Geometry Theorems 1 REMC Tutoring
- 1. Theorems Midpoint Theorem 𝐼𝑓 𝐵 𝑖𝑠 𝑡ℎ𝑒 𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶 ̅̅̅̅, 𝑡ℎ𝑒𝑛 𝐴𝐵 ̅̅̅̅ ≅ 𝐵𝐶 ̅̅̅̅ Segment Congruence Theorems Reflexive Symmetric Transitive 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑠 𝑡ℎ𝑒 𝐴𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑜𝑓 𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒, 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐, 𝑎𝑛𝑑 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝐴𝐵 ̅̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐼𝑓 𝐴𝐵 ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅, 𝑡ℎ𝑒𝑛 𝐶𝐷 ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐼𝑓 𝐴𝐵 ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅ 𝑎𝑛𝑑 𝐶𝐷 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅, 𝑡ℎ𝑒𝑛 𝐴𝐵 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅ 𝑎𝑛𝑑 𝐶𝐷 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐴𝐵 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ (𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑖𝑠 𝑢𝑠𝑒𝑑 𝑤ℎ𝑒𝑛 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛𝑜𝑡ℎ𝑒𝑟) Supplement Theorem 𝐼𝑓 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑎𝑖𝑟 𝑖𝑠 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 (𝑠𝑢𝑚 𝑜𝑓 𝑚∠1 𝑎𝑛𝑑 𝑚∠2 𝑒𝑞𝑢𝑎𝑙𝑠 180°) Complement Theorem 𝐼𝑓 𝑡ℎ𝑒 𝑢𝑛𝑐𝑜𝑚𝑚𝑜𝑛 𝑠𝑖𝑑𝑒𝑠 𝑓𝑟𝑜𝑚 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎𝑛𝑔𝑙𝑒𝑠 𝑓𝑜𝑟𝑚 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 (𝑠𝑢𝑚 𝑜𝑓 𝑚∠1 𝑎𝑛𝑑 𝑚∠2 𝑒𝑞𝑢𝑎𝑙𝑠 90°) (𝑝𝑢𝑟𝑝𝑙𝑒 𝑏𝑜𝑥 𝑠𝑦𝑚𝑏𝑜𝑙 𝑚𝑒𝑎𝑛𝑠 𝑏𝑙𝑎𝑐𝑘 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑎 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑜𝑟 90°)
- 2. Congruent Supplement Theorem (3 angles) Congruent Supplement Theorem (4 angles) 𝐼𝑓 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑜𝑛𝑒 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑤𝑖𝑡ℎ 𝑎 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑔𝑙𝑒, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑎𝑛𝑔𝑙𝑒. 𝐼𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒𝑠 𝑐𝑟𝑒𝑎𝑡𝑒 𝑡𝑤𝑜 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑝𝑎𝑖𝑟𝑠, 𝑎𝑛𝑑 𝑜𝑛𝑒 𝑎𝑛𝑔𝑙𝑒 𝑓𝑟𝑜𝑚 𝑒𝑎𝑐ℎ 𝑠𝑒𝑡 𝑖𝑠 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑠𝑒𝑡, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑛𝑜𝑡 𝑙𝑎𝑏𝑒𝑙𝑒𝑑 𝑎𝑠 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑙𝑠𝑜. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ∠2 ≅ ∠3 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠2 ≅ ∠3 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, ∠1 ≅ ∠4 Congruent Complement Theorem 𝐼𝑓 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑜𝑛𝑒 𝑖𝑠 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑤𝑖𝑡ℎ 𝑎 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑔𝑙𝑒, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑎𝑛𝑔𝑙𝑒. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, ∠1 ≅ ∠3 Vertical Angles Theorem 𝐼𝑓 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡. ∠1 ≅ ∠3 ∠2 ≅ ∠4
