Reasoning and Proof: practice solutions, Grade 10 – download the PDF

1 Continue the pattern ★★★

1. Each term is 6 more than the previous one. Next terms: \(23+6=29\) and \(29+6=35\).
2. The 10th term is \(5+9\cdot 6=59\).
3. It is inductive reasoning: we used the first four terms to guess a rule for all.

2 Hypothesis and conclusion ★★★

1. Hypothesis: a whole number is divisible by 10. Conclusion: it ends in 0.
2. Converse: “If a whole number ends in 0, then it is divisible by 10.”
3. Both are true: every multiple of 10 ends in 0, and every whole number ending in 0 is a multiple of 10 (for example 370 \(=10\cdot 37\)).

3 Inductive or deductive? ★★★

1. Inductive (a pattern from specific cases).
2. Deductive (a general fact applied to one case).
3. Inductive (a generalization from measurements).
4. Deductive (uses the definition of supplementary angles: \(180^\circ-70^\circ=110^\circ\)).

4 Name the property ★★★

1. Symmetric Property of Equality.
2. Transitive Property of Equality.
3. Reflexive Property of Congruence.
4. Division Property of Equality (divide both sides by 3).
5. Substitution Property of Equality.

5 Four related statements ★★★

Original: true (a square has four right angles).

Converse: “If a figure is a rectangle, then it is a square.” False: a \(2\text{ in}\times 3\text{ in}\) rectangle is not a square.

Inverse: “If a figure is not a square, then it is not a rectangle.” False: the same \(2\times 3\) rectangle is not a square but is a rectangle.

Contrapositive: “If a figure is not a rectangle, then it is not a square.” True, like the original.

6 Biconditional or not? ★★★

1. No. The converse is false: \(x=-4\) gives \(x^2=16\) too.
2. Yes. The converse (“If an integer is divisible by 2, then it is even”) is true, so: “An integer is even if and only if it is divisible by 2.”
3. Yes. A polygon is a triangle if and only if it has exactly three sides.

7 Justify each step ★★★

\(5x-9=26\) (Given). \(5x=35\) (Addition Property of Equality: add 9). \(x=7\) (Division Property of Equality: divide by 5).

Check: \(5\cdot 7-9=26\). So \(x=7\).

8 Find a counterexample ★★★

1. The number 2 is prime and even.
2. The number 6 is a multiple of 3 and is even (also 12, 18).

One counterexample is enough to show each statement false.

9 Trail midpoint ★★★

\(M\) is the midpoint, so \(AM=MB\) (definition of midpoint). Then \(3x+2=5x-10\) (Substitution). Subtract \(3x\): \(2=2x-10\). Add 10: \(12=2x\). Divide by 2: \(x=6\).

\(AM=3\cdot 6+2=20\) and \(MB=5\cdot 6-10=20\). By the Segment Addition Postulate \(AB=20+20=40\) m (about 131 ft).

10 Angle addition ★★★

Angle Addition Postulate: \((2x+10)+(3x-5)=125\). So \(5x+5=125\), \(5x=120\), \(x=24\).

\(m\angle ABD=2\cdot 24+10=58^\circ\) and \(m\angle DBC=3\cdot 24-5=67^\circ\). Check: \(58+67=125\).

11 Linear pair ★★★

Linear Pair Postulate: the angles are supplementary, so \((4x+12)+(2x+6)=180\). Then \(6x+18=180\), \(6x=162\), \(x=27\).

\(m\angle 1=4\cdot 27+12=120^\circ\) and \(m\angle 2=2\cdot 27+6=60^\circ\). Check: \(120+60=180\).

12 Reading vertical angles ★★★

\(m\angle 3=85^\circ\): vertical angles are congruent (Vertical Angles Theorem).

\(m\angle 2=180-85=95^\circ\): \(\angle 1\) and \(\angle 2\) are a linear pair.

\(m\angle 4=95^\circ\): vertical to \(\angle 2\). Check: \(85+95+85+95=360\).

