Free math worksheets

- Compare each angle visually with a square corner. - First name the vertex, and then give the total number of right angles.

1. At \(A\), sides \(AB\) and \(AC\) meet at a right angle. 2. The angles at \(B\) and \(C\) are not right angles. The triangle has \(1\) right angle in all.

The right angle is at \(A\). The triangle has \(1\) right angle in all.

- A right triangle has exactly one right angle. - Look for the corner that makes a square-corner turn, even if it is tilted. - Compare all three corners of each triangle before choosing.

1. Triangle a) has three acute angles, so it is not a right triangle. 2. Triangle b) has one obtuse angle, so it is not a right triangle. 3. In c), the two sides that meet at \(P\) are perpendicular, so \(\angle QPR\) is a right angle. Therefore c) is the right triangle.

c)

- Turning a triangle does not change its angle sizes. - A right angle can be tilted. - Compare the two sides that meet at each candidate corner.

1. In a), the two sides meeting at \(A\) are perpendicular even though both are slanted, so the triangle has a right angle at \(A\). 2. The angles in b) and c) are not right angles. Therefore a) is a right triangle.

a)

- Find the vertex where the right-angle mark is placed. - The middle letter in \(\angle PQR\) is its vertex. - A right triangle has one right-angle vertex.

1. The right-angle mark is drawn at vertex \(Q\). 2. Therefore \(\angle PQR\) is the right angle, and \(Q\) is the right-angle vertex.

\(Q\)

- A right angle can point in any direction. - Look for the pair of sides that make a square-corner turn. - The right-angle vertex is where those two sides meet.

1. The sides \(\overline{BA}\) and \(\overline{BC}\) make a square-corner turn, so they are perpendicular. 2. They meet at \(B\), so \(\angle ABC\) is the right angle.

\(B\)

- Use the definition of a right triangle. - Ask which statement directly tells you a right angle is present. - Equal side lengths do not determine a right angle.

1. A right triangle is defined as a triangle with one right angle. 2. An angle measuring \(90^\circ\) is a right angle. 3. Therefore choice b) guarantees a right triangle. The other facts do not.

b) One angle measures \(90^\circ\).

- Use the meaning of perpendicular sides. - Identify the vertex shared by the two perpendicular sides. - Connect the resulting angle type to the definition of a right triangle.

1. Perpendicular sides meet to form a right angle. 2. Sides \(\overline{AB}\) and \(\overline{AC}\) meet at vertex \(A\), so \(\angle BAC\) is a right angle. 3. A triangle with a right angle is a right triangle.

\(ABC\) is a right triangle because perpendicular sides \(\overline{AB}\) and \(\overline{AC}\) make a right angle at \(A\).

- A right triangle must contain exactly one right angle. - An acute angle is smaller than a right angle; an obtuse angle is larger. - Check the largest-looking angle in each triangle first.

1. Triangle a) has one right angle, so it is right. 2. Triangle b) has three acute angles, so it is not right. 3. Triangle c) has one obtuse angle, so it is not right.

a) right b) not right c) not right

- A right angle measures exactly \(90^\circ\). - Compare each marked measure with \(90^\circ\). - “Almost \(90^\circ\)” is not a right angle.

1. In a), the marked angle is \(89^\circ\), which is acute, and the other two angles are also acute. It has no right angle. 2. The marked angle in b) is \(90^\circ\), which is a right angle, so b) is a right triangle. 3. In c), the marked angle is \(91^\circ\), which is obtuse, and the other two angles are acute. It has no right angle.

b)

- Imagine rotating a familiar right triangle. - Rotation does not change angle size. - Check the angle at \(A\), not whether any side is horizontal.

1. Whether a triangle is right depends on its angle sizes, not its orientation on the page. 2. The two sides meeting at \(A\) are perpendicular, so \(\angle A\) is a right angle. 3. Therefore the triangle is right even though all of its sides are slanted.

Sofia is incorrect. A right triangle may be turned; this triangle still has a right angle at \(A\).

- Name the two triangles made by the diagonal. - Use the rectangle's right-angle corners. - Identify which original corner belongs to each triangle.

1. The diagonal divides the rectangle into triangles \(ABC\) and \(ACD\). 2. In triangle \(ABC\), sides \(\overline{AB}\) and \(\overline{BC}\) are perpendicular, so the right angle is at \(B\). 3. In triangle \(ACD\), sides \(\overline{CD}\) and \(\overline{DA}\) are perpendicular, so the right angle is at \(D\). 4. Therefore the diagonal forms \(2\) right triangles.