- 3. Right Angle Theorems Perpendicular Lines Intersect to Form Four Right Angles All Right Angles are Congruent Perpendicular Lines Will Form Four Congruent Adjacent Angles If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles 𝐼𝑓 𝑡𝑤𝑜 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟, 𝑡ℎ𝑒𝑛 𝑓𝑜𝑢𝑟 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡. 𝐼𝑓 𝑡𝑤𝑜 𝑜𝑟 𝑚𝑜𝑟𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑎𝑙𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡. 𝐼𝑓 𝑡𝑤𝑜 𝑙𝑖𝑛𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑡𝑜 𝑎𝑡 𝑎 90° 𝑎𝑛𝑔𝑙𝑒, 𝑡ℎ𝑒𝑛 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎𝑛𝑔𝑙𝑒𝑠 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑. 𝐼𝑓 𝑡𝑤𝑜 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑏𝑜𝑡ℎ 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑖𝑛 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑠 180°, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑒𝑎𝑐ℎ 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 90° 𝐼𝑓 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑎𝑖𝑟 𝑖𝑠 𝑓𝑜𝑟𝑚𝑒𝑑 𝑓𝑟𝑜𝑚 𝑡𝑤𝑜 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑏𝑜𝑡ℎ 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 90° 𝐹𝑜𝑢𝑟 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑐𝑟𝑒𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝐿𝑖𝑛𝑒𝑠 𝐴𝑙𝑙 𝑡ℎ𝑒 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑒𝑞𝑢𝑎𝑙 90° ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠4 𝑎𝑛𝑑 ∠1 𝑎𝑟𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑙𝑙 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒 𝑃𝑎𝑖𝑟𝑠 𝑎𝑟𝑒 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑛𝑑 𝑎𝑙𝑠𝑜 𝑡ℎ𝑒 𝑝𝑎𝑖𝑟𝑠 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦, 𝑤ℎ𝑖𝑐ℎ 𝑚𝑒𝑎𝑛𝑠 𝑒𝑎𝑐ℎ 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒. 𝐴𝑙𝑙 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒 𝑃𝑎𝑖𝑟𝑠 𝑎𝑟𝑒 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑛𝑑 𝑎𝑙𝑠𝑜 𝑐𝑟𝑒𝑎𝑡𝑒 𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟𝑠 𝑤ℎ𝑖𝑐ℎ 𝑚𝑒𝑎𝑛𝑠 𝑒𝑎𝑐ℎ 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒. Theorem Proofs Midpoint Theorem: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: 𝐴𝐵 ̅̅̅̅ ≅ 𝐵𝐶 ̅̅̅̅ 𝐵 𝑖𝑠 𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶 ̅̅̅̅ 𝑔𝑖𝑣𝑒𝑛 𝐴𝐵 = 𝐵𝐶 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑎 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵 ̅̅̅̅ ≅ 𝐵𝐶 ̅̅̅̅ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒
- 4. Segment Congruence Theorems: Reflexive Property: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: 𝐴𝐵 ̅̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝑔𝑖𝑣𝑒𝑛 𝐴𝐵 = 𝐴𝐵 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 𝐴𝐵 = 𝐴𝐵 𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒 𝐴𝐵 ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 Symmetric Property: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: 𝐶𝐷 ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅ 𝑔𝑖𝑣𝑒𝑛 𝐴𝐵 = 𝐶𝐷 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 𝐶𝐷 = 𝐴𝐵 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝐶𝐷 ̅̅̅̅ ≅ 𝐴𝐵 ̅̅̅̅ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 Transitive Property: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡: 𝐴𝐵 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ ≅ 𝐶𝐷 ̅̅̅̅ 𝑎𝑛𝑑 𝐶𝐷 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝑔𝑖𝑣𝑒𝑛 𝐴𝐵 = 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 = 𝐸𝐹 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 𝐴𝐵 = 𝐸𝐹 𝐴𝐵 = 𝐸𝐹 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑜𝑟, 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑠𝑢𝑏 𝐴𝐵 𝑓𝑜𝑟 𝐶𝐷) 𝐴𝐵 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑒𝑔𝑚𝑒𝑛𝑡 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑐𝑒 Supplement Theorem: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠1 𝑎𝑛𝑑 ∠2 𝑓𝑜𝑟𝑚 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑎𝑖𝑟 𝑔𝑖𝑣𝑒𝑛 𝑁𝑜𝑡𝑒: 𝐻𝑜𝑤 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟? 