Question

1 for each figure, identify any lines of symmetry the figure has.
2 in quadrilateral badc, ab = ad and bc = dc. the line ac is a line of symmetry for this quadrilateral.
a. based on the line of symmetry, explain why the diagonals ac and bd are perpendicular.
b. based on the line of symmetry, explain why angles abc and adc have the same measure.

Explanation:

a. When a figure is reflected over a line of symmetry (in this case, line \(AC\)), corresponding points are equidistant from the line of symmetry. Let the intersection of \(AC\) and \(BD\) be point \(O\). Since \(AB = AD\) and \(BC=DC\), and \(AC\) is the line of symmetry, the reflection of \(B\) over \(AC\) is \(D\). The line \(BD\) is the pre - image and image connection for the reflection over \(AC\). In a reflection, the line joining a point and its image is perpendicular to the line of symmetry. So, \(BD\perp AC\).
b. When we reflect the quadrilateral \(BADC\) over the line of symmetry \(AC\), point \(B\) maps to point \(D\) and point \(C\) maps to itself. The angle \(\angle ABC\) is the pre - image and \(\angle ADC\) is the image of the reflection over \(AC\). In a reflection, the measure of an angle and its image are equal. So, \(m\angle ABC=m\angle ADC\).

Answer:

a. Because in a reflection over line \(AC\) (the line of symmetry), the line segment \(BD\) (connecting a point \(B\) and its image \(D\)) is perpendicular to the line of symmetry \(AC\).
b. Because when reflecting the quadrilateral over line \(AC\) (the line of symmetry), \(\angle ABC\) and \(\angle ADC\) are pre - image and image of each other under the reflection, and reflection preserves angle measure.