\(2\) right triangles; the right angles are at \(B\) and \(D\).

- Check the four small triangles one at a time instead of counting all visible triangles. - Use the right-angle corners of the square as exact information. - Do not assume triangles cut from a slanted parallelogram are right triangles.

1. In the square, \(\triangle ABE\) contains the square's right angle at \(B\), and \(\triangle AEF\) contains the square's right angle at \(F\). Both are right triangles. 2. In the slanted parallelogram, neither \(\triangle BCD\) nor \(\triangle BDE\) has a right angle. Therefore the right triangles are \(\triangle ABE\) and \(\triangle AEF\).

\(\triangle ABE\) and \(\triangle AEF\)

- Focus on the angle at the center point \(O\). - Perpendicular segments make right angles where they cross. - Count only the four small triangles named by neighboring outer vertices and \(O\).

1. The vertical segment \(\overline{AC}\) and horizontal segment \(\overline{BD}\) are perpendicular at \(O\). 2. Each small triangle uses one part of \(\overline{AC}\) and one part of \(\overline{BD}\), so each has a right angle at \(O\). 3. The four small triangles are \(AOB\), \(BOC\), \(COD\), and \(DOA\). All \(4\) are right triangles.

\(4\) right triangles, each with its right angle at \(O\)

- Identify which two pieces meet at the corner of the shelf. - Describe the relationship between vertical and horizontal directions. - The diagonal brace is the third side of the triangle.

1. The post and shelf meet at vertex \(A\). 2. A vertical line and a horizontal line are perpendicular, so they form a right angle at \(A\). 3. The diagonal brace closes the triangle, so \(ABC\) is a right triangle with right angle at \(A\).

The post and shelf are perpendicular, so \(ABC\) is a right triangle. The right-angle vertex is \(A\).

- Use the right-angle mark in a). - Translate “perpendicular sides” into an angle fact in b). - Classification depends on angle properties, not orientation.

1. In a), the right-angle mark is at \(A\), so \(A\) is the right-angle vertex. 2. In b), perpendicular sides \(\overline{DE}\) and \(\overline{DF}\) meet at \(D\), so \(D\) is the right-angle vertex. 3. Both triangles contain one right angle, so both are right triangles. Their different orientations do not change their category.

a) right angle at \(A\) b) right angle at \(D\) Both are right triangles because each has one right angle.

- Ask what property defines a right triangle. - One side's direction does not determine the direction of the next side. - State the relationship the neighboring side must have with the vertical side.

1. A vertical side by itself does not determine any angle size. 2. For the triangle to be right at an endpoint of that side, the neighboring side must be perpendicular to the vertical side. 3. Therefore the triangle needs a right angle; merely having a vertical side is not enough.

Jordan's rule is wrong. A vertical side does not guarantee a right angle. A neighboring side must meet it perpendicularly so that the triangle contains a \(90^\circ\) angle.

- Translate “parallel” into a statement about direction. - Ask what being perpendicular to line \(m\) means for anything parallel to \(m\). - The two triangle sides meet at vertex \(A\).

1. Because \(\overline{AB}\) is parallel to line \(m\), \(\overline{AB}\) has the same direction as \(m\). 2. Side \(\overline{AC}\) is perpendicular to \(m\), so it is also perpendicular to any segment with the same direction as \(m\), including \(\overline{AB}\). 3. Sides \(\overline{AB}\) and \(\overline{AC}\) meet at \(A\), so \(\angle BAC\) is a right angle. Therefore \(ABC\) is a right triangle.

Yes. \(\overline{AB} \perp \overline{AC}\), so the triangle has a right angle at \(A\).

- Treat each triangle independently; the evidence is different in each panel. - Translate perpendicular sides into a right-angle fact. - For d), first use the parallel relationship to compare the direction of \(\overline{PQ}\) with line \(g\).

1. In a), the marked \(90^\circ\) angle shows directly that the triangle is right. 2. In b), perpendicular sides \(\overline{DE}\) and \(\overline{DF}\) make a right angle at \(D\), so b) is right. 3. In c), all three angles are acute, so c) has no right angle and is not right. 4. In d), \(\overline{PQ}\) has the same direction as line \(g\). Since \(\overline{PR}\) is perpendicular to \(g\), \(\overline{PR}\) is also perpendicular to \(\overline{PQ}\). Thus d) has a right angle at \(P\). Therefore a), b), and d) are right triangles.

a), b), and d)