𝑇𝑤𝑜 𝐴𝑛𝑔𝑙𝑒𝑠 𝑡ℎ𝑎𝑡: 𝐴𝑟𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑈𝑛𝑠ℎ𝑎𝑟𝑒𝑑 𝑠𝑖𝑑𝑒𝑠 𝑓𝑜𝑟𝑚 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑟𝑎𝑦𝑠 (180°) 𝑚∠1 + 𝑚∠2 = 180° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 Complement Theorem: 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎𝑛𝑔𝑙𝑒𝑠 ∠1 𝑎𝑛𝑑 ∠2 𝑓𝑜𝑟𝑚 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑔𝑖𝑣𝑒𝑛 𝑁𝑜𝑡𝑒: 𝐻𝑜𝑤 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠? 𝑃𝑎𝑖𝑟 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑆ℎ𝑎𝑟𝑒 𝑎 𝑉𝑒𝑟𝑡𝑒𝑥, 𝑆ℎ𝑎𝑟𝑒 𝑎 𝑆𝑖𝑑𝑒, 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑆𝑎𝑚𝑒 𝑃𝑙𝑎𝑛𝑒 𝐻𝑜𝑤 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒? 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 = 90° 𝑎𝑛𝑔𝑙𝑒 𝑚∠1 + 𝑚∠2 = 90° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 Alternative Complement Theorem: ∠1 𝑎𝑛𝑑 ∠2 𝑐𝑟𝑒𝑎𝑡𝑒 ∠𝐴𝐵𝐶, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
- 5. ∠𝐴𝐵𝐶 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠1 𝑎𝑛𝑑 ∠2 𝑓𝑜𝑟𝑚 ∠𝐴𝐵𝐶 𝑔𝑖𝑣𝑒𝑛 𝑚∠𝐴𝐵𝐶 = 90° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐵𝐶 𝐴𝑛𝑔𝑙𝑒 𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛 𝑃𝑜𝑠𝑡𝑢𝑙𝑎𝑡𝑒 𝑚∠1 + 𝑚∠2 = 90° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑠𝑢𝑏 90° 𝑖𝑛 𝑓𝑜𝑟 ∠𝐴𝐵𝐶 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorems: Congruent Supplement Theorem (3 angles): 𝑃𝑟𝑜𝑣𝑒 ∠2 ≅ ∠3 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑔𝑖𝑣𝑒𝑛 𝑚∠1 + 𝑚∠2 = 180° 𝑚∠1 + 𝑚∠3 = 180° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 180° = 𝑚∠1 + 𝑚∠3 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑚∠1 + 𝑚∠2 = 𝑚∠1 + 𝑚∠3 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝐼𝑓 𝑁 = 𝑃 𝑎𝑛𝑑 𝑃 = 𝑀 𝑡ℎ𝑒𝑛, 𝑁 = 𝑀 𝑚∠1 + 𝑚∠2 = 180° = 𝑚∠1 + 𝑚∠3 𝑁 = 𝑃 = 𝑀 𝑚∠2 = 𝑚∠3 𝑆𝑡𝑒𝑝𝑠: 𝑚∠1 + 𝑚∠2 = 𝑚∠1 + 𝑚∠3 𝑚∠1 + 𝑚∠2 − 𝑚∠1 = 𝑚∠1 + 𝑚∠3 − 𝑚∠1 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑃 = 𝑚∠1 (𝑃 𝑖𝑠 𝑟𝑒𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑡𝑜 𝑒𝑞𝑢𝑎𝑙 𝑚∠1) 𝑁 = 𝑀 𝑁 − 𝑃 = 𝑀 − 𝑃 ∠2 ≅ ∠3 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Supplement Theorem (4 angles): 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠2 ≅ ∠3 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑎𝑛𝑔𝑙𝑒𝑠 ∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑎𝑛𝑔𝑙𝑒𝑠 ∠1 ≅ ∠4 𝑔𝑖𝑣𝑒𝑛 𝑚∠1 + 𝑚∠2 = 180° 𝑚∠3 + 𝑚∠4 = 180° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 180° = 𝑚∠3 + 𝑚∠4 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠4 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝐼𝑓 𝑁 = 𝑃 𝑎𝑛𝑑 𝑃 = 𝑀, 𝑡ℎ𝑒𝑛 𝑁 = 𝑀 𝑚∠1 + 𝑚∠2 = 180° = 𝑚∠3 + 𝑚∠4 𝑁 = 𝑃 = 𝑀 ∠1 ≅ ∠4 𝑔𝑖𝑣𝑒𝑛 𝑚∠1 = 𝑚∠4 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑅𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑁𝑢𝑚𝑏𝑒𝑟 𝑃, 𝑃 = 𝑃 𝑃 𝑖𝑠 𝑟𝑒𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑡𝑜 𝑒𝑞𝑢𝑎𝑙 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑚∠1 𝑎𝑛𝑑 𝑚∠4 𝑚∠2 = 𝑚∠3 𝑆𝑡𝑒𝑝𝑠: 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠4 𝑚∠1 + 𝑚∠2 − 𝑚∠1 = 𝑚∠3 + 𝑚∠4 − 𝑚∠4 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑁 = 𝑀 𝑁 − 𝑃 = 𝑀 − 𝑃
