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

1 Continue the pattern ★★★

Look at the sequence \(5, 11, 17, 23, \dots\)

1. Describe the pattern and give the next two terms.
2. Make a conjecture for the 10th term.
3. Is this inductive or deductive reasoning?

2 Hypothesis and conclusion ★★★

Consider: “If a whole number is divisible by 10, then it ends in 0.”

1. Identify the hypothesis and the conclusion.
2. Write the converse.
3. Decide whether each statement is true or false.

3 Inductive or deductive? ★★★

Classify each argument.

1. The last five buses were late, so tomorrow’s bus will be late.
2. Every prime greater than 2 is odd, and 31 is a prime greater than 2, so 31 is odd.
3. You measure the angles of eight triangles, find each sum is \(180^\circ\), and decide all triangles work this way.
4. \(\angle A\) and \(\angle B\) are supplementary and \(m\angle A=70^\circ\), so \(m\angle B=110^\circ\).

4 Name the property ★★★

Name the property that justifies each statement.

1. If \(AB=CD\), then \(CD=AB\).
2. If \(x=y\) and \(y=7\), then \(x=7\).
3. \(\angle P\cong\angle P\).
4. If \(3x=18\), then \(x=6\).
5. If \(a=5\) and \(a+b=12\), then \(5+b=12\).

5 Four related statements ★★★

Start with: “If a figure is a square, then it is a rectangle.” Write the converse, inverse and contrapositive, and give the truth value of each, with a counterexample for any false one.

6 Biconditional or not? ★★★

Which statements can be rewritten as a true biconditional? Explain.

1. If \(x=4\), then \(x^2=16\).
2. If an integer is even, then it is divisible by 2.
3. If a polygon is a triangle, then it has exactly three sides.

7 Justify each step ★★★

Solve \(5x-9=26\) and give a reason for every step.

8 Find a counterexample ★★★

Give a counterexample to each false statement.

1. Every prime number is odd.
2. If a number is a multiple of 3, then it is odd.

9 Trail midpoint ★★★

On a trail map, a rest stop \(M\) is the midpoint of trail \(\overline{AB}\) (lengths in meters), as in the figure. Find \(x\) and the full length \(AB\). Give reasons.

10 Angle addition ★★★

Ray \(BD\) lies inside \(\angle ABC\). \(m\angle ABD=(2x+10)^\circ\), \(m\angle DBC=(3x-5)^\circ\), and \(m\angle ABC=125^\circ\). Find \(x\) and both smaller angles.

11 Linear pair ★★★

\(\angle 1\) and \(\angle 2\) are a linear pair with \(m\angle 1=(4x+12)^\circ\) and \(m\angle 2=(2x+6)^\circ\). Find both measures.

12 Reading vertical angles ★★★

Two lines intersect as shown, and \(m\angle 1=85^\circ\). Find \(m\angle 2\), \(m\angle 3\), \(m\angle 4\) and name the reason for each.

13 Complete the proof ★★★

Given: \(3(x-2)=2x+5\). Prove: \(x=11\). Supply the missing reason in each row.

14 True or false? ★★★

Decide whether each claim is true or false, and justify.

1. The contrapositive of a true conditional is always true.
2. The converse of a true conditional is always true.
3. A conditional and its inverse always have the same truth value.

15 Chain of congruent angles ★★★

\(\angle A\cong\angle B\), \(\angle B\cong\angle C\), and \(\angle C\cong\angle D\). If \(m\angle A=47^\circ\), find \(m\angle D\) and name the properties used.

16 Rewrite as if-then ★★★

Rewrite in if-then form and test the converse.

1. Every integer divisible by 6 is divisible by 3.
2. A triangle with three congruent angles is equiangular.

17 Congruent supplements ★★★

Given: \(\angle 1\) and \(\angle 2\) are supplementary, and \(\angle 3\) and \(\angle 2\) are supplementary. Prove: \(\angle 1\cong\angle 3\). Write a paragraph proof.

18 A prime pattern that fails ★★★

A student computes \(n^2-n+11\) for \(n=1,2,3,\dots,10\) and gets 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, all prime. The student conjectures that it is prime for every positive integer \(n\). Find a counterexample.

19 Midpoint converse ★★★

The definition of midpoint says: “\(M\) is the midpoint of \(\overline{AB}\) if and only if \(M\) is on \(\overline{AB}\) and \(AM=MB\).”

1. Split it into two conditionals.
2. A student claims “If \(AM=MB\), then \(M\) is the midpoint of \(\overline{AB}\).” Use \(A(0,0)\), \(B(4,0)\), \(M(2,3)\) to refute it.

20 Chain of conditionals ★★★

A building’s system works like this. (1) If the alarm rings, then the door is locked. (2) If the door is locked, then the lights go off. You notice the lights are on. What can you conclude about the alarm? Explain with the contrapositive.

21 Full two-column proof ★★★

Point \(B\) is between \(A\) and \(C\), \(AC=24\) ft, \(AB=3x-1\), \(BC=x+9\). Write a two-column proof that \(x=4\), then find \(AB\) and \(BC\).

22 Algebra with vertical angles ★★★

Two lines intersect, forming vertical angles \(\angle 1\) and \(\angle 3\) with \(m\angle 1=(5x-7)^\circ\) and \(m\angle 3=(3x+19)^\circ\). Find \(x\), \(m\angle 1\) and \(m\angle 2\), where \(\angle 2\) forms a linear pair with \(\angle 1\) (figure not drawn to scale).