You proved whether two triangles were congruent.

• Identify reflections, translations, and rotations.

• Verify congruence after a congruence transformation.

Identify Congruence Transformations

A. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

Each vertex and its imageare in the same position, just5 units right and 2 units down.

Answer: This is a translation.

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.

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.

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.

Understand You are asked to identify the type oftransformation—reflection, translation,or rotation. Then, you need to showthat the two figures are congruent.

Plan Use the Distance Formula to find themeasure of each side. Then show thatthe two triangles are congruent by SSS.

Verify Congruence after a Transformation

Solve Graph each figure. The transformationappears to be a translation 6 units rightand 2 units up. Find the measures of thesides of each triangle.

Verify Congruence after a Transformation

Verify Congruence after a Transformation

Answer: By SSS, ΔJKL ΔPQR.

Check Use the definition of a translation. Use a ruler to measure and comparethe corresponding sides of the triangles. The corresponding sides are congruent, so the triangles are congruent.

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

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

4-8 Triangles and Coordinate Proof