The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

• Move the line, the center of dilation, and set the scale factor.
• Move a point along the line and verify that the image points indeed form a line parallel to the preimage.

The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

• Move the segment, the center of dilation, and set the scale factor.
• Move a point along the segment and verify that the image points indeed form a segment parallel to the preimage.
• Check also that the ratio of the length of the image and the length of the preimage is indeed the scale factor. Note that the displayed lengths are only approximate, so the ratio will only be approximately equal to the scale factor.

Let $\angle B^{\prime }{A}^{\prime }{C}^{\prime }$ be the dilated image of $\angle BAC.$ According to the first property, $\overrightarrow{AC}$ is parallel to $\overrightarrow{A^{\prime }{C}^{\prime }}$ and $\overrightarrow{AB}$ is parallel to $\overrightarrow{A^{\prime }{B}^{\prime }}$

Let $M$ be the intersection point of $\overrightarrow{AB}$ and $\overrightarrow{A^{\prime }{C}^{\prime }},$ and focus on the parallel legs $\overrightarrow{AC}$ and $\overrightarrow{A^{\prime }{C}^{\prime }}.$ According to the Alternate Interior Angles Theorem, angles $\angle MAC$ and $\angle AM{A}^{\prime }$ are congruent.

Similarly, since the other legs are also parallel, angles $\angle B^{\prime }{A}^{\prime }{M}^{\prime }$ and $\angle AM{A}^{\prime }$ are congruent.

Since $\angle AM{A}^{\prime }$ is congruent to both $\angle BAC$ and $\angle B^{\prime }{A}^{\prime }{C}^{\prime },$ the transitive property of congruence implies that these two angles are congruent.

By the definition of congruence, this completes the proof that dilation preserves angle measures.