12 multiple-choice questions, progressively harder.
A, B, C, D are distinct and collinear in that order. AC = BD. What follows, and how?
Solution
Correct answer: A
Segment addition gives AC = AB + BC and BD = BC + CD.
Substitute into AC = BD: AB+BC=BC+CD.AB+BC=BC+CD.AB+BC=BC+CD.
Subtract BC from both sides to obtain AB = CD.
M and N are midpoints of AB and CD respectively, and AB = CD. Which route directly proves AM = CN?
Correct answer: C
The midpoint definitions and segment addition give AB = 2AM and CD = 2CN.
The equal wholes give 2AM=2CN.2AM=2CN.2AM=2CN.
Dividing by the nonzero number 2 proves the goal.
A proof begins, “To prove that l and m have at most one common point, assume they have two distinct common points.” Which additional hypothesis is necessary for the resulting uniqueness contradiction?
Correct answer: B
Two distinct points determine one line.
The contradiction is that two lines claimed to be distinct would have to be that same line. Coincident lines can share both points without a contradiction.
Two concentric circles in one plane have common center O and equal positive radii. Does the unequal-radius contradiction from the lesson apply?
Correct answer: D
The no-common-point argument needs unequal radii. Equal radii fail that hypothesis.
In the stated plane, the common center and equal radius define the same set of points. These concentric circles are the same circle, so they have common points without contradiction.
AB = AC is given, and B, A, C are collinear in that order. Which segment-addition equation is valid?
A, not B, is between the other two points.
The parts BA and AC join to form BC, so BA + AC = BC.
A proposed proof assumes the negation of its goal, then reaches another statement that has not yet been proved. Has it completed a contradiction proof?
An unproved statement may still be true.
The argument needs a demonstrated conflict, such as equality of quantities given to be unequal or a violation of an accepted uniqueness fact.
Selected angles ∠AOB\angle AOB∠AOB and ∠BOC\angle BOC∠BOC are congruent, with measures 5x−45x-45x−4 and 3x+183x+183x+18 degrees. Ray OB lies inside the selected angle AOC. What is m∠AOCm\angle AOCm∠AOC?
Congruence gives equal measures, so 5x - 4 = 3x + 18.
Subtract 3x and add 4 to get 2x = 22, so x = 11. Each part measures 51 degrees.
The interior-ray condition permits angle addition: 51+51=102∘.51+51=102^\circ.51+51=102∘.
Which pair of facts is sufficient for a direct proof that a point M is the midpoint of AB?
The midpoint definition has both a location condition and an equal-length condition.
Together the two stated facts establish the classification directly.
Two equal whole lengths each have the same length removed. A proof concludes the remainders are equal. Which chain of reasons is appropriate?
The algebraic subtraction step is valid for equal numbers.
To identify a difference with a particular segment length, the diagram hypotheses must say that the subtracted segment is actually part of that whole.
A, B, C, D are ordered on a line. A flow proof joins “AC = AB + BC” and “AB = CD” to obtain “AC = CD + BC.” Which dependence is shown?
The second equality permits replacing the term AB in the first equality with CD.
Both input statements are needed for that substitution step.
A theorem needed for a proposed proof is scheduled for a later chapter and has not been accepted as a foundation here. What should the proof do?
A proof must trace its reasons to definitions, accepted foundations, prior results, and valid algebra.
A later theorem is not available merely because its conclusion is plausible.
A contradiction proof concludes r = s while its givens include r not equal to s. What remains to finish the argument?
The equality conflicts with an unchanged given.
Since the conflict arose from the temporary negation, that assumption cannot hold together with the givens; the goal follows.
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