We are given that triangle ABC is a equilateral and D is the midpoint of line segment AB. How can we prove that triangle ACD is congruent to triangle BCD?
A=B
angle ADC = BCD
CD = CD
AC=BC
so now use SAS or SSS
To prove that triangle ACD is congruent to triangle BCD, we can use the property of an equilateral triangle that states all its sides are congruent.
Here's how we can prove this using the concept of congruent triangles:
1. Drawing: Start by drawing an equilateral triangle ABC and mark the midpoint of line segment AB as D.
2. Side AD = Side BD: Since D is the midpoint of AB, we know that side AD is congruent to side BD by definition.
3. Side CD = Side CD: The common side CD is congruent to itself by the reflexive property of congruence.
4. Side AC = Side BC: In an equilateral triangle, all sides are congruent. Therefore, side AC is congruent to side BC.
5. Angle ACD = Angle BCD: Both angles ACD and BCD are right angles, as CD is perpendicular to AB. Right angles are congruent.
By comparing the corresponding sides and angles of triangles ACD and BCD, we can conclude that triangle ACD is congruent to triangle BCD by the Side-Angle-Side (SAS) congruence criterion.
Hence, triangle ACD is congruent to triangle BCD, and we have successfully proven it.
To prove that triangle ACD is congruent to triangle BCD, we can use the Side-Angle-Side (SAS) congruence criterion. Here are the steps for the proof:
Step 1: Given that triangle ABC is equilateral, we know that all the sides of triangle ABC are congruent.
Step 2: Since D is the midpoint of AB, we can conclude that AD = DB, since D splits AB into two congruent segments.
Step 3: From step 1, we know that AC = BC, since triangle ABC is equilateral.
Step 4: Angle ACD is congruent to angle BCD. This is because they are vertical angles formed by intersecting lines AC and BC.
Step 5: Using the SAS congruence criterion, we have AD = DB (from step 2), AC = BC (from step 3), and angle ACD = angle BCD (from step 4). These satisfy the condition for triangle ACD to be congruent to triangle BCD.
Hence, triangle ACD is congruent to triangle BCD based on the SAS congruence criterion.