Write a two-column proof.
If FG ⊥ IJ and IG ≅ JG, then FI ≅ FJ
If IJ intersects FG at M, then angles IMG and JMG are right angles
MG is common to both triangles IMG and JMG.
So, via the Pythagorean Theorem, IM ≅ JM.
Now use that last fact, along with the common side FM to show that triangles IMF and JMF are congruent.
That means that IF≅JF
To write a two-column proof, we need to present the statements (or reasons) in one column and the corresponding justifications in the other column. Here's a step-by-step explanation of how to write the proof:
Statement | Justification
-----------------------------------------------------
1. FG ⊥ IJ | Given
2. IG ≅ JG | Given
3. ∠FIG ≅ ∠GJG | Corresponding parts of congruent triangles are congruent (CPCTC)
4. ΔFIG ≅ ΔGJG | SAS (side-angle-side) congruence
5. FI ≅ FJ | Corresponding parts of congruent triangles are congruent (CPCTC)
Now let's explain each step in more detail:
1. FG ⊥ IJ | Given
The given statement states that the line segment FG is perpendicular (⊥) to the line segment IJ.
2. IG ≅ JG | Given
The given statement states that the line segments IG and JG are congruent.
3. ∠FIG ≅ ∠GJG | Corresponding parts of congruent triangles are congruent (CPCTC)
Since IG ≅ JG, we can conclude that ∠FIG and ∠GJG are congruent by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.
4. ΔFIG ≅ ΔGJG | SAS (side-angle-side) congruence
By combining the information from step 1 (FG ⊥ IJ) and step 3 (∠FIG ≅ ∠GJG), we can conclude that ΔFIG and ΔGJG are congruent by the Side-Angle-Side (SAS) congruence theorem.
5. FI ≅ FJ | Corresponding parts of congruent triangles are congruent (CPCTC)
As ΔFIG ≅ ΔGJG (step 4), we can deduce that the corresponding parts of these congruent triangles are congruent. In this case, FI ≅ FJ.
By following these steps and providing clear explanations, you can construct a valid two-column proof for the given statement.