Proofs in Mathematics

63

Step 1: We need to find the product of 7 and 9. Step 2: Multiply: 7 × 9 = 63. Step 3: Verify: 7 × 9 = 7 × (10-1) = 70 - 7 = 63. Step 4: Check that both numbers are odd: 7 = 2×3 + 1 (odd), 9 = 2×4 + 1 (odd). Step 5: The product 63 = 2×31 + 1 is also odd, confirming the theorem that product of two odd numbers is odd.

2k + 1

Step 1: An odd number is one that leaves remainder 1 when divided by 2. Step 2: Any integer can be written as 2k (even) or 2k+1 (odd) for some integer k. Step 3: Check: if k=0, then 2k+1=1 (odd); if k=1, then 2k+1=3 (odd); if k=2, then 2k+1=5 (odd). Step 4: Expression 2k gives even numbers, k+1 doesn't guarantee odd numbers, 3k gives multiples of 3. Step 5: Therefore, 2k+1 is the correct general form for any odd natural number.

A line parallel to one side through opposite vertex

Step 1: In triangle ABC, we construct line CE parallel to side BA through vertex C. Step 2: We also extend side BC to point D to form a straight line BCD. Step 3: Using properties of parallel lines, we establish that alternate angles are equal: ∠CAB = ∠ACE. Step 4: Similarly, corresponding angles are equal: ∠ABC = ∠DCE. Step 5: The three angles ∠ACE + ∠BCA + ∠DCE form a straight line (180°), proving the theorem.

Alternate angles are equal

Step 1: When we construct a line parallel to one side of the triangle through the opposite vertex, we create alternate angles. Step 2: The property states that when a transversal cuts two parallel lines, alternate angles are equal. Step 3: In the proof, ∠CAB (angle of triangle) = ∠ACE (alternate angle with parallel construction). Step 4: This equality allows us to substitute triangle angles with angles on a straight line. Step 5: Since angles on a straight line sum to 180°, we prove that triangle angles also sum to 180°.