Question

triangle ABC with A at 0 comma 0, B at 2 comma 4 and C at 0 comma 2, the measure of angle C is 135 degrees, and the measure of angle B is 30 degrees, triangle DEF with D at 2 comma 0, E at 4 comma 4 and F at 4 comma 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

Answer:

$\triangle ABC \cong \triangle EDF$

by SSS congruence.

Step-by-step explanation:

Given the two triangles with following coordinates and angles:

$\underline{\triangle ABC:}$

A(0, 0)

B(2, 4)

C(0, 2)

$\angle C = 135^\circ\\\angle B = 30^\circ$

$\underline{\triangle D EF :}$

D(2, 0)

E(4, 4)

F(4, 2)

$\angle F = 135^\circ\\\angle D = 30^\circ$

Let us calculate the sides of the two triangles one by one.

Distance formula:

$D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$(x_1, x_2)$ and $(y_1, y_2)$ are the two coordinates of which distance is to be calculated.

$AB = \sqrt{(4-0)^2+(2-0)^2} = \sqrt{20}$

$BC = \sqrt{(0-2)^2+(2-4)^2} = \sqrt{8}$

$AC= \sqrt{(0-0)^2+(0-2)^2} = 2$

$DE = \sqrt{(4-2)^2+(4-0)^2} = \sqrt{20}$

$EF = \sqrt{(4-4)^2+(4-2)^2} = 2$

$DF= \sqrt{(4-2)^2+(2-0)^2} = \sqrt8$

Sides AB = DE

BC = DF and

AC = EF

All the three sides are equal here. So SSS (Side Side Side) congruence.

And angles:

$\angle C = \angle F =135^\circ\\\angle B =\angle D = 30^\circ$

Therefore,

$\triangle ABC \cong \triangle EDF$ by SSS congruence.

Answer 2

Answer:

by SSS congruence.

Step-by-step explanation:

Given the two triangles with following coordinates and angles:

A(0, 0)

B(2, 4)

C(0, 2)

D(2, 0)

E(4, 4)

F(4, 2)

Let us calculate the sides of the two triangles one by one.

Distance formula:

and  are the two coordinates of which distance is to be calculated.

Sides AB = DE

BC = DF and

AC = EF

All the three sides are equal here. So SSS (Side Side Side) congruence.

And angles:

Therefore,

by SSS congruence.

Step-by-step explanation: