\(2\)

- 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 \(B\).

- “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.

- Start with what parallel tells you about direction. - Perpendicular means making a right-angle turn from a direction. - Use the fact that ray \(r\) starts on line \(t\).

a) Because \(m\) and \(n\) are parallel, they have the same direction. A line perpendicular to \(m\) is therefore also perpendicular to \(n\). b) Ray \(r\) has the same direction as \(n\) and starts on \(t\), so \(t\) and \(r\) meet at a right angle. They are perpendicular. c) Parallel objects share a direction. Turning \(90^\circ\) from that direction gives the perpendicular direction, so the same perpendicular relationship carries to \(n\) and to ray \(r\).

a) perpendicular b) perpendicular c) Since \(m\), \(n\), and \(r\) share one direction, line \(t\) is perpendicular to both \(n\) and \(r\).