Question

1. in quadrilateral badc, ab = ad and bc = dc. the line ac is a line of symmetry for this quadrilateral. explain why the diagonals ac and bd are perpendicular.

Explanation:

Step1: Recall congruent - triangle properties

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

Step2: Analyze corresponding angles

Since $\triangle ABC\cong\triangle ADC$, $\angle BAC=\angle DAC$ and $\angle BCA=\angle DCA$.

Step3: Consider isosceles - triangle in $\triangle ABD$ and $\triangle BCD$

In $\triangle ABD$, because $AB = AD$, $\triangle ABD$ is isosceles. The line $AC$ is the angle - bisector of $\angle BAD$. Similarly, in $\triangle BCD$ with $BC = DC$, $AC$ is the angle - bisector of $\angle BCD$.

Step4: Use angle - sum property

Let the intersection of $AC$ and $BD$ be point $O$. In $\triangle ABO$ and $\triangle ADO$, $\angle BAO=\angle DAO$, $AB = AD$, and $AO=AO$. So, $\triangle ABO\cong\triangle ADO$ (SAS - Side - Angle - Side). Then $\angle AOB=\angle AOD$. Since $\angle AOB+\angle AOD = 180^{\circ}$, $\angle AOB=\angle AOD = 90^{\circ}$. So, $AC\perp BD$.

Answer:

The diagonals $AC$ and $BD$ are perpendicular because of the congruence of triangles formed by the equal - side lengths of the quadrilateral and the properties of isosceles triangles and angle - bisectors. The line $AC$ is a line of symmetry as it divides the quadrilateral into two congruent halves, and the perpendicularity of the diagonals is a result of the geometric relationships established through congruence proofs.