Question

proof write a two - column proof or a paragraph proof.

1. given $overline{bd} perp overline{ac}$, $overline{bd}$ bisects $overline{ac}$.

prove $angle abd$ and $angle bcd$ are complementary angles.

Explanation:

Step1: Use the definition of perpendicular lines

Since \( \overline{BD}\perp\overline{AC}\), by the definition of perpendicular lines, \( \angle BDA=\angle BDC = 90^{\circ}\). In \(\triangle ABD\), \(\angle A+\angle ABD+\angle ADB=180^{\circ}\) (angle - sum property of a triangle). Substituting \(\angle ADB = 90^{\circ}\), we get \(\angle A+\angle ABD=90^{\circ}\).

Step2: Use the definition of a bisector

Since \( \overline{BD}\) bisects \( \overline{AC}\), \(AD = DC\). In \(\triangle ABD\) and \(\triangle CBD\):

• \(AD = DC\) (given, \(BD\) bisects \(AC\))
• \(\angle ADB=\angle CDB = 90^{\circ}\) (from Step 1)
• \(BD=BD\) (common side)

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ABD\cong\triangle CBD\). Then, \(\angle A=\angle BCD\) (corresponding parts of congruent triangles are equal).

Step3: Substitute \(\angle A\) with \(\angle BCD\)

Since \(\angle A+\angle ABD = 90^{\circ}\) (from Step 1) and \(\angle A=\angle BCD\) (from Step 2), substituting \(\angle A\) with \(\angle BCD\) gives \(\angle BCD+\angle ABD=90^{\circ}\)

Answer:

Since \(\angle BCD+\angle ABD = 90^{\circ}\), \(\angle ABD\) and \(\angle BCD\) are complementary angles.