Write proof for the following
Given: line AC is congruent to line BD, line AD is congruent to BC
Prove: triangle DAX is congruent to triangle CBX
It's an isosceles trapezoid. From top left is labeled A. Top right labeled B. Bottom right labeled C. Bottom left labeled D. With X as the angle of AC and BD.
To prove that triangle DAX is congruent to triangle CBX, we will use the side-angle-side (SAS) congruence criterion.
Given the information:
1. line AC is congruent to line BD
2. line AD is congruent to line BC
We need to prove that:
triangle DAX is congruent to triangle CBX
Now, let's break down the proof step by step:
Step 1: Since line AC is congruent to line BD, we can conclude that angle AXD is congruent to angle BXC. This is because congruent line segments create congruent angles.
Step 2: From the given information, we know that line AD is congruent to line BC. This implies that angle DAX is congruent to angle CBX since congruent line segments create congruent angles.
Step 3: We have already established that angle AXD is congruent to angle BXC in Step 1.
Step 4: Now we have two pairs of congruent angles, AXD ≅ BXC and DAX ≅ CBX, along with the shared side AX ≅ BX.
Step 5: Using the SAS congruence criterion, where two triangles have two pairs of congruent sides and the included angle congruent, we conclude that triangle DAX is congruent to triangle CBX.
Therefore, it is proved that triangle DAX is congruent to triangle CBX based on the given information and the SAS congruence criterion.