Let's pay close attention to the angles formed by the parallel segments and the sides of △ QRS. Here, we have two pairs of corresponding angles at the endpoints of XY and RS. By the Corresponding Angles Theorem, these angles are congruent.
These are corresponding angles in △ QXY and △ QRS. Therefore, by the Angle-Angle Similarity Postulate, these triangles are similar.
△ QXY~△ QRS
Step 2, △ ABC≅△ QXY
Knowing that △ QXY and △ QRS are similar triangles, we can write proportions connecting their side lengths. Let's also write two of the given proportions.
cc
Proportions Knowing & Given
that △ ABC ~ △ QRS & Proportions [0.8em]
QX/QR=XY/RS & AB/QR=BC/RS [1em]
QY/QS=XY/RS & AC/QS=BC/RS
Let's rewrite the above proportions.
Proportion
Rearrange
QX/QR=XY/RS
QX=XYQR/RS
AB/QR=BC/RS
AB=BCQR/RS
QY/QS=XY/RS
QY=XYQS/RS
AC/QS=BC/RS
AC=BCQS/RS
Recall that, by construction, we have that XY=BC. With this information, we can rewrite the expressions for QX and QY.
c|c
QX=XY QR/RS & QY=XY QS/RS
⇓ & ⇓
QX= BC QR/RS & QY= BC QS/RS
Note that now QX and AB are equal to the same expression. Similarly, QY and AC are equal to the same expression. By the Transitive Property of Equality, we can say that QX=AB and QY=AC.
AB=BCQR/RS QX=BC QR/RS ⇒ AB=QX [4em]
AB=BCQR/RS QX=BC QR/RS ⇒ AB=QX
Let's mark this information on the diagram.
We can see that three sides of △ ABC are congruent to three sides of △ QXY.
By the Side-Side-Side Congruence Postulate, the two triangles are congruent.
△ ABC≅△ QXY
Step 3, △ ABC~△ QRS
Since corresponding angles of congruent triangles are congruent, we know that ∠ A ≅ ∠ Q. Let's show this information on the diagram and turn our attention back to △ ABC and △ QRS.
It is given that the sides of these triangles are proportional. We now know that an angle of △ ABC is congruent to the corresponding angle in △ QRS. By the Side-Angle-Side Similarity Theorem, the triangles are similar.
△ ABC~△ QRS
