Question

1. in quadrilateral badc, ab = ad and bc = dc. the line ac is a line of symmetry for this quadrilateral. based on the line of symmetry, explain why angles acb and acd have the same measure. (lesson 1-10)

Explanation:

A line of symmetry divides a figure into two congruent parts. When we reflect the quadrilateral \(BADC\) over the line \(AC\), point \(B\) maps to point \(D\) (since \(AB = AD\)) and point \(C\) maps to itself. The angle \(\angle ACB\) maps to \(\angle ACD\) under this reflection. Congruent transformations (reflections) preserve angle - measures.

Answer:

Since \(AC\) is a line of symmetry, the reflection over \(AC\) maps \(\angle ACB\) to \(\angle ACD\). A reflection is a congruence transformation, which means it preserves the measure of angles. So, \(\angle ACB\) and \(\angle ACD\) have the same measure.