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Download 4.3: The Rectangle, Square, and Rhombus The Rectangle
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Transcript
4.3: The Rectangle, Square, and Rhombus A B D C The Rectangle: Definition – A rectangle is a parallelogram that has a right angle. ▭ “rect” Corollary 4.3.1 All angles of a rectangle are right angles Corollary 4.3.2 The diagonals of a rectangle are congruent M N Q P Given: ▭𝑀𝑁𝑃𝑄 with diagonals 𝑀𝑃 and 𝑁𝑄 Prove: 𝑀𝑃 ≅ 𝑁𝑄 Proof: 1. 2. 3. 4. 5. 6. 7. 8. ▭𝑀𝑁𝑃𝑄 with diagonals 𝑀𝑃 and 𝑁𝑄 𝑀𝑁𝑃𝑄 is a parallelogram 𝑀𝑁 ≌ 𝑄𝑃 𝑀𝑄 ≌ 𝑀𝑄 ∠𝑁𝑀𝑄 and ∠𝑃𝑄𝑀 are right angles ∠𝑁𝑀𝑄 ≌ ∠PQM ∆𝑁𝑀𝑄 ≌ ∆𝑃𝑄𝑀 𝑀𝑃 ≅ 𝑁𝑄 1. Given 2. By definition a rect. is a parallelogram with one right angle 3. opposite sides of a parallelogram 4. Identity 5. Corollary 4.3.1 6. Right angles 7. SAS 8. CPCTC The Square: Definition – A square is a rectangle that has two congruent adjacent sides A B D C Corollary 4.3.3 All sides of a square are congruent. The Rhombus: Definition – A rhombus is a parallelogram with two congruent adjacent sides Corollary 4.3.4 All sides of a rhombus are congruent. Theorem 4.3.5 The diagonals of a rhombus are perpendicular. Given: Rhombus ABCD with diagonals 𝐴𝐶 and 𝐷𝐵 Prove: 𝐴𝐶 ⊥ 𝐷𝐵 Picture Proof C D E A B Corollary 4.3.6 The diagonals of a rhombus (or square) are perpendicular bisectors of each other. The Pythagorean Theorem C D A B 4.4: The Trapezoid Definition – A trapezoid is a quadrilateral with exactly two parallel sides. “trap” 𝐴 base 𝐴 𝐵 leg leg base 𝐷 𝐴 𝐵 median 𝐷 𝐶 𝐶 𝐷 𝐷 𝐵 𝐶 𝐵 Right trap Isosceles trap Definition – An altitude of a trapezoid is a line segment from one base of the trapezoid perpendicular to the opposite base (or to an extension of that base). 𝐴 𝐴 𝐵 𝐶 𝐷 𝐶 Given Trapezoid 𝑅𝑆𝑇𝑉 with 𝑅𝑉 ≌ 𝑆𝑇 and 𝑅𝑆 ∥ 𝑉𝑇 . Picture Proof Theorem 4.4.1 The base angles of an isosceles trapezoid are congruent. Prove: ∠𝑉 ≌ ∠𝑇 and ∠𝑅 ≌ ∠𝑆 . 𝑅 𝑉 𝑌 𝑆 𝑍 Proof: By drawing 𝑅𝑌 ⊥ 𝑉𝑇 and 𝑆𝑍 ⊥ 𝑉𝑇, we see that 𝑅𝑌 ≌ 𝑆𝑍 (Th. 4.1.6). By HL, ∆𝑅𝑌𝑉 ≌ ∆𝑆𝑍𝑇, so ∠𝑉 ≌ ∠𝑇 (CPCTC). ∠𝑅 ≌ ∠𝑆 because these angles are supplementary to congruent angles (∠𝑉 𝑎𝑛𝑑∠𝑇) 𝑇 Corollary 4.4.2 The diagonals of an isosceles trapezoid are congruent. 𝐵 𝐴 𝐶 𝐷 Theorem 4.4.3 The length of the median of a trapezoid equals one-half the sum of the length of the two bases. 𝐴 Theorem 4.4.4 The median of a trapezoid is parallel to each base. 𝑀 𝐷 𝐵 𝑁 𝐶 Theorem 4.4.5 If two base angles of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. Given Trapezoid 𝑅𝑆𝑇𝑉 with 𝑅𝑆 ∥ 𝑉𝑇 and ∠𝑉 ≌ ∠𝑇 . 𝑅 𝑆 Prove: 𝑅𝑆𝑇𝑉 is a n isosceles trapezoid. Plan: Draw auxiliary altitudes 𝑅𝑋𝑎𝑛𝑑 𝑆𝑌. Because 𝑅𝑋 ≌ 𝑆𝑌 by Thm 4.1.6, we can show that ∆𝑅𝑋𝑉 ≌ ∆𝑆𝑌𝑇 by AAS. Then 𝑅𝑉 ≌ 𝑆𝑇 by CPCTC and 𝑅𝑆𝑇𝑉 is an isosceles trapezoid. 𝑉 𝑋 𝑇 𝑌 Theorem 4.4.6 If the diagonals of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. 𝐴 𝐷 𝐵 𝐸 𝐹 𝐶 Theorem 4.4.7 If three (or more) parallel lines intercept congruent line segments on one transversal, then they intercept congruent line segments on any transversal. Given Parallel lines a, b and c, cut by transversal t so that 𝐴𝐵 ≌ 𝐵𝐶 . 𝑡 Prove: 𝐷𝐸 ≌ 𝐸𝐹. 𝑎 Plan: Through D and E, draw 𝐷𝑅 ∥ 𝐴𝐵 and 𝐸𝑆 ∥ 𝐴𝐵. In each parallelogram formed, 𝐷𝑅 ≌ 𝐴𝐵 and 𝐸𝑆 ≌ 𝐵𝐶. Given 𝐴𝐵 ≌ 𝐵𝐶 it follows that 𝐷𝑅 ≌ 𝐸𝑆. By AAS, we can show ∆𝐷𝐸𝑅 ≌ ∆𝐸𝐹𝑆; Then by CPCTC 𝐷𝐸 ≌ 𝐸𝐹. 𝑏 𝑚 𝐴 𝐷 𝐵 𝐸 𝑅 𝑐 𝐹 𝐶 𝑆
