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

- 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.

- Distance between parallel lines is measured along a perpendicular segment. - A point halfway between them has half the total distance to each line. - For point \(Q\), add the distance to \(a\) and the distance between the lines.

1. Half of \(6\,\text{cm}\) is \(6 \div 2=3\,\text{cm}\), so \(P\) is \(3\,\text{cm}\) from \(a\). 2. To travel perpendicularly from \(Q\) to \(b\), first travel \(1\,\text{cm}\) to \(a\), then \(6\,\text{cm}\) to \(b\). The total distance is \(1+6=7\,\text{cm}\).

1. \(3\,\text{cm}\) 2. \(7\,\text{cm}\)

- Picture a shoebox or a book. - Count vertices, then edges, then faces. - Look for edges that run in the same direction.

1. A rectangular prism has \(8\) vertices, \(12\) edges, and \(6\) faces. 2. Each face is a rectangle. 3. The edges form three sets of four. In each set, the four edges are parallel and have the same length.

a) \(8\) vertices, \(12\) edges, and \(6\) faces b) Rectangles c) \(4\) edges in each set

- 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 \(\angle XSZ\)

- Look at all the regions around the intersection point. - Think about whether opposite angles appear equal.

1. The two lines are perpendicular. 2. They divide the region around \(S\) into four equal angles. Each angle is a right angle, so \(4\) right angles are formed.

There are \(4\) right angles around \(S\).

- Remember that the line continues on both sides of \(S\). - Count the angle on each side of the ray separately.

1. The line extends from \(S\) in two opposite directions. 2. The perpendicular ray forms a right angle with each direction of the line. Therefore, there are \(2\) right angles.

There are \(2\) right angles at the meeting point.

- 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).

- Picture a shoebox and focus on edges that point in the same direction. - Count the edges parallel to one bottom edge. - Think about the three directions: length, width, and height.

1. A rectangular prism has four parallel edges in each direction: length, width, and height. 2. The \(12\) edges can be grouped into \(3\) sets of \(4\) parallel edges. 3. One selected edge belongs to a set of \(4\), so the other \(3\) edges in that set are parallel to it.

a) \(4\) edges b) \(3\) sets c) \(3\) other edges

- Picture the prism as a tall box with a square top and bottom. - Count the edges around the two square faces. - Then count the edges connecting the squares.

1. The two square faces have \(4\) distinct edges each, so there are \(2 \times 4=8\) edges of length \(5\,\text{cm}\). 2. Four edges connect the two square faces, so there are \(4\) edges of length \(12\,\text{cm}\). 3. A rectangular prism has \(8\) vertices.

a) \(8\) edges b) \(4\) edges c) \(8\) vertices

- 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 \(\overline{TS}\).