Question

prove the following using a two - column, statement/reason format. given: ac = bc; d is the midpoint of ab prove: δadc ≅ δbdc use a t st reasons 1. 1. given 2. 3 2. given 3. 3. base angles in isosceles triangles are equal

Explanation:

Step1: Analyze Given Information

We know \( AC = BC \), so \( \triangle ABC \) is isosceles with \( AC = BC \). The base angles of an isosceles triangle are equal. In \( \triangle ABC \), the sides \( AC \) and \( BC \) are equal, so the base is \( AB \), and the base angles are \( \angle A \) and \( \angle B \). So by the property of isosceles triangles (base angles are equal), we have \( \angle A=\angle B \). Wait, but let's check the problem's dropdown. Wait, the reason for step 3 is "Base angles in isosceles triangles are equal". So the statement for step 3 should be the equality of the base angles. Since \( AC = BC \), the triangle \( \triangle ABC \) has \( AC = BC \), so the base is \( AB \), so the base angles are \( \angle A \) and \( \angle B \). So the correct statement for step 3 (with reason "Base angles in isosceles triangles are equal") is \( \angle A=\angle B \). Wait, but let's re - examine. Wait, the given is \( AC = BC \), so \( \triangle ABC \) is isosceles with legs \( AC \) and \( BC \), base \( AB \). So base angles are \( \angle A \) (opposite \( BC \)) and \( \angle B \) (opposite \( AC \)). So by the isosceles triangle base angles theorem, \( \angle A=\angle B \). So the correct option from the dropdown (the options are \( \angle B = \angle C \), \( \angle A=\angle B \), \( \angle A=\angle C \)) is \( \angle A=\angle B \).

Step2: Construct the Two - Column Proof

Let's build the two - column proof properly:

Statements	Reasons
2. \( D \) is the mid - point of \( AB \)	2. Given
3. \( \angle A=\angle B \)	3. Base angles in isosceles triangles are equal (since \( AC = BC \), \( \triangle ABC \) is isosceles with base \( AB \))
4. \( AD = BD \)	4. Definition of mid - point (since \( D \) is the mid - point of \( AB \))
5. \( DC = DC \)	5. Reflexive property of congruence
6. \( \triangle ADC\cong\triangle BDC \)	6. SSS (Side - Side - Side) congruence criterion (\( AC = BC \), \( AD = BD \), \( DC = DC \)) or SAS (since \( AC = BC \), \( \angle A=\angle B \), \( AD = BD \))

But focusing on the dropdown in the problem, the third statement (with reason "Base angles in isosceles triangles are equal") should be \( \angle A=\angle B \).

Answer:

\( \angle A = \angle B \)