13 Complete the proof ★★★

2. Distributive Property. 3. Subtraction Property of Equality (subtract \(2x\) from both sides). 4. Addition Property of Equality (add 6).

Check: \(3(11-2)=27\) and \(2\cdot 11+5=27\).

14 True or false? ★★★

1. True. A conditional and its contrapositive are logically equivalent.
2. False. “If a number is divisible by 6, then it is divisible by 3” is true, but its converse fails for 9.
3. False. “If an animal is a dog, then it is a mammal” is true, but “If an animal is not a dog, then it is not a mammal” is false (a cat).

15 Chain of congruent angles ★★★

\(\angle A\cong\angle B\) and \(\angle B\cong\angle C\) give \(\angle A\cong\angle C\) (Transitive Property of Congruence). Using it again with \(\angle C\cong\angle D\) gives \(\angle A\cong\angle D\). Congruent angles have equal measures, so \(m\angle D=47^\circ\).

16 Rewrite as if-then ★★★

1. “If an integer is divisible by 6, then it is divisible by 3.” Converse: “If an integer is divisible by 3, then it is divisible by 6.” False; 9 is a counterexample.
2. “If a triangle has three congruent angles, then it is equiangular.” Converse: “If a triangle is equiangular, then it has three congruent angles.” True, so this is a definition and gives a biconditional.

17 Congruent supplements ★★★

Since \(\angle 1\) and \(\angle 2\) are supplementary, \(m\angle 1+m\angle 2=180^\circ\). Since \(\angle 3\) and \(\angle 2\) are supplementary, \(m\angle 3+m\angle 2=180^\circ\). By substitution, \(m\angle 1+m\angle 2=m\angle 3+m\angle 2\). Subtracting \(m\angle 2\) from both sides gives \(m\angle 1=m\angle 3\), so \(\angle 1\cong\angle 3\) by the definition of congruent angles.

18 A prime pattern that fails ★★★

Try \(n=11\): \(11^2-11+11=121=11\cdot 11\), which is not prime. So \(n=11\) is a counterexample and the conjecture is false.

Lesson: ten successes do not prove a general rule.

19 Midpoint converse ★★★

1. “If \(M\) is the midpoint of \(\overline{AB}\), then \(M\) is on \(\overline{AB}\) and \(AM=MB\)” and “If \(M\) is on \(\overline{AB}\) and \(AM=MB\), then \(M\) is the midpoint.”
2. \(AM^2=2^2+3^2=13\) and \(MB^2=(4-2)^2+3^2=13\), so \(AM=MB=\sqrt{13}\). But \(M\) is not on \(\overline{AB}\) (its \(y\)-coordinate is 3), so it is not the midpoint. The hypothesis is true and the conclusion false: counterexample.

20 Chain of conditionals ★★★

Contrapositive of (2): if the lights do not go off, then the door is not locked. The lights are on, so the door is not locked. Contrapositive of (1): if the door is not locked, then the alarm did not ring. Therefore the alarm did not ring. This is deductive reasoning.

21 Full two-column proof ★★★

Statements and reasons: 1. \(B\) is between \(A\) and \(C\); \(AC=24\) (Given). 2. \(AB+BC=AC\) (Segment Addition Postulate). 3. \((3x-1)+(x+9)=24\) (Substitution). 4. \(4x+8=24\) (Simplify, combine like terms). 5. \(4x=16\) (Subtraction Property of Equality). 6. \(x=4\) (Division Property of Equality).

\(AB=3\cdot 4-1=11\) ft and \(BC=4+9=13\) ft; \(11+13=24\).

22 Algebra with vertical angles ★★★

Vertical angles are congruent: \(5x-7=3x+19\). So \(2x=26\) and \(x=13\).

\(m\angle 1=5\cdot 13-7=58^\circ\) (and \(m\angle 3=3\cdot 13+19=58^\circ\)). \(\angle 2\) is its linear pair, so \(m\angle 2=180-58=122^\circ\).