- 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 small angle mark inside the triangle. - Identify the common endpoint of the two sides touching that mark.

1. The marked angle at \(Q\) measures \(90^\circ\). 2. A triangle with a right angle is a right triangle, so \(PQR\) is right at \(Q\).

Yes. It is a right triangle, with its right angle at \(Q\).

- 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 \(O\)

- 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.

- 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 \(A\).

- Check the angle at each corner rather than the direction of one side. - Compare the two slanted sides with the diagonals of grid squares. - Would a square paper corner fit at any vertex?

1. In a), \(PQ\) is vertical, but neither side meeting it is horizontal, and none of the corners is a right angle. 2. In b), from \(X\) to \(Y\) is 2 squares right and 2 up, while from \(X\) to \(Z\) is 2 squares left and 2 up. These are the two sloping directions of a square's diagonals and meet at a square corner at \(X\). 3. So b) is a right triangle, while a) is not.

b) is a right triangle. From \(X\), one side goes 2 squares right and 2 up; the other goes 2 squares left and 2 up. Those equal, opposite diagonal directions make a square corner at \(X\). In a), the other sides do not meet its vertical side at a right angle.

- Compare the corner angles in each drawing with a square corner. - In b), test each corner before you decide whether the tilted triangle is right. - In c), think about what must change at \(G\), not at the other vertices.

1. In a), the marked angle at \(A\) is \(90^\circ\), so the triangle is right. 2. In b), a square corner placed at \(D\) fits between \(DE\) and \(DF\), so b) is right even though both sides are slanted. 3. In c), no two sides meet at a right angle, so c) is not right. 4. To make a right angle at \(G\), move \(J\) three squares left so it is straight above \(G\). Then \(GJ\) is vertical and \(GH\) is horizontal.

a) Right triangle; the angle at \(A\) is right. b) Right triangle; the angle at \(D\) is right. c) Not a right triangle as shown. Move \(J\) \(3\) squares left to make a right angle at \(G\).

- Picture the two parallel lines and the line crossing them. - What can you work out about the angle where \(t\) meets \(h\)? - If a point slides along a straight line, does that line's direction change?

1. Since \(g\) and \(h\) are parallel, line \(t\), which is perpendicular to \(g\), is also perpendicular to \(h\). 2. Side \(\overline{PQ}\) lies on \(h\), and side \(\overline{PR}\) lies on \(t\). They meet at \(P\) at a right angle, so \(PQR\) is a right triangle. 3. Moving \(R\) to any other point on \(t\) (except \(P\)) changes a side length but not the right angle between \(t\) and \(h\).

Yes. The right angle is at \(P\) because \(t\) is perpendicular to \(g\) and therefore also to the parallel line \(h\). Moving \(R\) along \(t\), but not to \(P\), does not change the angle between \(t\) and \(h\), so it stays \(90^\circ\).

- 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 \(A\).