GED Math: Four-Sided Plane Figures
To name a four-sided figure, collect the properties from the description or drawing and match them to the strictest definition that fits. Exactly one pair of parallel sides: trapezoid. Two pairs of parallel sides (opposite sides parallel and equal): parallelogram. A parallelogram with four equal sides: rhombus. A parallelogram with four right angles: rectangle. Four equal sides and four right angles: square. If the opposite angles are equal and the opposite sides are equal but there are no right angles, the figure is a parallelogram (unless all four sides are equal, which makes it a rhombus). Remember that one name can include another: a square is also a rectangle, a rhombus, and a parallelogram.
1. Four-Sided Plane Figures and the 360° Rule
A plane figure is a flat shape made of line segments. A four-sided plane figure (a quadrilateral) is named by the properties of its sides and angles: whether sides are equal, whether sides are parallel, and whether angles are right angles. The most important fact about every quadrilateral is that its interior angles add to 360°, so if you know three angles you can find the fourth. In drawings, matching tick marks on sides mean those sides are equal, arrowheads mean parallel sides, and a small square at a corner means a right (90°) angle. Test tip: do not decide a figure's name by how it looks. Read the properties that are given, then decide.
Worked Example: Three angles of a four-sided figure measure 85°, 95°, and 110°. Find the fourth angle.
85 + 95 + 110 = 290, so the fourth angle is 360 − 290 = 70°.
1. What is the sum of the interior angles of any four-sided plane figure?
Explanation: Every quadrilateral has interior angles that total 360°.
2. Three angles of a four-sided figure are 85°, 95°, and 110°. What is the fourth angle?
Explanation: 360 − (85 + 95 + 110) = 360 − 290 = 70.
3. Three angles of a four-sided figure are 120°, 75°, and 90°. What is the fourth angle?
Explanation: 120 + 75 + 90 = 285, and 360 − 285 = 75.
4. The angles of a four-sided figure are x, 2x, 3x, and 4x. What is the largest angle?
Explanation: 10x = 360, so x = 36. The largest angle is 4(36) = 144.
5. In a drawing, a small square at a corner means:
Explanation: A small square marks a 90° angle.
6. In a drawing, two sides each have one matching tick mark. This means:
Explanation: Matching tick marks show equal side lengths (arrowheads show parallel sides).
2. Rectangles and Squares
A rectangle is a four-sided figure with four right angles. Its opposite sides are the same length and parallel. A square is a special rectangle: it also has four equal sides, so it has four equal angles and four equal sides. Every square is a rectangle, but not every rectangle is a square. Useful formulas: perimeter of a rectangle = 2(length + width); area of a rectangle = length × width; area of a square = side × side. Because a rectangle's angles are all 90°, the angle sum works out: 4 × 90° = 360°.
Worked Example: A rectangle is 12 cm long and 7 cm wide. Find its perimeter and area.
Perimeter = 2(12 + 7) = 38 cm. Area = 12 × 7 = 84 cm2.
1. Which four-sided figures always have four right angles?
Explanation: A rectangle has four right angles by definition, and a square is a rectangle with equal sides.
2. Which statement is true?
Explanation: A square meets all the rules for a rectangle (four right angles, opposite sides equal and parallel) plus equal sides.
3. Rectangle RSUT has opposite sides parallel and four right angles. What is the measure of ∠R?
Explanation: Every angle of a rectangle (and square) is 90°.
4. A rectangle is 12 cm long and 7 cm wide. What is its perimeter?
Explanation: 2(12 + 7) = 2(19) = 38.
5. A rectangle is 12 cm long and 7 cm wide. What is its area?
Explanation: Area = 12 × 7 = 84.
6. A square has a side of 6 inches. What is its area?
Explanation: Area = 6 × 6 = 36.
7. A square has a perimeter of 28 feet. What is the length of one side?
Explanation: All four sides are equal, so side = 28 ÷ 4 = 7.
