Parallel lines are two or more lines that are always the same distance apart and never meet. They have the same slope and will never intersect, no matter how far they are extended.

The symbol for parallel lines is "||". If two lines are parallel, we can write it as line AB || line CD.

Here are some key properties of parallel lines:

- Parallel lines have the same slope.
- Corresponding angles formed by a transversal (a line that intersects two or more lines) are congruent.
- Alternate interior angles and alternate exterior angles are congruent.
- Consecutive interior angles are supplementary, meaning they add up to 180 degrees.

In geometry, we often encounter parallel lines in the context of parallel shapes such as parallelograms and trapezoids.

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. The opposite angles of a parallelogram are also equal.

A trapezoid is a quadrilateral with one pair of parallel sides. The parallel sides are called the bases of the trapezoid.

When studying parallel lines and shapes, it's important to understand the following concepts:

- Definition of parallel lines and the symbol used to denote parallel lines.
- Properties of parallel lines, including the relationships between angles formed by a transversal.
- Identifying and working with parallel shapes such as parallelograms and trapezoids.
- Solving problems involving parallel lines and shapes, including finding missing angles and side lengths.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.