A triangle is a closed, three-sided polygon. The sum of the interior angles of a triangle is always 180 degrees.

There are several types of triangles based on their side lengths and angles:

**Equilateral Triangle:**All three sides are of equal length, and all three angles are of equal measure (60 degrees).**Isosceles Triangle:**Two sides are of equal length, and the two angles opposite those sides are of equal measure.**Scalene Triangle:**All three sides have different lengths, and all three angles have different measures.**Right Triangle:**One of the angles is a right angle (90 degrees).**Acute Triangle:**All three angles are less than 90 degrees.**Obtuse Triangle:**One of the angles is greater than 90 degrees.

Here are some important formulas related to triangles:

**Perimeter:**The perimeter of a triangle is the sum of the lengths of its three sides: P = a + b + c**Area:**The area of a triangle can be calculated using the formula: A = (1/2) * base * height**Pythagorean Theorem:**In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides: c^2 = a^2 + b^2

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, for a triangle with sides a, b, and c, where c is the longest side, the following inequality holds: a + b > c

Triangles also have several geometric properties, such as the altitude, median, and centroid, which are important for solving various problems.

Now that you've learned about triangles, try solving the following practice problems:

- Find the area of a triangle with a base of 6 units and a height of 8 units.
- Determine the type of triangle with side lengths of 5, 5, and 5 units.
- If one angle of a triangle is 45 degrees, and another angle is 60 degrees, what is the measure of the third angle?

Good luck with your triangle studies!

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.