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Rectangles, Rhombi, and Squares: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Convex quadrilateral $ABCD$ has both pairs of opposite sides congruent. Its diagonals are congruent and perpendicular. Which proof order is valid?

Answer choices for question 1 A Prove square from congruent diagonals alone B Prove rectangle from perpendicular diagonals alone C Use the square conclusion to prove its opposite sides congruent D Prove parallelogram, then rectangle and rhombus, then square
2

A parallelogram has congruent diagonals. In proving that it is a rectangle, why does SSS give equal consecutive angles?

Answer choices for question 2 A Every SSS pair of triangles has only right angles B The congruent triangles include those angles as corresponding parts C The diagonal intersection must be perpendicular D Consecutive parallelogram angles are always congruent
3

Parallelogram $ABCD$ has perpendicular diagonals meeting at $O$ . Triangles $AOB$ and $AOD$ are compared to prove that it is a rhombus. Which information uses the parallelogram hypothesis?

Answer choices for question 3 A $AO=AO$ B The given diagonals are perpendicular C $BO=OD$ D The shared endpoint of the diagonals is named $O$
4

A parallelogram has diagonal lengths $3t+8$ cm and $5t-6$ cm, where $t>2$ . For which $t$ does this information establish a rectangle?

Answer choices for question 4 A $7$ B $1$ C $4$ D $14$
5

In rhombus $ABCD$ , angle $BAC$ measures $(2u+7)^\circ$ and the whole angle at $B$ measures $(5u+13)^\circ$ . Find $u$ .

Answer choices for question 5 A $13$ B $17$ C $19$ D $27$
6

A convex quadrilateral has congruent, perpendicular diagonals. Which statement describes what can be concluded?

Answer choices for question 6 A A square is guaranteed B A rectangle is guaranteed, but a rhombus is impossible C A rhombus is guaranteed, but a rectangle is impossible D A square is possible, but it is not guaranteed
7

In a rectangle, a diagonal bisects one of its endpoint vertex angles. Which conclusion follows?

Answer choices for question 7 A The rectangle is a square B Every rectangle automatically has this property C The rectangle cannot be a rhombus D Its diagonals must have different lengths
8

A rhombus has whole diagonals of $16$ cm and $30$ cm. What is its side length?

Answer choices for question 8 A $34$ cm B $23$ cm C $17$ cm D $8.5$ cm
9

Convex quadrilateral $ABCD$ has diagonals meeting at $O$ . All four segments $AO$ , $BO$ , $CO$ , and $DO$ have length $5$ cm. No angle measure is given. What is guaranteed?

Answer choices for question 9 A A rectangle, but not necessarily a square B A square C A rhombus, but not necessarily a rectangle D No parallelogram classification is possible
10

Which biconditional is true for convex quadrilaterals?

Answer choices for question 10 A A quadrilateral is a rectangle if and only if its diagonals are congruent B A quadrilateral is a rhombus if and only if its diagonals are perpendicular C A parallelogram is a square if and only if its diagonals are congruent and perpendicular D A parallelogram is a square if and only if its diagonals bisect each other
11

A convex quadrilateral has one pair of opposite sides both parallel and congruent, and a diagonal bisects a whole angle at its endpoint. What is guaranteed?

Answer choices for question 11 A A square, with no other case possible B A rectangle, with rhombus impossible C No parallelogram classification is possible D A rhombus, with square still possible
12

Which extra condition is sufficient to turn a known rhombus into a square?

Answer choices for question 12 A Its diagonals are perpendicular B Its two whole diagonals are congruent C Its diagonals bisect each other D A diagonal bisects an endpoint angle

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