Given: angle EBA is congruent to angle CBD, measure of angle ABD is = to 90 degrees.
Prove: angle EBA is the complement of the angle ABC.
Write a proof....
I have no idea how to proceed...help!
Thank you!
all the angles add up to 180.
Since angle ABD is 90, the other two must add up to 90.
So, they are complementary angles.
To prove that angle EBA is the complement of angle ABC, we need to show that the sum of angles EBA and ABC equals 90 degrees.
Proof:
Given:
1. Angle EBA is congruent to angle CBD.
2. Measure of angle ABD is equal to 90 degrees.
To prove:
Angle EBA is the complement of angle ABC.
Proof:
1. Draw a line segment BD.
2. Place point C on the same side of line segment BD as point E.
3. Construct line segment AC.
Statement | Reason
---------------------------------------------------------
1. Angle EBA is congruent to angle CBD | Given
2. Angle ABD is equal to 90 degrees | Given
3. Line segment BD and AC intersect at point B | Construction
4. Angle EBA + angle ABC = angle ABD | Angle Addition Postulate
5. Angle EBA + angle CBD = angle ABD | Given
6. Angle EBA + angle ABC = angle EBA + angle CBD | Transitive property of equality
7. Angle ABC = angle CBD | Subtract angle EBA from both sides (Subtraction Property of Equality)
8. Angle EBA + angle ABC = angle EBA + angle ABC | Substitution property
9. Angle EBA + angle ABC = angle ABD | Transitive property of equality
10. Angle EBA + angle ABC = 90 degrees | Substitution for angle ABD
11. Therefore, angle EBA is the complement of angle ABC | Definition of Complementary Angles
Therefore, we've shown that angle EBA is the complement of angle ABC.
To prove that angle EBA is the complement of angle ABC, we need to show that the sum of their measures equals 90 degrees.
Let's break down the steps to prove this:
Step 1: Given that angle EBA is congruent to angle CBD and the measure of angle ABD is 90 degrees.
Step 2: Since angle EBA is congruent to angle CBD, we can say that angle EBA = angle CBD.
Step 3: From step 2, we know that angle EBA + angle CBD = angle EBA + angle EBA (since angle EBA = angle CBD).
Step 4: Simplifying step 3, we have 2 * angle EBA = angle EBA + angle EBA.
Step 5: Combining like terms, 2 * angle EBA = 2 * angle EBA.
Step 6: Canceling out 2 on each side, we get angle EBA = angle EBA.
Step 7: Therefore, angle EBA is equal to itself.
Step 8: From step 1, we know that angle ABD = 90 degrees.
Step 9: Since angle ABD = angle ABC + angle CBD (from triangle ABD), we can substitute the measure of angle ABC to angle ABD - angle CBD.
Step 10: Substituting the known values, 90 degrees = angle ABC + angle CBD - angle CBD.
Step 11: Simplifying, 90 degrees = angle ABC.
Step 12: Therefore, angle EBA is the complement of angle ABC, as the sum of their measures is 90 degrees.
In this proof, we used the given information and fundamental principles of geometry, such as the congruent angles and the sum of angles in a triangle being 180 degrees. By breaking down the problem and using deductive reasoning, we were able to establish that angle EBA is indeed the complement of angle ABC.