- 6. ∠2 ≅ ∠3 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 Congruent Complementary Theorem: 𝑃𝑟𝑜𝑣𝑒 ∠2 ≅ ∠3 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑔𝑖𝑣𝑒𝑛 𝑚∠1 + 𝑚∠2 = 90° 𝑚∠1 + 𝑚∠3 = 90° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 90° = 𝑚∠1 + 𝑚∠3 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑚∠1 + 𝑚∠2 = 𝑚∠1 + 𝑚∠3 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝐼𝑓 𝑁 = 𝑃 𝑎𝑛𝑑 𝑃 = 𝑀 𝑡ℎ𝑒𝑛, 𝑁 = 𝑀 𝑚∠1 + 𝑚∠2 = 90° = 𝑚∠1 + 𝑚∠3 𝑁 = 𝑃 = 𝑀 𝑚∠2 = 𝑚∠3 𝑆𝑡𝑒𝑝𝑠: 𝑚∠1 + 𝑚∠2 = 𝑚∠1 + 𝑚∠3 𝑚∠1 + 𝑚∠2 − 𝑚∠1 = 𝑚∠1 + 𝑚∠3 − 𝑚∠1 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑃 = 𝑚∠1 (𝑃 𝑖𝑠 𝑟𝑒𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑡𝑜 𝑒𝑞𝑢𝑎𝑙 𝑚∠1) 𝑁 = 𝑀 𝑁 − 𝑃 = 𝑀 − 𝑃 ∠2 ≅ ∠3 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 Vertical Angles Theorem: 𝑃𝑟𝑜𝑣𝑒 ∠1 ≅ ∠3 ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑛𝑔𝑙𝑒𝑠 ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑛𝑔𝑙𝑒𝑠 𝑔𝑖𝑣𝑒𝑛 𝑁𝑜𝑡𝑒: 𝐻𝑜𝑤 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐴𝑛𝑔𝑙𝑒𝑠? 𝑃𝑎𝑖𝑟 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑆ℎ𝑎𝑟𝑒 𝑎 𝑉𝑒𝑟𝑡𝑒𝑥, 𝑎𝑟𝑒 𝑁𝑜𝑡 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠, 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑓𝑜𝑟𝑚𝑒𝑑 𝑓𝑟𝑜𝑚 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑛𝑔 𝐿𝑖𝑛𝑒𝑠 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟 𝑚∠1 + 𝑚∠2 = 180° 𝑚∠2 + 𝑚∠3 = 180° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 180° = 𝑚∠2 + ∠3 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑚∠1 + 𝑚∠2 = 𝑚∠2 + 𝑚∠3 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑜𝑝𝑒𝑟𝑦 𝐼𝑓 𝑁 = 𝑃 𝑎𝑛𝑑 𝑃 = 𝑀 𝑡ℎ𝑒𝑛, 𝑁 = 𝑀 𝑚∠1 + 𝑚∠2 = 180° = 𝑚∠2 + 𝑚∠3 𝑁 = 𝑃 = 𝑀 𝑚∠1 = 𝑚∠3 𝑆𝑡𝑒𝑝𝑠: 𝑚∠1 + 𝑚∠2 = 𝑚∠2 + 𝑚∠3 𝑚∠1 + 𝑚∠2 − 𝑚∠2 = 𝑚∠2 + 𝑚∠3 − 𝑚∠2 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦: 𝑃 = 𝑚∠2 (𝑃 𝑖𝑠 𝑟𝑒𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑡𝑜 𝑒𝑞𝑢𝑎𝑙 𝑚∠2) 𝑁 = 𝑀 𝑁 − 𝑃 = 𝑀 − 𝑃 ∠1 ≅ ∠3 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠
- 7. Perpendicular Lines Theorems: Perpendicular Lines Intersect to Form Four Right Angles Theorem 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1, ∠2, ∠3, 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝐿𝑖𝑛𝑒𝑠 𝑡 𝑎𝑛𝑑 𝑠 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 (𝑡 ⊥ 𝑠) 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝐿𝑖𝑛𝑒𝑠 𝑚∠1 = 90° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑚∠1 + 𝑚∠2 = 180° 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑁𝑜𝑡𝑒: 𝑁𝑒𝑒𝑑 𝑡𝑜 𝐼𝑑𝑒𝑛𝑡𝑖𝑓𝑦 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑠 𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟 90° + 𝑚∠2 = 180° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 (𝑠𝑢𝑏 90° 𝑓𝑜𝑟 𝑚∠1) 𝑚∠2 = 90° 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑆𝑢𝑏𝑠𝑡𝑟𝑎𝑐𝑡 90° 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛) ∠2 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 ∠1 ≅ ∠3 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐴𝑛𝑔𝑙𝑒 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑁𝑜𝑡𝑒: 𝑁𝑒𝑒𝑑 𝑡𝑜 𝐼𝑑𝑒𝑛𝑡𝑖𝑓𝑦 ∠1 𝑎𝑛𝑑 ∠3 𝑎𝑠 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐴𝑛𝑔𝑙𝑒𝑠, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑚∠1 = 𝑚∠3 