I II X X Statements Reasons 1. 𝐴𝐵 ≅ 𝐴𝐶 1. Given 2. 𝑅𝐵 ≅ 𝑅𝐶 2. Given 1. 𝐴𝐵 ≅ 𝐴𝐶 1. Given II 2. 𝑅𝐵 ≅ 𝑅𝐶 2. Given X X 3. 𝐴𝑅 ≅ 𝐴𝑅 3. ∡𝑍𝑋𝐽≅∡𝑍𝑌𝐾 3. Reflexive Postulate 4. ∆𝐴𝐵𝑅≅∆𝐴𝐶𝑅 4. 𝑆𝑆𝑆≅𝑆𝑆𝑆 5. Corresponding parts of congruent triangles are congruent 5. ∡𝐴𝑅𝐵≅∡𝐴𝑅𝐶 6. Angles on a line 6. ∡𝐵𝑅𝑆 𝑎𝑛𝑑 ∡𝐴𝑅𝐵 are supplements ∡𝐶𝑅𝑆 𝑎𝑛𝑑 ∡𝐴𝑅𝐶 are supplements 7. ∡𝑃≅∡𝑃 7. Complements of congruent angles are congruent 7. ∡𝐵𝑅𝑆≅∡𝐶𝑅𝑆 8. 𝑅𝑆 ≅ 𝑅𝑆 8. Reflexive Postulate 9. ∆𝑅𝐵𝑆≅∆𝑅𝐶𝑆 9. 𝑆𝐴𝑆≅𝑆𝐴𝑆 10. Corresponding parts of congruent triangles are congruent 10. 𝑆𝐵 ≅ 𝑆𝐶

I II Statements Reasons 1. 𝑱𝑲 ≅ 𝑱𝑳 1. Given 2. 𝑱𝑿 ≅ 𝑱𝒀 2. Given I 1. 𝑱𝑲 ≅ 𝑱𝑳 1. Given 2. 𝑱𝑿 ≅ 𝑱𝒀 2. Given I 3. ∡𝑲≅∡𝑳 3. In a triangle (KJL) angles opposite congruent sides are congruent 4. ∡𝑱𝑿𝒀≅∡𝑱𝒀𝑿 4. In a triangle (JXY) angles opposite congruent sides are congruent 6. Angles on a line 6. ∡𝑱𝑿𝒀 𝒂𝒏𝒅 ∡𝑱𝑿𝑲 are supplements ∡𝑱𝒀𝑿 𝒂𝒏𝒅 ∡𝑱𝒀𝑳 are supplements 7. ∡𝑱𝑿𝑲≅∡𝑱𝒀𝑳 7. Complements of congruent angles are congruent 8. ∆𝑱𝑲𝑿≅∆𝑱𝑳𝒀 8. 𝑨𝑨𝑺≅𝑨𝑨𝑺 9. 𝑲𝑿 ≅ 𝑳𝒀 9. Corresponding parts of congruent triangles are congruent

Lesson 8 - Properties of Parallelograms Unit 3 Lesson 8 - Properties of Parallelograms

No, not based on the immediate evidence. Is 𝐴𝑇 congruent to 𝐽𝑀 ? No, not based on the immediate evidence. What relationship does 𝐴𝑇 have to 𝐽𝑀 ? 𝐴𝑇 ∥ 𝐽𝑀 Is ∡𝑇 congruent to ∡J, why? Yes, they are alternate interior angles. ∆𝐴𝐺𝑇≅∆𝑀𝑌𝐽 S𝐴𝐴≅𝑆𝐴𝐴 𝐽𝑌

- a quadrilateral with both pairs of opposite sides parallel quad Prior knowledge: - prefix meaning four *** four-sided figure - meaning side - a quadrilateral with both pairs of opposite sides parallel quad lateral

Note: A and C can be the obtuse angles or the acute angles 1 2 4 3 A B C D 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 X Opposite sides and opposite angles are congruent Statements Reasons 1. 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 1. Given D 2. 𝐴𝐵 ∥ 𝐷𝐶 and 𝐴𝐷 ∥ 𝐵𝐶 2. Opposite sides of a parallelogram are parallel 3. ∡1 ≅ ∡2 and ∡3 ≅ ∡4 3. When parallel lines are cut by a transversal the alternate interior angles are congruent 4. 𝐷𝐵 ≅ 𝐷𝐵 4. Reflexive postulate 5. ∆𝐴𝐵𝐷≅∆𝐶𝐷𝐵 5. 𝐴𝑆𝐴≅𝐴𝑆𝐴 6. CPCTC 6. 𝐴𝐷 ≅ 𝐶𝐵 and 𝐴𝐵 ≅ 𝐶𝐷 7. ∡𝐴 ≅ ∡C 7. CPCTC Therefore, in a parallelogram the opposite sides and opposite angles are congruent

l V 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 The diagonals bisect each other Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB ∥ DC 2. Opposite sides of a parallelogram are parallel 3. ∡1 ≅ ∡2 and ∡3 ≅ ∡4 3. When parallel lines are cut by a transversal the alternate interior angles are congruent 4. AB ≅ DC 4. Opposite sides of a parallelogram are congruent 5. ∆ABE≅∆CDE 5. ASA≅ASA 6. CPCTC 6. BE ≅ DE and AE ≅ CE 7. A bisector divides a segment into two congruent segments 7. The diagonals bisect each other

II I X Statements Reasons Quadrilateral ABCD, with opposite sides congruent X 𝑨𝑩𝑪𝑫 𝒊𝒔 𝒂 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍𝒐𝒈𝒓𝒂𝒎 I Statements Reasons 1. Quadrilateral ABCD, with opposite sides congruent 1. Given 2. 𝑫𝑩 ≅ 𝑫𝑩 2. Reflexive postulate 3. ∆𝑨𝑩𝑫≅∆𝑪𝑫𝑩 3. 𝑺𝑺𝑺≅𝑺𝑺𝑺 4. ∡𝟏 ≅ ∡2 and ∡𝟑 ≅ ∡4 4. CPCTC 5. When two lines are cut by a transversal, making the alternate interior angles are congruent, the lines are parallel. 5. 𝑨𝑩 ∥ 𝑫𝑪 and 𝑨𝑫 ∥ 𝑩𝑪 6. 𝑨𝑩𝑪𝑫 𝒊𝒔 𝒂 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍𝒐𝒈𝒓𝒂𝒎 6. A quadrilateral with two pairs of opposite sides parallel is a parallelogram

Homework – Page 38