1. **Problem:** Write the inverse of the conditional: "If Deanna eats the last cookie, then Leon will buy more." 2. **Inverse** means negating both hypothesis and conclusion: If Deanna does not eat the last cookie, then Leon will not buy more. 3. **Problem:** Write the converse of the conditional: "If x + 1 = 9, then x = 8." 4. **Converse** means swapping hypothesis and conclusion: If x = 8, then x + 1 = 9. 5. **Problem:** Which statement shows the Transitive property? 6. The Transitive property states: If $a = b$ and $b = c$, then $a = c$. 7. Checking options: - a) If AB = CD and AB = EF, then CD = EF (No, this is not transitive) - b) If $m \angle A = m \angle B$ and $m \angle B = m \angle C$, then $m \angle A = m \angle C$ (Yes, this is transitive) - c) If AB = CD, then CD = AB (Symmetric property) - d) If AB + BC = DE + BC, then AB = DE (Subtraction property) **Answer:** b 8. **Problem:** If $\angle 1$ and $\angle 2$ form a linear pair and $m \angle 2 = 64$, find $m \angle 1$. 9. Linear pair angles sum to 180 degrees: $$m \angle 1 + m \angle 2 = 180$$ 10. Substitute $m \angle 2 = 64$: $$m \angle 1 + 64 = 180$$ 11. Solve for $m \angle 1$: $$m \angle 1 = 180 - 64 = 116$$ 12. **Problem:** Which law is similar to the Transitive property? 13. The Law of Syllogism is similar to the Transitive property. **Answer:** d 14. **Problem:** Use the Law of Detachment to write a valid conclusion for: - i) If two angles are vertical, then they are congruent. - ii) $\angle 1$ and $\angle 2$ are vertical. 15. Law of Detachment states: If "If p then q" is true and p is true, then q is true. 16. Since $\angle 1$ and $\angle 2$ are vertical (p is true), then they are congruent (q). **Conclusion:** $\angle 1$ is congruent to $\angle 2$. 17. **Problem:** Choose reasons to complete the proof: Given: $AB \cong CD$; $CD \cong EF$ Prove: $AB \cong EF$ | Statements | Reasons | |----------------------------|------------------------------| | 1) $AB \cong CD$; $CD \cong EF$ | 1) Given | | 2) $AB = CD$; $CD = EF$ | 2) Definition of congruent segments | | 3) $AB = EF$ | 3) Transitive Property | | 4) $AB \cong EF$ | 4) Definition of congruent segments | 18. **Problem:** Which three properties hold true for congruence of segments? 19. The properties are reflexive, symmetric, and transitive. **Answer:** b 20. **Problem:** If $\angle 1$ and $\angle 2$ form a linear pair and $m \angle 1 = 2x + 16$ and $m \angle 2 = 50x + 60$, find $m \angle 2$. 21. Since linear pair angles sum to 180 degrees: $$ (2x + 16) + (50x + 60) = 180 $$ 22. Simplify: $$ 52x + 76 = 180 $$ 23. Solve for $x$: $$ 52x = 180 - 76 $$ $$ 52x = 104 $$ $$ x = \frac{104}{52} = 2 $$ 24. Find $m \angle 2$: $$ m \angle 2 = 50(2) + 60 = 100 + 60 = 160 $$ 25. **Problem:** If $m \angle 3 = 108$, find $m \angle 7$. 26. From the diagram, $\angle 3$ and $\angle 7$ are vertical angles, so they are congruent. $$ m \angle 7 = m \angle 3 = 108 $$ 27. **Problem:** What is the value of $y$? 28. From the diagram with parallel lines and transversals, corresponding angles are equal. Given angles: - $126^\circ$ - $(6z + 6)^\circ$ - $(5y - 4)^\circ$ - $x^\circ$ 29. Since $(6z + 6)^\circ$ and $126^\circ$ are corresponding angles, set equal: $$ 6z + 6 = 126 $$ 30. Solve for $z$: $$ 6z = 120 $$ $$ z = 20 $$ 31. Using $z=20$, find $x$ from the diagram (not given explicitly, so assume $x$ is related to $z$ or $y$). 32. Use alternate interior angles or corresponding angles to relate $x$ and $y$. 33. From the diagram, $(5y - 4)^\circ$ and $x^\circ$ are equal (corresponding angles): $$ 5y - 4 = x $$ 34. Also, $x$ and $(6z + 6)$ are supplementary or equal depending on the diagram. Assuming $x = 126$ (from corresponding angle), then: $$ 5y - 4 = 126 $$ 35. Solve for $y$: $$ 5y = 130 $$ $$ y = \frac{130}{5} = 26 $$ **Final answers:** - Inverse: If Deanna does not eat the last cookie, then Leon will not buy more. - Converse: If $x=8$, then $x+1=9$. - Transitive property statement: b - $m \angle 1 = 116$ - Law similar to Transitive: d - Law of Detachment conclusion: $\angle 1$ is congruent to $\angle 2$ - Proof reasons: 1) Given, 2) Definition of congruent segments, 3) Transitive Property, 4) Definition of congruent segments - Properties of congruence: b - $m \angle 2 = 160$ - $m \angle 7 = 108$ - $y = 26$