12 multiple-choice questions, progressively harder.
A, B, C are collinear in that order. AB = 9 and BC = 15 centimeters. Which conclusion and reason are correct?
Solution
Correct answer: B
The point order makes B lie between A and C.
Segment addition gives AC=9+15=24 cm.AC=9+15=24\text{ cm}.AC=9+15=24 cm.
Goal 1 is to prove PM = MQ from the given that M is the midpoint of PQ. Goal 2 is to prove that circles with the same center and unequal positive radii have no common point. Which pair of starting plans is valid?
Correct answer: D
The midpoint definition supplies equal part lengths, so it gives a direct route to the positive equality goal.
For the no-common-point goal, assuming a common point supplies an object to which both circle definitions apply. Their unequal radius conditions then conflict.
These are valid starting plans, not claims that each goal allows only one proof method.
You know AB = CD and BC = DE. Which equality follows by adding equals?
Correct answer: A
Add the left sides of the two given equalities and add their right sides.
Equal quantities added to equal quantities yield equal sums. The other rearrangements do not follow from these givens.
A proof has AC = AB + BC and AB = DE. Which replacement is justified?
Correct answer: C
The given AB = DE permits replacing AB with DE.
No fact identifies BC or the whole AC with DE.
A ray OB lies inside the selected angle AOC. The two parts measure 29 and 48 degrees. Which reason establishes the total measure?
The interior-ray condition lets the two selected parts form the whole angle.
Angle addition gives 29+48=77∘.29+48=77^\circ.29+48=77∘.
A proof concludes that the unambiguously named selected angles ∠X\angle X∠X and ∠Y\angle Y∠Y are congruent from m∠X=m∠Ym\angle X=m\angle Ym∠X=m∠Y. Which reason is needed?
Congruence of selected angles means their measures are equal.
The final step translates the numerical equality into a statement about the angle objects.
A two-column proof has AC = AB + BC, BD = BC + CD, and AB = CD. Which additional property is useful before concluding AC = BD?
Substitution changes the first sum to CD + BC.
Commutativity puts it in the same order as the expression for BD. Symmetry and transitivity then give AC = BD.
Two distinct lines are assumed to share distinct points P and Q. Which accepted fact produces a contradiction?
Each supposed line passes through both P and Q.
Uniqueness says there is only one such line, so they cannot be distinct.
Two circles share center O and have unequal radii. In a contradiction proof, what does assuming a common point P imply?
Membership of P in each circle invokes the circle definition.
The resulting two equalities force the radii to equal each other, contradicting their given inequality.
Which reason is missing from this step: “The lengths a and b are equal, so a divided by c equals b divided by c”?
Division of both sides by the same number preserves equality when that number is nonzero.
Division by zero is undefined, so the reason needs that condition.
A paragraph proof has been moved into a flow diagram. One necessary equality is omitted from every box. What should be done?
A proof format must preserve the logical steps.
An arrow does not establish an unstated equality, so the missing support must be supplied.
You want to prove two segments congruent. Which is a useful planning question?
Segment congruence means equal lengths, so that equality would establish the goal.
Planning backward identifies a useful intermediate target; the final proof still needs a forward justification from the givens.
Reset this practice set?
This clears every answer you have given and starts the set again from question 1.