Question

given that the two triangles shown are congruent, explain one way to verify that the corresponding angles of the two triangles are congruent. a translate triangle 2 up 2 units, and to the left 6 units b translate triangle 2 up 2 units, and to the right 6 units c translate triangle 2 down 2 units, and to the left 6 units d translate triangle 2 down 2 units, and to the right 6 units

Explanation:

Step1: Analyze the translation direction and distance

To verify the corresponding angles are congruent, we can translate one triangle to overlap with the other. Looking at the positions, to move triangle 2 (the lower one) to match triangle 1 (the upper one), we need to move it up (positive y - direction) and to the right (positive x - direction) or analyze the vertical and horizontal shifts. Wait, actually, let's check the vertical and horizontal distances. The vertical distance between the two triangles: the lower triangle's top vertex and the upper triangle's top vertex. If we translate triangle 2 up 2 units (to match the y - coordinate) and to the right 6 units (to match the x - coordinate), or wait, no, let's see the options. Wait, the correct translation: Let's assume triangle 2 is the lower one. To get to triangle 1, we need to move triangle 2 up 2 units (since the vertical difference is 2) and to the right 6 units? Wait, no, looking at the options, option D: translate triangle 2 down 2 units, and to the right 6 units? Wait, no, maybe I got the triangles reversed. Wait, the problem says "translate triangle 2...". Let's check the coordinates. Suppose triangle 1 is at the top right, triangle 2 is at the bottom left. To make them overlap, if we translate triangle 2 up 2 units (to increase y - coordinate) and to the right 6 units (to increase x - coordinate)? Wait, no, the options: A: up 2, left 6; B: up 2, right 6; C: down 2, left 6; D: down 2, right 6. Wait, maybe the correct translation is to move triangle 2 up 2 units (so y increases by 2) and to the right 6 units (x increases by 6) to overlap with triangle 1. When two congruent figures are translated to overlap, their corresponding angles will coincide, hence be congruent. So the correct option is B? Wait, no, wait the original problem: the two triangles are congruent. To verify corresponding angles are congruent, we can translate one to map onto the other. Let's check the vertical and horizontal shifts. If triangle 2 is the lower triangle, to get to triangle 1, we need to move up (since triangle 1 is above triangle 2) by 2 units and to the right (since triangle 1 is to the right of triangle 2) by 6 units. So the translation is: translate triangle 2 up 2 units, and to the right 6 units (option B). When we translate a figure, the shape and size (and hence angles) remain the same. So by translating triangle 2 to overlap with triangle 1, the corresponding angles will coincide, proving they are congruent.

Step2: Confirm the translation

By translating triangle 2 up 2 units (changing the y - coordinate of each vertex by + 2) and to the right 6 units (changing the x - coordinate of each vertex by + 6), triangle 2 will exactly overlap with triangle 1. Since translation is a rigid transformation (preserves shape, size, and angle measures), the corresponding angles of the two triangles will be congruent.

Answer:

B. translate triangle 2 up 2 units, and to the right 6 units