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Properties of Congruence and Similarity Name_________________________________________ Hour___________ Take a minute to recall what you know about congruent figures. Congruence: Figure 1 These two shapes ARE Similar. Using your centimeter ruler and protractor, measure all of the sides and angles. Label the measures of the sides and the measures of the angles on the picture. A D B E F H G C 1. What do you notice about the measurements of the corresponding angles? 2. What do you notice about the measurements of the corresponding sides? 3. ̅̅̅̅ and 𝑠𝑖𝑑𝑒 𝐸𝐹 ̅̅̅̅ , are they congruent? When you compare 𝐴𝐵 4. Can you find any relationship between 𝑠𝑖𝑑𝑒 ̅̅̅̅ 𝐴𝐵 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 ̅̅̅̅ 𝐸𝐹 ? 5. Does this relationship hold (is it true) for all other corresponding sides? Figure 2 The following two shapes are similar. Measure the following sides and angles. Label these measurements on your picture. 1. A B X W How are the shapes in Figure 1 different from the shapes here in Figure 2? Z This means that other shapes can be similar as well as long as certain rules are true. D C 2. What do you notice about the measurements of the corresponding angles? 3. What do you notice about the measurements of corresponding sides? 4. ̅̅̅̅ 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 𝑋𝑌 ̅̅̅̅ are they congruent? When you compare 𝑠𝑖𝑑𝑒 𝐴𝐵 5. Explain the relationship between 𝑠𝑖𝑑𝑒 ̅̅̅̅ 𝐴𝐵 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 ̅̅̅̅ 𝑋𝑌. 6. Does this relationship hold (is it true) for all other corresponding sides? (Measure) Figure 3 These two figures are NOT similar figures. 7. Explain WHY the two triangles above are not similar. 8. Summary – What makes two figures similar? (Write a general rule you have discovered to be true.) Y Similarity Practice 1 Name_______________________________ Hour____________ Ready Solve each proportion for the given variable. 1. 15 4. 126 𝑦 𝑘 40 = 12 = 14 3 2. 16 5. 2𝑥+1 40 = 2 30 = 𝑦 3. 𝑥 𝑥+2 42.3 2 6. 5 5 144 = 56.4 𝑥 = 21−𝑥 Set Is each given similarity statement true or false? Take measurements to decide if needed. EXPLAIN your answer. 7. 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐻 C F G B E H D A F G C 8. 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐻 B D A H E State whether each of the following illustrates two similar figures. How do you know? 9. 10. C 8 1 B 6 F 2 50◦ 130◦ 1 3 6 40◦ 140◦ A 3 E E 50◦ 4 4 3 130◦ 2 D 2 140◦ 40◦ 6 Go! Assume that ∆𝑃𝐿𝑈 ~ ∆𝐴𝐵𝐶. 𝑭𝒊𝒏𝒅 𝑨𝑪 𝒂𝒏𝒅 𝑩𝑪 (𝑙𝑎𝑏𝑒𝑙 𝑖𝑛 𝑎𝑛𝑠𝑤𝑒𝑟) 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝐴𝐵. M at h Com poser 1. 1. 5 ht t p: / / www. m at hcom poser . com U C 5 6 4 P 11. 13. L A B (not to scale) 𝐴𝐵 = 1 𝐴𝐵 = 4 12. 𝐴𝐵 = 3 14. 𝐴𝐵 = 𝑥 ∆𝐶𝐴𝑇~∆𝐷𝑂𝑇. 𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑒𝑎𝑐ℎ 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡. 15. ∠𝐶 ≅ ____________ 16. ∠𝐶𝑇𝐴 ≅ ___________ 17. ∠𝐷𝑇𝑂 ≅ __________ 18. ∠𝐴 ≅ _____________ 19. ∠ 𝐷 ≅ __________ 20. ∠𝑂 ≅ _____________