12 multiple-choice questions, progressively harder.
The angles at one base of a convex isosceles trapezoid measure (x+14)∘(x+14)^\circ(x+14)∘ and (3x−36)∘(3x-36)^\circ(3x−36)∘. What is xxx?
Solution
Correct answer: C
Equal base angles give
The two base angles are each 39 degrees.
A convex isosceles trapezoid has diagonal lengths, in centimeters, 5x+75x+75x+7 and 8x−208x-208x−20. What is each diagonal length?
Correct answer: A
The diagonals are congruent.
Each expression then gives 52 cm.
A trapezoid has bases of 151515 cm and 414141 cm. Its midsegment length is (3x+1)(3x+1)(3x+1) cm. What is xxx?
Correct answer: D
The average base length is 28 cm.
A trapezoid has one base of 11 cm and a midsegment of 20 cm. What is the other base length?
Correct answer: B
Let the other base length be b centimeters.
Convex quadrilateral ABCDABCDABCD has exactly one opposite parallel pair, ABABAB and CDCDCD. Its legs ADADAD and BCBCBC are congruent. What follows?
The exactly-one condition makes the equal-legs converse valid. It gives an isosceles trapezoid, not necessarily a parallelogram or rhombus.
Reflection in the common perpendicular bisector of an isosceles trapezoid's bases swaps AAA with BBB and DDD with CCC. What is the image of segment ACACAC?
Reflect each endpoint: A goes to B, and C goes to D. Therefore AC goes to BD, explaining the diagonal-length equality.
Convex kite ABCDABCDABCD has equal pairs meeting at AAA and CCC. Diagonals meet at OOO. The lengths AO,OC,BOAO,OC,BOAO,OC,BO are 101010 cm, 777 cm, and 242424 cm. How long is ABABAB?
The kite theorem gives a right angle at O. Use triangle AOB, whose legs are 10 cm and 24 cm.
OC belongs to a different right triangle.
Convex kite ABCDABCDABCD has equal pairs meeting at AAA and CCC. Its angles at AAA and CCC are 52∘52^\circ52∘ and 108∘108^\circ108∘. What is the angle at BBB?
The angles at B and D are congruent. The quadrilateral angle sum leaves 200 degrees for those two angles.
In convex kite ABCDABCDABCD, ACACAC joins the equal-side meeting vertices. Angle BACBACBAC measures 23∘23^\circ23∘, and angle BCABCABCA measures 31∘31^\circ31∘. What is the whole angle at BBB?
Use triangle ABC. Its angles at A and C are already the supplied half-angles.
This is the whole kite angle at B.
A convex kite has equal pairs meeting at AAA and CCC. Diagonals meet at OOO, with AOAOAO shorter than OCOCOC. Which reflection is guaranteed?
AC is the perpendicular bisector of BD and is the guaranteed axis. Reflection in BD would have to swap A and C, which their unequal distances from BD prevent.
How many rotational symmetries does a nonsquare rhombus have, including the identity?
A nonsquare rhombus has the identity and a half-turn. A quarter-turn would require right angles and would make it a square.
In a convex quadrilateral, one supplied equality is AB=BCAB=BCAB=BC.
The other supplied equality is BC=CDBC=CDBC=CD.
Why do these alone fail to establish the kite definition?
The stated equalities reuse BC. They show three equal sides but give no matching condition for AD. The kite definition must use each side once in two adjacent pairs.
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