8. A rectangle has a perimeter of 50 and a width of 10. What is its length?
Explanation: 2(L + 10) = 50, so L + 10 = 25 and L = 15.
3. Parallelograms and Rhombuses
A parallelogram is a four-sided figure whose opposite sides are parallel and equal in length. Its opposite angles are equal, and any two neighboring (consecutive) angles add to 180°. A rhombus is a parallelogram with four equal sides; it has the same angle rules (opposite angles equal, neighbors add to 180°). Rectangles and squares are also parallelograms, because their opposite sides are parallel. To find missing angles, use the fact that opposite angles are equal and neighbors sum to 180°, or simply that the four angles sum to 360°. Area of a parallelogram = base × height, where the height is the perpendicular distance between the parallel sides (not the slanted side).
Worked Example: In parallelogram ABCD, ∠B = 110°. Find the other angles.
The opposite angle ∠D equals 110°. The two remaining angles are equal, so 2x + 110 + 110 = 360, 2x = 140, x = 70°. So ∠A = ∠C = 70°. Check: neighbors 110 + 70 = 180.
1. In parallelogram ABCD (shown), what is the measure of ∠D?
Explanation: Opposite angles of a parallelogram are equal, so ∠D = ∠B = 110°.
2. In the same parallelogram, what is the measure of ∠A?
Explanation: ∠A and ∠B are neighbors, so they add to 180: 180 − 110 = 70.
3. In a parallelogram, one angle measures 65°. What is the measure of an angle next to it?
Explanation: Neighboring angles add to 180: 180 − 65 = 115.
4. In a rhombus, one angle measures 50°. What is the measure of the angle opposite it?
Explanation: Opposite angles of a rhombus are equal.
5. In a rhombus, one angle measures 50°. What is the measure of each of the other two angles?
Explanation: The other two angles are equal: (360 − 50 − 50)/2 = 130.
6. Opposite angles of a parallelogram measure (2x + 10)° and (3x − 20)°. What is the value of x?
Explanation: Opposite angles are equal: 2x + 10 = 3x − 20, so x = 30 (each angle is 70°).
7. Two neighboring angles of a parallelogram measure (3x + 5)° and (2x + 15)°. What is the smaller angle?
Explanation: 3x + 5 + 2x + 15 = 180 gives x = 32. The angles are 101° and 79°, so the smaller is 79°.
8. A parallelogram has a base of 10 cm and a perpendicular height of 4 cm. What is its area?
Explanation: Area = base × height = 10 × 4 = 40.
4. Trapezoids
A trapezoid is a four-sided figure with exactly one pair of parallel sides (called the bases). The definition does not say anything about the lengths of the other sides or the sizes of the angles. In a trapezoid, the two angles along the same non-parallel side (leg) are supplementary: they add to 180°. If the legs are equal (an isosceles trapezoid), the base angles on each base are equal. A right trapezoid has two right angles. Area of a trapezoid = ½ × (base1 + base2) × height. Test tip: for a figure to be a trapezoid, one pair of opposite sides must be parallel. Equal side lengths alone are not enough.
Worked Example: In trapezoid ABCD, AB ∥ DC and ∠D = 65°. Find ∠A.
∠A and ∠D are on the same leg between the parallel sides, so they add to 180°: ∠A = 180 − 65 = 115°.
1. What is the key property of a trapezoid?
Explanation: A trapezoid has exactly one pair of parallel sides.
2. In trapezoid ABCD (shown), AB ∥ DC and ∠D = 65°. What is ∠A?
Explanation: The angles on the same leg between parallel sides add to 180: 180 − 65 = 115.
3. In trapezoid EFGH, EF ∥ HG with right angles at E and H. ∠F is 30° more than twice ∠G. What is ∠F?
Explanation: ∠F + ∠G = 180 (same leg FG). F = 2G + 30, so 3G + 30 = 180, G = 50, and F = 130.
4. An isosceles trapezoid has a base angle of 72°. What is the other angle at the same base?
Explanation: In an isosceles trapezoid, the two angles on the same base are equal.
