CCGPS Geometry1- Similarity, Congruence & Proofs1.8 – Notes

Name: ______Date: ______

Congruence and CPCTC in the Coordinate Plane

Transformations seen in Coordinate Algebra:

Translation: A translation is sometimes called a slide. In a translation, the figure is moved horizontally and/or vertically.

For example, triangle ABC is translated by 2 units to the right.

Reflection: A reflection creates a mirror image of the original figure over a reflection line.

Rotation: A rotation moves all points of a figure along a circular arc about a point. Rotations aresometimes called turns.

For example, triangle ABC is rotated about O through 90º in an anticlockwise direction.

Looking at the triangles in the above examples, does the size or shape ever change after performing each translation?

Each of these transformations is known as a rigid motion, or isometry. A rigid motionis a transformation done to a figure that maintains the figure’sshape and size or its segment lengths and angle measures. Hence, congruent triangles are formed.

Next week, we will look at dilations and similar triangles where the shape stays the same, but the size changes. These are called non-rigid motions.

Problem 1: is the image of. Write the translation rule.

a) b)

Problem 2: Find the line of reflection between the pre-image and the image.

a.b. c.

Problem3: Identify the type transformation(s) that have taken place. Then, determine if it is a rigid transformation, meaning the 2 triangles are congruent.

a) b)

Problem 4: A truss is a structure used in building bridges. The bridge truss pictured below is made up of5 triangles. Describe the transformations that have taken place, and determine whether thetriangles are congruent in terms of rigid and non-rigid motions.

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

  • If two triangles are congruent, then corresponding sides are congruent and corresponding angles are congruent.

5) Find AB and 6) Find NP and

CPCTC in Real World Situations:

Problem 7: A and B are on the edges of a ravine. What is AB?

Problem 8: A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK?

CPCTC on the Coordinate Plane:

Problem 9: You can also use CPCTC when triangles are on the coordinate plane.

Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)

Prove:

Step 1 Find the lengths of the sides of each triangle.

Use the Distance Formula if necessary.

Step 2 Determine if the triangles are congruent.

Step 3 If so, can you use CPCTC to say both angles are congruent?

Page 1 and 2 Adapted From: Walch Education Resources: CCGPS Analytic Geometry Teacher Resource Binder