This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Let's prove that the diagonals of a rectangle meet at right angles if and only if it is a square. This requires proving two directions. Let ABCD be a rectangle with diagonals AC and BD intersecting at point O. Part 1: If the diagonals of a rectangle meet at right angles, then it is a square. Step 1: State the given conditions and properties of a rectangle. Given that ABCD is a rectangle and its diagonals AC and BD intersect at O such that ∠AOB = 90°. In a rectangle, the diagonals are equal in length (AC = BD) and bisect each other (AO = OC and BO = OD). Step 2: Deduce equality of segments. Since AC = BD and the diagonals bisect each other, it follows that AO = OC = BO = OD. Step 3: Prove congruence of adjacent triangles. Consider triangle AOB and triangle BOC. • AO = CO (from Step 2) • BO = BO (common side) • ∠AOB = 90° (given) • ∠BOC = 180° - ∠AOB = 180° - 90° = 90° (angles on a straight line) Therefore, triangle AOB is congruent to triangle BOC by the SAS (Side-Angle-Side) congruence criterion. Step 4: Conclude that the rectangle is a square. Since triangle AOB is congruent to triangle BOC, their corresponding sides are equal. Thus, AB = BC. A rectangle with adjacent sides equal (AB = BC) is a square. Part 2: If a rectangle is a square, then its diagonals meet at right angles. Step 1: State the given conditions and properties of a square. Given that ABCD is a square. In a square, all sides are equal (AB = BC = CD = DA). A square is also a rectangle, so its diagonals are equal (AC = BD) and bisect each other (AO = OC and BO = OD). Step 2: Deduce equality of segments. Since AC = BD and the diagonals bisect each other, it follows that AO = OC = BO = OD. Step 3: Prove congruence of adjacent triangles. Consider triangle AOB and triangle AOD. • AO = AO (common side) • BO = DO (from Step 2) • AB = AD (sides of a square) Therefore, triangle AOB is congruent to triangle AOD by the SSS (Side-Side-Side) congruence criterion. Step 4: Conclude that the diagonals meet at right angles. Since triangle AOB is congruent to triangle AOD, their corresponding angles are equal. Thus, ∠AOB = ∠AOD. Also, B, O, D are collinear, so ∠AOB + ∠AOD = 180° (angles on a straight line). Substituting ∠AOB for ∠AOD, we get 2 × ∠AOB = 180°. Therefore, ∠AOB = 90°. This means the diagonals of a square meet at right angles. Since both directions have been proven, the diagonals of a rectangle meet at right angles if and only if it is a square. What's next? 📸