12 multiple-choice questions, progressively harder.
For every nondegenerate triangle ABC, which statement must hold?
Solution
Correct answer: B
The triangle inequality compares two sides together with the third:
Equality would be flat. No equality between individual sides is required.
Which positive side triple can make a nondegenerate triangle?
Correct answer: D
Compare the two shorter lengths with the longest.
The 4, 7, 11 triple is flat. The shorter lengths in each other triple add to less than its longest length.
Three positive rods have lengths 7, 9, and 16 units. What happens when you try to close them?
Correct answer: A
The shorter rods exactly span the longest:
The two shorter rods lie along the longest, producing a flat arrangement. This is not a nondegenerate triangle.
Two triangle sides have lengths 8 and 15 units. What is the complete range of the third side x?
Correct answer: C
The lower endpoint is the difference and the upper endpoint is the sum:
Thus
Both endpoints are flat and excluded.
Two triangle sides measure 5 and 13 centimeters. What is the smallest possible integer length of the third side?
The third side must lie strictly between the difference and the sum:
The first integer above 8 is 9. The boundary length 8 would make a flat arrangement.
Two triangle sides measure 6 and 11 units. What is the largest possible integer length of the third side?
The upper boundary is
The third side must be strictly smaller, so the largest integer is 16. It also exceeds the lower boundary of 5.
P is off line RT. Which comparison between the route R to P to T and segment RT is certain?
RPT is a nondegenerate triangle. Its two route legs satisfy
P being on either side of the line gives the same strict comparison.
A, P, and B are collinear in that order. AP measures 4 units and PB measures 7 units. What is AB?
P lies between A and B, so segment addition applies:
This route ties the direct segment because it travels straight forward.
Why can 0, 4, and 4 not be side lengths of a nondegenerate triangle?
A nondegenerate triangle has positive side lengths. Zero is not positive, so the proposed zero length fails that condition.
Equal positive side lengths are allowed, and integer lengths are not required.
In the extension proof, B, A, and D lie in that order, and AD is copied from AC. Which angle equality follows from the isosceles triangle ACD?
The given equal sides are AC and AD. Their opposite base angles in triangle ACD are equal:
The full angle BCD includes an additional positive part, so it is not equal to the base angle.
A nondegenerate quadrilateral ABCD has consecutive sides AB, BC, and CD of lengths 3, 5, and 8 units. Which strict bound must side DA satisfy?
The other three sides form a broken path from A to D of length
The polygon inequality gives
Equality would require a flat forward path.
For proposed lengths 7, 12, and 19, checking that 12 plus 19 exceeds 7 does not settle feasibility. Which comparison still fails?
You must compare the two shorter lengths with the longest. Here
The required strict comparison fails at equality, so these lengths are flat.
Reset this practice set?
This clears every answer you have given and starts the set again from question 1.