Question 1173654: Provide a two column proof or paragraph proof.
Given MA and AI are perpendicular: RS and AI are perpendicular : K is the midpoint of MT.
Prove: Triangles MAK and TIK are congruent. (If didn't work prove Triangles MKI and TKA are congruent)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's provide a two-column proof for this geometry problem.
Given:
MA ⊥ AI
RS ⊥ AI
K is the midpoint of MT
Prove:
ΔMAK ≅ ΔTIK
Two-Column Proof:
Statement Reason
1. MA ⊥ AI Given
2. ∠MAI = 90° Definition of perpendicular lines
3. RS ⊥ AI Given
4. ∠RIS = 90° Definition of perpendicular lines
5. ∠MAI = ∠RIS Both are 90° angles
6. ∠TIK = ∠MAK Vertical angles are congruent
7. K is the midpoint of MT Given
8. MK = TK Definition of midpoint
9. ∠TIK = ∠MAK Vertical angles are congruent
10. ΔMAK ≅ ΔTIK Angle-Side-Angle (ASA) Congruence Theorem (∠MAK, MK, ∠MKA and ∠TIK, TK, ∠TKI)
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Paragraph Proof:
We are given that MA is perpendicular to AI and RS is perpendicular to AI. This means that angles MAI and RIS are both 90 degrees, making them congruent. We are also given that K is the midpoint of MT, which means that line segments MK and TK are congruent. Since vertical angles are congruent, angle TIK is congruent to angle MAK. With angle MAK congruent to angle TIK, MK congruent to TK and angle MKA congruent to angle TKI, we can apply the Angle-Side-Angle (ASA) congruence theorem. Therefore, triangles MAK and TIK are congruent.
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