1. **Write the inverse of the conditional:** "If Deanna eats the last cookie, then Leon will buy more." The inverse negates both the hypothesis and conclusion: If Deanna does not eat the last cookie, then Leon will not buy more. 2. **Write the converse of the conditional:** "If $x + 1 = 9$, then $x = 8$." The converse switches the hypothesis and conclusion: If $x = 8$, then $x + 1 = 9$. 3. **Which statement shows the Transitive property?** The Transitive property states: If $a = b$ and $b = c$, then $a = c$. Looking at options: - a) If $AB = CD$ and $AB = EF$, then $CD = EF$ (Not transitive, because $AB$ equals both but $CD$ and $EF$ are not directly related) - b) If $m\angle A = m\angle B$ and $m\angle B = m\angle C$, then $m\angle A = m\angle C$ (This matches the transitive property) - c) If $AB = CD$, then $CD = AB$ (Symmetric property) - d) If $AB + BC = DE + BC$, then $AB = DE$ (Subtraction property) **Answer: b)** 4. **If $\angle 1$ and $\angle 2$ form a linear pair and $m\angle 2 = 64$, find $m\angle 1$.** Linear pairs are supplementary, so their measures add to 180: $$m\angle 1 + m\angle 2 = 180$$ Substitute $m\angle 2 = 64$: $$m\angle 1 + 64 = 180$$ Subtract 64 from both sides: $$m\angle 1 = 180 - 64$$ $$m\angle 1 = 116$$ 5. **Which law is similar to the Transitive property?** - a) Law of Deductive Reasoning - b) Law of Detachment - c) Law of Conditional - d) Law of Syllogism The Law of Syllogism states: If $p \to q$ and $q \to r$, then $p \to r$, which is similar to transitive property. **Answer: d)** 6. **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. iii) Conclusion: Since $\angle 1$ and $\angle 2$ are vertical angles and vertical angles are congruent, by Law of Detachment: $\angle 1 \cong \angle 2$. 7. **Choose the reasons that 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 | 8. **Which three properties hold true for congruence of segments?** - a) addition, subtraction, multiplication - b) reflexive, symmetric, transitive - c) reflexive, multiplication, division - d) reflexive, symmetric, substitution The three main properties of congruence are reflexive, symmetric, and transitive. **Answer: b)** 9. **If $\angle 1$ and $\angle 2$ form a linear pair and $m\angle 1 = 2x + 16$ and $m\angle 2 = 50x + 60$, find $m\angle 2$.** Since they form a linear pair, their measures add to 180: $$ (2x + 16) + (50x + 60) = 180 $$ Simplify: $$ 2x + 16 + 50x + 60 = 180 $$ $$ 52x + 76 = 180 $$ Subtract 76: $$ 52x = 180 - 76 $$ $$ 52x = 104 $$ Divide both sides by 52: $$ x = \frac{\cancel{52}x}{\cancel{52}} = \frac{104}{52} $$ $$ x = 2 $$ Find $m\angle 2$: $$ m\angle 2 = 50x + 60 = 50(2) + 60 = 100 + 60 = 160 $$ 10. **If $m\angle 3 = 108$, find $m\angle 7$.** From the diagram with parallel lines and transversal, $\angle 3$ and $\angle 7$ are corresponding angles, so they are congruent: $$ m\angle 7 = m\angle 3 = 108 $$ 11. **What is the value of $y$?** Given angles: 126°, $(6z + 6)°$, $(5y - 4)°$, and $x°$ with three parallel lines and a transversal. Assuming $(6z + 6)°$ and $(5y - 4)°$ are alternate interior or corresponding angles equal to 126° or related. From the diagram, $(6z + 6)°$ and 126° are alternate interior angles, so: $$ 6z + 6 = 126 $$ Subtract 6: $$ 6z = 120 $$ $$ z = 20 $$ Now, $(5y - 4)°$ and $x°$ are related; if $x = 126$ (corresponding angle), then: $$ 5y - 4 = 126 $$ Add 4: $$ 5y = 130 $$ Divide by 5: $$ y = 26 $$ **Final answers:** 1) Inverse: If Deanna does not eat the last cookie, then Leon will not buy more. 2) Converse: If $x = 8$, then $x + 1 = 9$. 3) Transitive property: b) 4) $m\angle 1 = 116$ 5) Law similar to Transitive: d) 6) $\angle 1 \cong \angle 2$ 7) Reasons: 1) Given, 2) Definition of congruent segments, 3) Transitive Property, 4) Definition of congruent segments 8) Properties: b) 9) $m\angle 2 = 160$ 10) $m\angle 7 = 108$ 11) $y = 26$