Quadrilaterals and Polygons: math practice, Grade 10 – download the PDF

1 Interior angle sums ★★★

Find the sum of the interior angles of (a) a hexagon, (b) an octagon, (c) a 15-gon.

2 A regular nonagon ★★★

A tile has the shape of a regular 9-gon. Find each exterior angle and each interior angle.

3 Angles of a parallelogram ★★★

In parallelogram \(PQRS\), \(\angle P=68^\circ\). Find the other three angles and justify.

4 Opposite sides ★★★

In parallelogram \(ABCD\), \(AB=3x+2\) and \(CD=5x-6\) (in centimeters). Find \(x\) and the length of \(\overline{AB}\).

5 True or false? ★★★

Decide whether each statement is true or false, and justify with a reason or a counterexample.

1. Every square is a rectangle.
2. Every rectangle is a rhombus.
3. The diagonals of every parallelogram are congruent.

6 Diagonals that bisect ★★★

The diagonals of parallelogram \(JKLM\) meet at \(N\). Given \(JN=2x+3\) and \(NL=4x-7\), find \(x\) and the length of diagonal \(\overline{JL}\).

7 A garden bed ★★★

A garden bed is a trapezoid with parallel sides of \(12\text{ ft}\) and \(20\text{ ft}\). A fence runs across the bed through the midpoints of the two slanted sides. How long is this fence?

8 Angles in a pentagon ★★★

The five interior angles of a convex pentagon measure \(x^\circ,\ (x+10)^\circ,\ (x+20)^\circ,\ (x+30)^\circ,\ (x+40)^\circ\). Find all five angles.

9 How many sides? ★★★

Each interior angle of a regular polygon measures \(156^\circ\). How many sides does it have?

10 A rhombus from its diagonals ★★★

The diagonals of a rhombus measure \(10\text{ cm}\) and \(24\text{ cm}\). Find the side length, the perimeter, and the area.

11 A rectangular screen ★★★

In rectangle \(ABCD\), \(AC=7x-5\) and \(BD=4x+10\). (a) Find \(x\) and the diagonal length. (b) The short side is \(18\text{ cm}\). Find the long side.

12 Angles of a kite ★★★

In kite \(ABCD\) with \(AB=AD\) and \(CB=CD\), \(\angle A=48^\circ\) and \(\angle B=112^\circ\). Find \(\angle D\), \(\angle C\), and the angle that diagonal \(\overline{AC}\) makes with side \(\overline{AB}\).

13 Isosceles trapezoid angles ★★★

In isosceles trapezoid \(ABCD\) with bases \(\overline{AB}\) and \(\overline{DC}\), the base angles at \(A\) and \(B\) measure \(65^\circ\). Find \(\angle C\) and \(\angle D\), and check the angle sum.

14 Which condition is enough? ★★★

For each description of quadrilateral \(WXYZ\), say whether it proves a parallelogram, and name the criterion or explain why not.

1. \(WX=YZ=7\) and \(\overline{WX}\parallel\overline{YZ}\).
2. Sides in order measure \(5,\ 8,\ 5,\ 8\).
3. Sides in order measure \(5,\ 5,\ 8,\ 8\).
4. \(\overline{WX}\parallel\overline{YZ}\) only.

15 Coordinate proof: parallelogram ★★★

Given \(A(-2,-1)\), \(B(4,1)\), \(C(6,6)\), \(D(0,4)\). (a) Prove that \(ABCD\) is a parallelogram. (b) Is it a rectangle?

16 Coordinate proof: rectangle ★★★

Show that \(P(0,0)\), \(Q(4,2)\), \(R(3,4)\), \(S(-1,2)\) are the vertices of a rectangle, then find its area.

17 Coordinate proof: rhombus ★★★

Let \(A(-3,0)\), \(B(0,4)\), \(C(3,0)\), \(D(0,-4)\). (a) Prove that \(ABCD\) is a rhombus. (b) Find its area. (c) Why is it not a square?

18 Coordinate proof: trapezoid ★★★

Let \(A(0,0)\), \(B(8,0)\), \(C(6,4)\), \(D(2,4)\). (a) Show that \(ABCD\) is an isosceles trapezoid. (b) Find the midsegment length and the area.

19 Algebra with a parallelogram ★★★

(a) In parallelogram \(ABCD\), \(\angle A=(3x+15)^\circ\) and \(\angle C=(5x-25)^\circ\). Find all four angles. (b) A parallelogram-shaped banner has a perimeter of \(76\text{ in}\), and its longer side is \(6\text{ in}\) more than its shorter side. Find both side lengths.

20 A mystery polygon ★★★

The interior angles of a convex polygon add up to \(1980^\circ\). (a) How many sides does it have? (b) How many diagonals can be drawn from one vertex? (c) How many diagonals does it have in all?

21 Joining the midpoints ★★★

Take \(A(0,0)\), \(B(8,2)\), \(C(10,8)\), \(D(2,6)\), and let \(M\), \(N\), \(P\), \(Q\) be the midpoints of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{DA}\). (a) Find the four midpoints. (b) Prove that \(MNPQ\) is a parallelogram. (c) Explain why this works for any quadrilateral.