(03.01 mc)

triangle abc is translated by the rule (x + 1, y + 1) and then dilated by a scale factor of 2 centered at the origin. which statement describes the properties of triangles abc and a''b''c'' after the transformations?

a. ∠c and ∠c'' are congruent after both the translation and the dilation.

b. ∠c and ∠c'' are congruent after the dilation, but not after the translation.

c. segment ac and segment a double prime c double prime are congruent after both the translation and the dilation.

d. segment ac and segment a double prime c double prime are congruent after the dilation, but not after the translation.

Option A. ∠C and ∠C'' are congruent after both the translation and the dilation

Step-by-step explanation:

we know that

The translation is a rigid transformation. After the translation the image and the pre-image are congruent

The dilation is a non rigid transformation. After the dilation the image and the pre-image are similar

Remember that

If two figures are similar, then its corresponding angles are congruent and the ratio of its corresponding sides is proportional

If two figures are congruent, then its corresponding angles and corresponding sides are congruent

therefore

∠C and ∠C'' are congruent after both the translation and the dilation

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)