- Line segment PQ and line segment RS intersect and form a right angle (90 degrees) where they cross. What is the relationship between line segment PQ and line segment RS?
Answer: They are perpendicular because they intersect at a right angle
Two line segments that cross and form a 90-degree angle are perpendicular, which matches segments PQ and RS. Choice A describes lines that never meet, which isn't the case here since they do intersect. Choice C describes an obtuse angle, not the right angle formed here. Choice D describes segments that lie exactly on top of each other, which is different from segments that simply cross at a point.
- Lisa is sorting quadrilaterals into groups. She puts all shapes with at least one pair of parallel sides in Group 1, and all shapes with at least one right angle in Group 2. A rectangle would belong in which group(s)?
Answer: Both Group 1 and Group 2, because of both features
The correct answer is C because a rectangle has both parallel sides and right angles, so it belongs in both groups. Choice A leaves out the right-angle feature. Choice B leaves out the parallel-sides feature. Choice D is incorrect because being a specific shape does not exclude a rectangle from having both of the listed features.
- Line AB passes through the points (0,2) and (4,6). Line CD passes through the points (0,5) and (4,9). Which statement correctly describes the relationship between lines AB and CD?
Answer: The lines are parallel
The lines are parallel because both lines rise the same amount over the same distance across (4 up for every 4 over), but they start at different points, so they never meet. Perpendicular is incorrect because the lines do not meet at a right angle. Intersecting but not perpendicular is incorrect because these lines never meet at all. Coincident is incorrect because the two lines start at different points, so they are not the same line.
- Examine the quadrilaterals shown in the figure. Which quadrilateral has both parallel sides AND perpendicular sides?
Answer: Quadrilateral P has both parallel and perpendicular sides
Quadrilateral P is a rectangle, which has two pairs of parallel sides (opposite sides are parallel) and four right angles (perpendicular sides meet at 90-degree angles). Quadrilateral Q is a parallelogram with no right angles, so it has parallel sides but no perpendicular sides. Quadrilateral R is a trapezoid with no right angles, so it has one pair of parallel sides but no perpendicular sides.
- Look at the triangles shown in the diagram below. Which triangle has perpendicular sides?
Answer: Triangle P because it has two sides that meet at a right angle
Triangle P has a right angle marked with a square symbol, showing two sides meet at exactly 90°. When two lines meet at 90°, they are perpendicular. Triangle Q has equal angles but no right angles (choice B). Triangle R has an obtuse angle, not a right angle (choice C). Triangle S has only acute angles, not right angles (choice D).
- A triangle has angles measuring 30°, 60°, and 90°. A small square symbol at one vertex marks the 90° angle. Which statement best describes this triangle?
Answer: Right triangle because it has a 90-degree angle
A triangle with a 90° angle is a right triangle, regardless of its other two angle measures. Choice B is incorrect because a triangle with a 90° angle cannot have all angles less than 90°. Choice C is incorrect because an obtuse triangle has an angle greater than 90°, not equal to it. Choice D is incorrect because a triangle with angles of 30°, 60°, and 90° does not have three equal sides. - Five points, A, B, C, D, and E, form three triangles: ABC, ADE, and BCE. In triangle ABC, sides BA and BC meet at a right angle at vertex B. In triangle BCE, sides CB and CE meet at a right angle at vertex C. In triangle ADE, no two sides meet at a right angle. Which statement correctly identifies the triangles that have perpendicular sides?
Answer: Only triangles ABC and BCE have sides meeting at right angles
Triangle ABC has a right angle where sides BA and BC meet at vertex B. Triangle BCE has a right angle where sides CB and CE meet at vertex C. Triangle ADE has no two sides that meet at a right angle. So only triangles ABC and BCE have perpendicular sides. Choice A leaves out triangle ABC. Choice B mistakenly includes ADE, which has no right angle. Choice D mistakenly includes all three triangles.
- Look at the quadrilateral shown in the figure below. Which statement best describes this figure based on its angles and line relationships?
Answer: It is a rectangle because it has four right angles and opposite sides are parallel
The figure shows a rectangle with four right angles (90°) marked and opposite sides that are parallel. Choice A correctly identifies both the right angles and parallel sides. Choice B is wrong because the figure does have right angles. Choice C is wrong because the figure has two pairs of parallel sides, not one. Choice D is wrong because while the sides may be equal, the defining characteristic shown is the right angles.
- Study the triangles in the figure. Based on the angle markings shown, which statement correctly classifies these triangles?
Answer: Both Triangle X and Triangle Y are right triangles
Looking at the angle markings, Triangle X has a right angle marked with a square symbol at one vertex, making it a right triangle. Triangle Y also has a right angle marked with a square symbol at one of its vertices, making it a right triangle as well. Both triangles have exactly one 90-degree angle, which is the defining characteristic of right triangles.
