Question

Two triangles are shown on the coordinate plane. Triangle ABC with A at (0, 0), B at (2, 4), and C at (0, 2). The measure of angle C is 135 degrees, and the measure of angle B is 30 degrees. Triangle DEF with D at (2, 0), E at (4, 4), and F at (4, 2). The measure of angle F is 135 degrees, and the measure of angle D is 30 degrees. Name the congruent triangles and justify the reason for congruence.

Answer 1

Triangles ABC and DEF are congruent because they have two pairs of corresponding angles that are equal and all corresponding sides are proportional with a ratio of 1:1.

Step-by-step explanation:

To determine if triangles ABC and DEF are congruent, we need to compare their sides and angles. Since both triangles have angles of 30 degrees and 135 degrees, they share at least two angles of the same measure. By the Angle-Angle (AA) similarity postulate, we know that the triangles are similar. However, for congruence, we need to establish that all corresponding sides are proportional with a ratio of 1:1. Let's calculate the lengths of the sides for both triangles using the distance formula.

• For triangle ABC:
  • Side AB is from (0, 0) to (2, 4), so its length is √((2-0)² + (4-0)²) = √(4+16) = √20.
  • Side AC is from (0, 0) to (0, 2), so its length is 2.
  • Side BC is from (2, 4) to (0, 2), so its length is √((2-0)² + (4-2)²) = √(4+4) = √8.
• For triangle DEF:
  • Side DE is from (2, 0) to (4, 4), so its length is √((4-2)² + (4-0)²) = √(4+16) = √20.
  • Side DF is from (2, 0) to (4, 2), so its length is 2.
  • Side EF is from (4, 4) to (4, 2), so its length is 2.

Since both triangles have the same side lengths of √20, 2, and √8, we can conclude that they are congruent by the Side-Side-Side (SSS) postulate. Therefore, triangle ABC is congruent to triangle DEF.