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 90° = 𝑚∠3 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 (𝑠𝑢𝑏 90° 𝑓𝑜𝑟 𝑚∠1) ∠3 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑚∠1 + 𝑚∠4 = 180° 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 (𝐿𝑖𝑛𝑒𝑎𝑟 𝑃𝑎𝑖𝑟) 90° + 𝑚∠4 = 180° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 (𝑠𝑢𝑏 90° 𝑓𝑜𝑟 𝑚∠1) 𝑚∠4 = 90° 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑆𝑢𝑏𝑠𝑡𝑟𝑎𝑐𝑡 90° 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛) ∠4 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 All Right Angles are Congruent Theorem 𝑃𝑟𝑜𝑣𝑒 ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 𝑡 ⊥ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠2 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠3 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠4 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝐿𝑖𝑛𝑒𝑠 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑡𝑜 𝐹𝑜𝑟𝑚 𝐹𝑜𝑢𝑟 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑚∠1 = 90° 𝑚∠2 = 90° 𝑚∠3 = 90° 𝑚∠4 = 90° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑚∠1 = 𝑚∠2 = 𝑚∠3 = 𝑚∠4 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠
- 8. Perpendicular Lines Will Form Four Congruent Adjacent Angles Theorem 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 ≅ ∠2 𝑡 ⊥ 𝑠 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠2 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠3 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 ∠4 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝐿𝑖𝑛𝑒𝑠 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑡𝑜 𝐹𝑜𝑟𝑚 𝐹𝑜𝑢𝑟 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 𝐴𝑙𝑙 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ∠4 𝑎𝑛𝑑 ∠1 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑁𝑜𝑡𝑒: 𝐻𝑜𝑤 𝑡𝑜 𝐼𝑑𝑒𝑛𝑡𝑖𝑓𝑦 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠? 𝑆ℎ𝑎𝑟𝑒 𝑎 𝑉𝑒𝑟𝑡𝑒𝑥, 𝑆ℎ𝑎𝑟𝑒 𝑎 𝑆𝑖𝑑𝑒, 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑆𝑎𝑚𝑒 𝑃𝑙𝑎𝑛𝑒 If Two Angles are Both Congruent and Supplementary, Then Each Angle Will be a Right Angle Theorem 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 ≅ ∠2 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑔𝑖𝑣𝑒𝑛 𝑚∠1 = 𝑚∠2 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 𝑚∠1 + 𝑚∠2 = 180° 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐴𝑛𝑔𝑙𝑒𝑠 𝑚∠1 + 𝑚∠1 = 180° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑠𝑢𝑏 𝑚∠1 𝑓𝑜𝑟 𝑚∠2 2 ∗ 𝑚∠1 = 180° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑠𝑢𝑏 2 ∗ 𝑚∠1 𝑓𝑜𝑟 𝑚∠1 + 𝑚∠1) 𝑚∠1 2 = 180° 2 𝑚∠1 = 90° 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ∠1 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑚∠2 = 90° 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 (𝑠𝑢𝑏 𝑚∠2 𝑓𝑜𝑟 𝑚∠1) ∠2 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 If Two Angles of a Linear Pair are Congruent, Then They Are Right Angles Theorem 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒𝑠 ∠1 ≅ ∠2 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑎𝑖𝑟 𝑔𝑖𝑣𝑒𝑛 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠 𝐼𝑓 𝑇𝑤𝑜 𝐴𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝐵𝑜𝑡ℎ 𝐶𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑛𝑑 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦, 𝑇ℎ𝑒𝑛 𝐸𝑎𝑐ℎ 𝐴𝑛𝑔𝑙𝑒 𝑊𝑖𝑙𝑙 𝑏𝑒 𝑎 𝑅𝑖𝑔ℎ𝑡 𝐴𝑛𝑔𝑙𝑒 𝑇ℎ𝑒𝑜𝑟𝑒𝑚