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Trapezoids and Kites: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

In convex trapezoid $ABCD$ , $AB$ is parallel to $CD$ . Let $M,N$ be leg midpoints and $P$ the midpoint of $AC$ . If $MP$ is $11$ cm and $PN$ is $6$ cm, what is the length of base $AB$ ?

Answer choices for question 1 A 12 cm B 22 cm C 17 cm D 34 cm
2

In convex trapezoid $ABCD$ , bases $AB,CD$ are parallel. $M$ is the midpoint of $AD$ and $P$ is the midpoint of $AC$ . Extend $MP$ to meet $BC$ at $N$ . Which reason proves $N$ is the midpoint of $BC$ ?

Answer choices for question 2 A A line through one midpoint always bisects every side it crosses B Every trapezoid has diagonals that bisect each other C The triangle-midsegment theorem makes MP parallel to AB, and AA similarity puts N halfway along BC D MP is perpendicular to both bases
3

Convex kite $ABCD$ has equal pairs meeting at $A$ and $C$ , and diagonal intersection $O$ . After SSS proves that $AC$ bisects angle $A$ , which congruence proves $BO=DO$ ?

Answer choices for question 3 A Triangles ABO and CBO by SSS B Triangles ABO and ADO by SAS C Triangles ABC and CBA by ASA D Triangles AOD and COD by SAS
4

Convex kite $ABCD$ has equal pairs meeting at $A$ and $C$ . Its diagonals meet at $O$ , and $AO=OC$ . What is the strongest guaranteed classification among these choices?

Answer choices for question 4 A Square in every case B Rectangle in every case C A kite that cannot be a parallelogram D Rhombus, possibly a square
5

Two perpendicular segments $AC$ and $BD$ meet internally at $O$ . Which length data give a convex kite that is not a rhombus?

Answer choices for question 5 A AO 5, OC 5, BO 6, DO 6 B AO 2, OC 2, BO 9, DO 9 C AO 3, OC 12, BO 4, DO 4 D AO 8, OC 8, BO 15, DO 15
6

In the equal-legs proof, convex trapezoid $ABCD$ has exactly one parallel pair, longer base $AB$ , and congruent legs. Through $C$ draw $CE$ parallel to $AD$ , meeting $AB$ at $E$ . Which triangle becomes isosceles, and why?

Answer choices for question 6 A Triangle ACD, because AC equals CD B Triangle EBC, because EC equals AD and AD equals BC C Triangle ABC, because AB equals BC D Triangle ADE, because AE equals DE
7

Why does a non-parallelogram isosceles trapezoid fail to have a half-turn symmetry?

Answer choices for question 7 A A half-turn cannot preserve any convex quadrilateral B Its legs are unequal C Every reflection axis forbids rotations D A half-turn would interchange its unequal bases
8

A convex trapezoid has chosen base lengths $a$ and $b$ . Its midsegment splits it into two smaller trapezoids. What is the midsegment length of the smaller trapezoid adjacent to the base of length $a$ ?

Answer choices for question 8 A $\frac{3a+b}{4}$ B $\frac{a+b}{4}$ C $\frac{a+3b}{4}$ D $\frac{a+b}{2}$
9

A convex kite has two different adjacent-pair side lengths. Why can it not have a second reflection axis?

Answer choices for question 9 A Every convex quadrilateral has at most one reflection axis B Perpendicular diagonals prevent all reflections C A second reflection would necessarily move every point to itself D A reflection must fix each unique equal-side meeting vertex, so its axis must be the line joining them
10

In a convex kite, the given side pairs are $AB,AD$ and $BC,CD$ . Which triangle matching is justified by those pairs and one common diagonal, without further conditions?

Answer choices for question 10 A $\triangle ABC\cong\triangle ADC$ B $\triangle ABC\cong\triangle CAD$ C $\triangle ABD\cong\triangle CDB$ D $\triangle ABO\cong\triangle CBO$ for the diagonal intersection $O$
11

A parallelogram $ABCD$ has congruent angles at chosen base $AB$ . Which conclusion is forced?

Answer choices for question 11 A It must have four congruent sides B It is a rectangle, possibly a square C It cannot be an isosceles trapezoid D Its diagonals must be perpendicular
12

Convex kite $ABCD$ has equal pairs meeting at $A$ and $C$ . The side lengths $AB$ and $BC$ are $10$ cm and $17$ cm. Diagonal $BD$ is $16$ cm long. How long is $AC$ ?

Answer choices for question 12 A 25 cm B 31 cm C 21 cm D 42 cm

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