Given: Line cd is the perdendicular bisector of line ab
e is on line cd
Prove EB=ea
To prove that EB equals ea, we need to show that triangle EAB is an isosceles triangle. Here's how we can do that:
Step 1: Draw a diagram
Draw lines AB and CD intersecting at point E. Place point e on line CD.
Step 2: Determine the characteristics of a perpendicular bisector
A perpendicular bisector is a line that intersects another line at a 90-degree angle and divides it into two equal parts.
Step 3: Analyze based on the given information
From the given information, we know that line CD is the perpendicular bisector of line AB. This means that line CD intersects line AB at a 90-degree angle (forming a right angle) and divides it into two equal parts.
Step 4: Identify the midpoint
Since line CD is the perpendicular bisector, it divides line AB into two equal parts. Let's call the intersection point of line CD and line AB as point M. Point M will be the midpoint of line AB.
Step 5: Apply the properties of an isosceles triangle
Now, let's consider triangle EAB. We want to prove that EB equals ea. Since line CD is the perpendicular bisector of line AB, M is the midpoint of line AB. Therefore, ME = MA and ME = MB.
Step 6: Use the given information
We know that point E lies on line CD. Therefore, line ME coincides with line CE.
Step 7: Apply the transitive property of equality
Based on the above information, we have:
ME = MA (from Step 5)
ME = MB (from Step 5)
ME = MA = MB (using the transitive property of equality)
ME = MA = MB = CE (by the properties of the perpendicular bisector)
Step 8: Apply the reflexive property of equality
Using the reflexive property of equality, we can state that:
CE = EA
Step 9: Apply the transitive property of equality again
By using the transitive property of equality, we can conclude:
EB = EA
Hence, we have proven that EB equals ea.