An angle is a geometric figure formed by two rays or lines that extend from a common point, called the vertex.

There are several types of angles based on their measure and relationship with each other:

**Acute Angle:**An angle that measures between 0 and 90 degrees.**Right Angle:**An angle that measures exactly 90 degrees.**Obtuse Angle:**An angle that measures between 90 and 180 degrees.**Straight Angle:**An angle that measures exactly 180 degrees.**Reflex Angle:**An angle that measures between 180 and 360 degrees.

Angles are typically denoted using the symbol ∠ followed by three points, with the middle point representing the vertex. For example, ∠ABC represents an angle with vertex at point B.

Angles are measured in degrees, with a full rotation around a point measuring 360 degrees. Degrees can be further divided into minutes (') and seconds (").

When two lines intersect, they form pairs of angles with specific relationships:

**Vertical Angles:**Two non-adjacent angles formed by intersecting lines. They are congruent (have the same measure).**Adjacent Angles:**Two angles that share a common vertex and side, but do not overlap. Their measures add up to 180 degrees (form a linear pair) in a straight line.**Complementary Angles:**Two angles whose measures add up to 90 degrees.**Supplementary Angles:**Two angles whose measures add up to 180 degrees.

There are several important properties and formulas related to angles, including:

**Sum of Interior Angles of a Polygon:**The sum of the interior angles of a polygon with n sides is given by (n-2) * 180 degrees.**Exterior Angle of a Polygon:**The exterior angle of a polygon is equal to the sum of the two remote interior angles.**Angles in a Triangle:**The sum of the interior angles of a triangle is always 180 degrees.

There are various tools used to measure and draw angles, including protractors, angle rulers, and geometric software applications.

Now that you have learned about angles, it's time to practice! Try solving the following problems:

- Find the complement of an angle measuring 25 degrees.
- Determine the type of angle formed by the hands of a clock at 3:00 PM.
- Calculate the measure of each interior angle of a regular hexagon.

Good luck with your angle practice!

.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.