Question

1. 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 acb and acd have the same measure.

Explanation:

Step1: Analyze the property of line of symmetry

A line of symmetry divides a figure into two congruent parts. If \(AC\) is a line of symmetry for quadrilateral \(BADC\), then when we reflect the figure over \(AC\), point \(B\) maps to point \(D\) (since \(AB = AD\)) and point \(C\) maps to itself.

Step2: Use the property of congruent triangles

Since \(AB = AD\), \(BC=DC\), and \(AC = AC\) (common side), by SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle ADC\).

Step3: Conclude the relationship between \(AC\) and \(BD\)

For two congruent triangles \(\triangle ABC\) and \(\triangle ADC\) with line of symmetry \(AC\), the diagonals \(AC\) and \(BD\) are perpendicular. Because the line of symmetry is the perpendicular bisector of the segment joining the corresponding points (here \(B\) and \(D\)).

Step4: Analyze the angles \(\angle ACB\) and \(\angle ACD\)

Since \(\triangle ABC\cong\triangle ADC\), corresponding parts of congruent triangles are equal. So, \(\angle ACB=\angle ACD\)

Answer:

a. Since \(AC\) is a line of symmetry of quadrilateral \(BADC\) with \(AB = AD\) and \(BC = DC\), when we reflect the quadrilateral over \(AC\), the two - halves ( \(\triangle ABC\) and \(\triangle ADC\)) coincide. The line of symmetry \(AC\) is the perpendicular bisector of \(BD\) (as \(B\) and \(D\) are symmetric about \(AC\)), so \(AC\perp BD\).
b. Because \(\triangle ABC\cong\triangle ADC\) (by SSS, since \(AB = AD\), \(BC = DC\), \(AC=AC\)), \(\angle ACB\) and \(\angle ACD\) are corresponding angles of congruent triangles. So, \(\angle ACB=\angle ACD\) (corresponding parts of congruent triangles are equal)