Question

triangle abc and triangle def are shown on the coordinate plane.
which statement correctly names the congruent triangles and justifies the reason for congruence?
△abc≅△efd by hl
△abc≅△def by hl
△abc≅△efd by aas
△bac≅△def by aas

Explanation:

Step1: Identify the right - angled triangles

Both \(\triangle ABC\) and \(\triangle DEF\) are right - angled triangles (marked with right - angle symbols).

Step2: Check the hypotenuse and leg (HL) congruence

In right - angled triangles, the Hypotenuse - Leg (HL) congruence criterion states that if the hypotenuse and a leg of one right - angled triangle are equal to the hypotenuse and a leg of another right - angled triangle, then the two triangles are congruent.
From the figure, the hypotenuse \(AC = DF=4.5\) (given in the figure). Also, by counting the units on the coordinate plane (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or by visual inspection of the grid), we can find that the legs (the non - hypotenuse sides) are also equal in length.

Answer:

\(\triangle ABC\cong\triangle DEF\) by \(HL\)