An isosceles triangle is a type of triangle that has two sides of equal length. These two equal sides are called the legs, and the third side is called the base. The angles opposite the legs are also equal in measure, and they are called the base angles.

**Two sides are equal:**In an isosceles triangle, two sides (legs) are of the same length.**Two angles are equal:**The angles opposite the equal sides are congruent (of the same measure).**The base angles:**The angles formed by the base and each of the congruent sides are equal in measure.**The base:**The side opposite the vertex where the two legs meet is called the base.

Here are some important formulas and theorems related to isosceles triangles:

**Area of an Isosceles Triangle:**The area (A) of an isosceles triangle can be calculated using the formula: A = (1/2) * base * height**Perimeter of an Isosceles Triangle:**The perimeter (P) of an isosceles triangle can be found by adding the lengths of all three sides: P = side1 + side2 + base**Isosceles Triangle Theorem:**If two sides of a triangle are congruent, then the angles opposite those sides are also congruent.**Converse of the Isosceles Triangle Theorem:**If two angles of a triangle are congruent, then the sides opposite those angles are also congruent.

Let's work through some example problems related to isosceles triangles:

**Find the area of an isosceles triangle with a base of 8 cm and a height of 6 cm.****Determine the measure of the base angles in an isosceles triangle if each of the congruent sides has a length of 5 inches.**

Solution: Using the area formula A = (1/2) * base * height, we can substitute the given values: A = (1/2) * 8 cm * 6 cm = 24 square cm

Solution: Since the sides are congruent, the base angles will also be congruent. We can use the fact that the sum of the angles in a triangle is 180 degrees, so each base angle will be (180 - measure of the vertex angle)/2 = (180 - 40)/2 = 70 degrees.

Isosceles triangles are a special type of triangle with two sides of equal length and two congruent angles. Understanding the properties, formulas, and theorems related to isosceles triangles is important in geometry and trigonometry.

.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.