Proofs in Mathematics — Telangana (SSC) Class 9 Mathematics Solutions (Free)

Step 1: Given triangle ABC Step 2: Draw a line through C parallel to AB Step 3: Since AB ∥ to the line through C: Alternate interior angles are equal Step 4: ∠BAC = ∠ACE (where E is on the parallel line, on one side of C) Step 5: ∠ABC = ∠DCB (where D is on the parallel line, on other side of C) Step 6: The angle on the parallel line = ∠ACD + ∠DCB + ∠ACE Step 7: But angles on a straight line sum to 180° Step 8: ∠ACE + ∠ACB + ∠BCD = 180° Step 9: ∠BAC + ∠ACB + ∠ABC = 180° Final Answer: Sum of angl…

Step 1: Let triangle ABC where AB = AC (equal sides) Step 2: We need to prove ∠ABC = ∠ACB (base angles) Step 3: Construct the angle bisector of ∠BAC meeting BC at D Step 4: In triangles ABD and ACD: AB = AC (given) ∠BAD = ∠CAD (AD is angle bisector) AD = AD (common side) Step 5: By SAS congruence: △ABD ≅ △ACD Step 6: Therefore ∠ABD = ∠ACD Step 7: Which means ∠ABC = ∠ACB Final Answer: Base angles of isosceles triangle are equal

Step 1: Direct statement: If n² is even, then n is even Step 2: Contrapositive: If n is odd, then n² is odd Step 3: Assume n is odd, so n = 2k + 1 for some integer k Step 4: n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1 Step 5: n² = 2m + 1 where m = 2k² + 2k Step 6: Therefore n² is odd Step 7: Contrapositive is true, so original statement is true Final Answer: If n² is even, then n is even (proved by contrapositive)

Step 1: Assume there are finitely many primes: p₁, p₂, p₃, ..., pₙ Step 2: Consider the number N = p₁ × p₂ × p₃ × ... × pₙ + 1 Step 3: N is not divisible by p₁ (leaves remainder 1) Step 4: N is not divisible by p₂ (leaves remainder 1) Step 5: Similarly, N is not divisible by any of the listed primes Step 6: Therefore, N is either prime itself, or has a prime factor not in our list Step 7: This contradicts our assumption that p₁, ..., pₙ are ALL primes Step 8: Therefore, there must be infinitely …

Step 1: Axiom: A fundamental statement accepted as true without proof, universal to all mathematics. Example: Things equal to the same thing are equal to each other. Step 2: Postulate: A fundamental assumption specific to a domain (like geometry), accepted without proof. Example: A line can be drawn between any two points. Step 3: Theorem: A statement that is proved using axioms, postulates, and previously proved theorems. Example: Sum of angles in a triangle = 180°. Step 4: Corollary: A theo…