Subject
Mathematics study guides
Algebra, geometry, and proofs — solved step by step with interactive graphs and figures.
13 explained questions, each with the answer, a worked explanation, and something interactive to help it stick.
- Which composition of transformations will create a pair of similar, but not congruent, triangles?Any composition that includes a dilation—for example, a rotation followed by a dilation. A dilation scales the figure, making it a different size (not congruent) while keeping the same shape and angle measures (still similar). Rigid motions alone only ever produce congruent triangles.Read the guide
- Which pair of undefined terms is used to define a ray?Point and line. A ray is defined using the two undefined terms point and line: it begins at an endpoint (a point) and extends without end in one direction along a line. Plane is not needed.Read the guide
- Compare and Contrast the Absolute Value of a Real Number with That of a Complex NumberBoth are non-negative distances from zero. A real number's absolute value |x| is a one-dimensional distance along the number line. A complex number's absolute value (modulus) |a+bi| = sqrt(a^2+b^2) is a two-dimensional distance from the origin in the complex plane, found using the Pythagorean theorem.Read the guide
- Which undefined term is used to define an angle?Point. The three undefined terms in geometry are point, line, and plane. An angle is two rays sharing a common endpoint called the vertex, and that vertex is a point. "Ray" is a defined term built from points, so it is not the answer.Read the guide
- Which pair of undefined terms is used to define the term parallel lines?Point and line. Parallel lines are defined using the undefined terms point and line: two lines (each made up of points) that lie in the same plane and never intersect, no matter how far they are extended.Read the guide
- Which Is a Property of an Angle?An angle is formed by two rays (its sides) that share a common endpoint called the vertex, and its size is measured in degrees. The defining property is two rays meeting at a single vertex — that is what identifies a figure as an angle.Read the guide
- Which Undefined Term Can Contain Parallel Lines?A plane. Of geometry's three undefined terms — point, line, and plane — only a plane, a flat surface extending infinitely in two dimensions, can contain two parallel lines that lie in the same surface and never intersect. A single line cannot contain parallel lines.Read the guide
- Given m∠A + m∠B = m∠B + m∠C, Prove m∠C = m∠A: Paragraph ProofStarting from m∠A + m∠B = m∠B + m∠C, subtract m∠B from both sides by the Subtraction Property of Equality to get m∠A = m∠C. Then apply the Symmetric Property of Equality to conclude m∠C = m∠A.Read the guide
- The Definition of a Circle Uses the Undefined Term ______Plane. A circle is defined as the set of all points in a plane that are a fixed distance (the radius) from a center point. Among the usual choices (arc, line, plane, ray), "plane" is the undefined term used; "point" is also undefined but is typically not offered.Read the guide
- Which is the endpoint of a ray?The endpoint of a ray is its single starting point — the fixed point where the ray begins before extending infinitely in one direction. In ray AB, the first named point, A, is the endpoint, and the ray passes through B forever.Read the guide
- Which Description Is Represented by a Discrete Graph?The description that counts whole, separate items is represented by a discrete graph. For example, "Kiley bought a platter for $19 and several matching bowls at $8 each" is discrete, because the number of bowls can only be whole numbers, giving separated points rather than a connected line.Read the guide
- What is true about the sum of the two polynomials 6s²t − 2st² and 4s²t − 3st²?The sum is a binomial with a degree of 3. Adding (6s²t − 2st²) + (4s²t − 3st²) combines like terms to give 10s²t − 5st². It has two terms (a binomial), and each term has degree 3 (2 + 1).Read the guide
- Which graph shows the solution to the system of linear inequalities?The correct graph is the one where the shaded regions of both inequalities overlap. That darker, double-shaded area is the solution. Use solid boundary lines for ≤ or ≥ and dashed lines for < or >.Read the guide