given that angle ACF ~=~ angle ECF and measure of angle ABF = measure of angle EDF = 90, write a two-column proof to show that Triangle ABF ~=~ triangle EDF
Statements Reasons
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1. Angle ACF ~=~ Angle ECF / Given
2. Measure of Angle ABF = 90° / Given
3. Measure of Angle EDF = 90° / Given
4. Triangle ABF is a right triangle / Definition of a right triangle
5. Triangle EDF is a right triangle / Definition of a right triangle
6. Triangle ABF has a right angle / Definition of a right triangle
7. Triangle EDF has a right angle / Definition of a right triangle
8. Triangle ABF and Triangle EDF have one angle of the same measure / Angles congruent to a right angle are congruent
9. Triangle ABF and Triangle EDF are similar / Definition of similar triangles (AA Similarity)
10. Triangle ABF ~=~ Triangle EDF / Definition of congruent triangles
To prove that triangle ABF is congruent to triangle EDF, we will use the following information:
Given:
1. Angle ACF is congruent to angle ECF.
2. Angle ABF measures 90 degrees.
3. Angle EDF measures 90 degrees.
To prove:
Triangle ABF is congruent to triangle EDF.
Proof:
Statements | Reasons
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1. ABF is a right triangle. | Given
2. EDF is a right triangle. | Given
3. Angle ACF is congruent to angle ECF. | Given
Now, let's prove that triangle ABF is congruent to triangle EDF using the Side-Angle-Side (SAS) congruence criterion:
4. AF is congruent to DF. | Common hypotenuse of right triangles ABF and EDF.
5. Angle BAF is congruent to angle EDF. | Right angles are congruent.
6. Triangle ABF is congruent to triangle EDF. | SAS criterion (sides AF and DF are congruent, angle BAF and angle EDF are congruent).
Hence, triangle ABF is congruent to triangle EDF.
where is C?
How are ABF and EDF related? They share a vertex, but there are lots of ways that can happen.