5. An isosceles trapezoid has a base angle of 72°. What is each angle on the other base?
Explanation: Each angle on the other base pairs with a 72° angle along a leg: 180 − 72 = 108.
6. In order for figure EFGH to be a trapezoid, which of the following must be true?
Explanation: A trapezoid needs a pair of parallel sides. Side lengths or angle equality alone do not make a trapezoid.
7. A trapezoid has bases of 8 and 14 and a height of 5. What is its area?
Explanation: Area = ½(8 + 14)(5) = ½(22)(5) = 55.
5. Identifying a Figure from Its Properties
To name a four-sided figure, collect the properties from the description or drawing and match them to the strictest definition that fits. Exactly one pair of parallel sides: trapezoid. Two pairs of parallel sides (opposite sides parallel and equal): parallelogram. A parallelogram with four equal sides: rhombus. A parallelogram with four right angles: rectangle. Four equal sides and four right angles: square. If the opposite angles are equal and the opposite sides are equal but there are no right angles, the figure is a parallelogram (unless all four sides are equal, which makes it a rhombus). Remember that one name can include another: a square is also a rectangle, a rhombus, and a parallelogram.
Worked Example: A four-sided figure has sides 8, 12, 8, and 12. Opposite angles are equal and there are no right angles. What is it?
Opposite sides are equal and opposite angles are equal, so it is a parallelogram. It is not a rectangle (no right angles) and not a rhombus (the sides are not all equal). The answer is parallelogram.
1. A four-sided figure has sides 8, 12, 8, and 12. Opposite angles are equal and none are right angles. What is the figure?
Explanation: Opposite sides and angles are equal with no right angles: parallelogram.
2. A four-sided figure has four equal sides and angles of 60°, 120°, 60°, and 120°. What is the figure?
Explanation: Four equal sides with no right angles is a rhombus.
3. A four-sided figure has four equal sides and four right angles. What is the figure?
Explanation: Equal sides and right angles together define a square.
4. A four-sided figure has exactly one pair of parallel sides. What is it?
Explanation: Exactly one pair of parallel sides is a trapezoid.
5. Which figures have opposite angles that are always equal?
Explanation: Every parallelogram, including rectangles, rhombuses, and squares, has equal opposite angles.
6. Which four-sided figures have all four sides equal in length?
Explanation: A rhombus has four equal sides, and a square is a special rhombus.
7. Can a four-sided figure have exactly three right angles?
Explanation: Three right angles use 270°, so the fourth angle is 360 − 270 = 90°. That makes four right angles.
8. A four-sided figure has four right angles and opposite sides 9 and 5. What is it?
Explanation: Four right angles with unequal adjacent sides: a rectangle, not a square.
Common Mistakes to Avoid
- Naming a figure by how it looks instead of by the given properties.
- Forgetting that the angles of any four-sided figure add to 360°.
- Thinking every rectangle is a square. A square needs four equal sides.
- In a parallelogram, mixing up opposite angles (equal) with neighboring angles (add to 180°).
- Saying a trapezoid has two pairs of parallel sides. It has exactly one pair.
- Using the slanted side as the height when finding the area of a parallelogram or trapezoid. The height is perpendicular to the bases.
- Adding up only three angles and forgetting to subtract from 360°.
- Forgetting the 2 in the perimeter formula 2(length + width).
Frequently Asked Questions
What is the sum of the interior angles of a four-sided figure?
Always 360°.
What makes a figure a parallelogram?
Both pairs of opposite sides are parallel and equal. Its opposite angles are also equal.
What is the difference between a rhombus and a square?
Both have four equal sides, but a square also has four right angles.
What is a trapezoid?
A four-sided figure with exactly one pair of parallel sides.
How do I find a missing angle?
Use the properties of the figure (opposite angles equal, neighboring angles add to 180°, or the 360° total) and write an equation if needed.
Is a square a rectangle?
Yes. A square has four right angles and opposite sides that are equal and parallel, so it is a special rectangle.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.