Question

ac⊥bd is given.
d is the midpoint of ac is given.
∠a≅∠c is given.
△abd≅△cbd by asa
△abc is an isosceles triangle
∠bda≅∠bdc because all right angles are congruent.
ab≅cb by definition of a midpoint
ad≅cd by definition of a midpoint
ab≅cb by cpctc
∠abd≅∠cbd because all right angles are congruent.

Explanation:

Step1: Analyze the given information

We know that \(AC\perp BD\), so \(\angle ADB=\angle CDB = 90^{\circ}\). \(D\) is the mid - point of \(AC\), so \(AD = CD\) (by the definition of a mid - point). Also, \(\angle A=\angle C\) (given).

Step2: Use the ASA (Angle - Side - Angle) congruence criterion

In \(\triangle ABD\) and \(\triangle CBD\), we have \(\angle A=\angle C\), \(AD = CD\), and \(\angle ADB=\angle CDB\). So, \(\triangle ABD\cong\triangle CBD\) (by ASA).

Step3: Use the CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle ABD\cong\triangle CBD\), then \(AB = CB\) (by CPCTC).

Step4: Use the definition of an isosceles triangle

A triangle is isosceles if at least two of its sides are congruent. Since \(AB = CB\) in \(\triangle ABC\), \(\triangle ABC\) is an isosceles triangle.

Answer:

\(\triangle ABC\) is an isosceles triangle.