Question

given: abcd is a parallelogram, \\(\overline{ac}\\) bisects \\(\angle bcd\\) and \\(\overline{db}\\) bisects \\(\angle abc\\). prove: \\(\overline{ac} \perp \overline{bd}\\). step statement reason 1 abcd is a parallelogram \\(\overline{ac}\\) bisects \\(\angle bcd\\) \\(\overline{db}\\) bisects \\(\angle abc\\) given 2 \\(\angle bca \cong \angle dca\\) an angle bisector divides an angle into two congruent angles 3 \\(\overline{ca} \cong \overline{ac}\\) reflexive property 4 \\(\overline{bc} \cong \overline{ad}\\) opposite sides of a parallelogram are congruent 5 \\(\overline{bd} \cong \overline{db}\\) reflexive property 6 \\(\overline{ac}\\) and \\(\overline{bd}\\) bisect each other the diagonals of a parallelogram bisect each other 7 \\(\overline{ae} \cong \overline{ce}\\) a segment bisector divides a segment into two congruent segments 8 \\(\overline{be} \cong \overline{ed}\\) a segment bisector divides a segment into two congruent segments try type of statement note: \\(\overline{bd}\\) and \\(\overline{ac}\\) are segments.

Explanation:

Step1: Define angle from bisector

$\angle CBD \cong \angle ABD$ (An angle bisector divides an angle into two congruent angles)

Step2: Use parallel side property

$\angle BCA \cong \angle CAD$ (Alternate interior angles of parallel sides $\overline{BC} \parallel \overline{AD}$ are congruent)

Step3: Substitute congruent angles

$\angle DCA \cong \angle CAD$ (From Step2 and $\angle BCA \cong \angle DCA$)

Step4: Derive congruent sides

$\overline{AD} \cong \overline{CD}$ (Sides opposite congruent angles in $\triangle ADC$ are congruent)

Step5: Identify rhombus property

$ABCD$ is a rhombus (A parallelogram with adjacent congruent sides is a rhombus, since $\overline{AD} \cong \overline{CD}$ and $\overline{AD} \cong \overline{BC}$, $\overline{AB} \cong \overline{CD}$)

Step6: Conclude perpendicular diagonals

$\overline{AC} \perp \overline{BD}$ (Diagonals of a rhombus are perpendicular to each other)

Answer:

$\overline{AC} \perp \overline{BD}$ is proven.