Question

7 from unit 1, lesson 15
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: Recall property of line - of - symmetry

A line of symmetry divides a figure into two congruent halves. Let the intersection of $AC$ and $BD$ be point $O$. Since $AC$ is a line of symmetry, $\triangle ABO$ and $\triangle ADO$ are congruent.

Step2: Analyze angles in congruent triangles

In congruent $\triangle ABO$ and $\triangle ADO$, $\angle AOB=\angle AOD$. Also, $\angle AOB+\angle AOD = 180^{\circ}$ (linear - pair of angles). So, $\angle AOB=\angle AOD = 90^{\circ}$, which means $AC\perp BD$.

Step3: Recall angle - related property of line - of - symmetry

Since $AC$ is the line of symmetry of quadrilateral $BADC$, $\triangle ABC$ and $\triangle ADC$ are congruent.

Step4: Identify corresponding angles in congruent triangles

In congruent $\triangle ABC$ and $\triangle ADC$, $\angle ACB$ and $\angle ACD$ are corresponding angles. Corresponding angles of congruent triangles are equal. So, $\angle ACB=\angle ACD$.

Answer:

a. Since $AC$ is the line of symmetry, $\triangle ABO\cong\triangle ADO$. $\angle AOB$ and $\angle AOD$ are a linear - pair and equal (from congruence), so $\angle AOB=\angle AOD = 90^{\circ}$, thus $AC\perp BD$.
b. Since $AC$ is the line of symmetry, $\triangle ABC\cong\triangle ADC$. $\angle ACB$ and $\angle ACD$ are corresponding angles of congruent triangles, so they have the same measure.