• Unit 3: Triangles, Congruence, and Other Relationships

    In this unit, we will explore different methods for comparing and identifying identical triangles depending on the information we have available. Whether we compare pieces of furniture, automobiles, or pieces of a candy bar, we will have to make sure that objects are the same size and shape.

    We will also study triangles and their parts to predict their measurements and distances between them. For example, we can use the process of triangulation to calculate a distance when we have two known locations. This can be useful when we need to calculate precise measurements for a land survey on a building construction site, determine distances for a race, or calculate the correct angles for a rocket launch.

    Completing this unit should take you approximately 5 hours.

    • Upon successful completion of this unit, you will be able to:

      • classify triangles by lengths and angles;
      • determine the measures of a triangle's angles (interior and exterior);
      • determine whether two triangles are congruent;
      • use theorems to calculate missing lengths of a triangle; and
      • calculate the measure of a bisected angle.

    • 3.1: Triangle Classification

      Before we can begin comparing triangles, we need to be able to classify triangles based on their angles and sides.

      • Triangle Classification URL

        Read this article and watch the videos to learn how to classify triangles based on their angles and sides. Pay attention to the list of different types of triangles. It may be helpful to make a list of these different classifications of triangles and their characteristics.

    • 3.2: Triangle Sum Theorem

      The triangle sum theorem states that the sum of the interior angles in a triangle add to 180 degrees. This theorem is useful when we need to solve for an unknown angle in a given triangle.

    • 3.3: Exterior Angle Theorem

      The exterior angle theorem states that an exterior angle of a triangle equals the sum of its nonadjacent interior angles. Like the triangle sum theorem, it allows us to determine unknown angles in a given triangle.

    • 3.4: Congruent Figures

      Congruent figures are figures that are exactly the same shape and size.

      • Congruent Figures URL

        Read this article and watch the video. Pay attention to the steps you need to take to determine if two figures are congruent.

    • 3.5: Congruent Triangles and SSS

      We can use the side-side-side (SSS) postulate for triangles to determine whether triangles are congruent. The SSS postulate states that if the three sides of two triangles are congruent, then the triangles are congruent.

    • 3.6: Other Triangle Congruence Postulates

      While the SSS postulate is useful, sometimes we do not have enough information about the sides of two triangles to use it. We can use other postulates depending on the given information for a pair of triangles. These additional postulates include side-angle-side (SAS), angle-side-angle (ASA), and angle-angle-side (AAS).

    • 3.7: Calculating Congruent Triangles

      Now that we have explored the different ways to determine whether two triangles are congruent, we will use what we learned to solve congruence problems.

    • 3.8: Isosceles Triangles

      Isosceles triangles are triangles that have at least two congruent sides.

      • Isosceles Triangles URL

        Read this article and watch the videos. Pay attention to the terminology that it introduces about isosceles triangles. The base angle theorem states that the base angles of an isosceles triangle must be congruent. Do not focus on the proof. Pay attention to the examples of how to recognize congruent angles, measure vertex angles, and measure base angles.

        Then, complete review questions 1–5 and check your answers.

    • 3.9: Equilateral Triangles

      Equilateral triangles are the last special type of triangle we explore in this unit. In an equilateral triangle, all sides and angles are congruent.

      • Equilateral Triangles URL

        Read this article and watch the videos. Pay attention to the equilateral triangle theorem and examples 3–5.

        Then, complete review questions 1–4 and check your answers.