1. Then/Now You proved whether two triangles were congruent. Identify reflections, translations, and rotations. Verify congruence after a congruence transformation.

2. Concept

3. Example 1 Identify Congruence Transformations A. Identify the type of congruence transformation shown as a reflection, translation, or rotation. Each vertex and its image are in the same position, just 5 units right and 2 units down. Answer: This is a translation.

4. Example 1 Identify Congruence Transformations B. Identify the type of congruence transformation shown as a reflection, translation, or rotation. Each vertex and its image are the same distance from the origin. The angles formed by each pair of corresponding points and the origin are congruent. Answer: This is a rotation.

5. Example 1 Identify Congruence Transformations C. Identify the type of congruence transformation shown as a reflection, translation, or rotation. Each vertex and its image are the same distance from the x-axis. Answer: This is a reflection.

6. Example 3 Verify Congruence after a Transformation Triangle PQR with vertices P(4, 2), Q(3, –3), and R(5, –2) is a transformation of ΔJKL with vertices J(–2, 0), K(–3, –5), and L(–1, –4). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation. UnderstandYou are asked to identify the type of transformation—reflection, translation, or rotation. Then, you need to show that the two figures are congruent. Plan Use the Distance Formula to find the measure of each side. Then show that the two triangles are congruent by SSS.

7. Example 3 Verify Congruence after a Transformation SolveGraph each figure. The transformation appears to be a translation 6 units right and 2 units up. Find the measures of the sides of each triangle.

8. Example 3 Verify Congruence after a Transformation

9. Example 3 Verify Congruence after a Transformation Answer:By SSS, ΔJKL  ΔPQR. CheckUse the definition of a translation. Use a ruler to measure and compare the corresponding sides of the triangles. The corresponding sides are congruent, so the triangles are congruent.

10. Example 1 Position and Label a Triangle Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position the triangle in the first quadrant. Position and label right triangle XYZ with leg d units long on the coordinate plane. Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long. 4-8 Triangles and Coordinate Proof

11. Example 1 Position and Label a Triangle Since triangle XYZ is a right triangle, the x-coordinate of Y is 0. We cannot determine the y-coordinate, so call it b. Answer: 4-8 Triangles and Coordinate Proof

12. Concept 4-8 Triangles and Coordinate Proof