Question

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.

A. ΔABC and ΔDEF by SAS (Side-Angle-Side)
B. ΔABC and ΔDEF by ASA (Angle-Side-Angle)
C. ΔABC and ΔDEF by SSS (Side-Side-Side)
D. ΔABC and ΔDEF are not congruent

Answer 1

Triangles ▲ABC and ▲DEF are congruent by SSS congruence criterion as their corresponding sides are of equal length, calculated using the distance formula, and the angles at B and F are both 30 degrees.

Step-by-step explanation:

To determine if triangles ▲ABC and ▲DEF are congruent, we analyze each triangle's given angles and corresponding sides. By plotting the given points, we find the following lengths of the sides using the distance formula: AB=√((2-0)^2+(4-0)^2)= √20, BC=√((0-0)^2+(2-4)^2)= √4, and since A and C are on the same x-coordinate, AC is simply the vertical distance between them, which is 2. Analyzing ▲DEF in the same way, we conclude DE=√((4-2)^2+(4-0)^2)=√20, EF=√((4-4)^2+(2-0)^2)=√4, and DF, being on the same y-coordinate, is 2.

With the lengths of corresponding sides equal (AB=DE, BC=EF, and AC=DF) and the angles at B and F both being 30 degrees, we can confirm that the triangles are congruent by the SSS (Side-Side-Side) criterion. Hence, the answer is option C, ▲ABC and ▲DEF by SSS.