3.2: Other Quadrilaterals
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In this section we will consider other quadrilaterals with special properties: the rhombus, the rectangle, the square, and the trapezoid.


A rhombus is a parallelogram in which all sides are equal (Figure 3.2.1). It has all the properties of a parallelogram plus some additional ones as well. Let us draw the diagonals AC and BD (Figure 3.2.2). By Theorem 3.2.3 of section 3.1 the diagonals bisect each other. Hence
△ADE≅△CDE≅△CBE≅△ABE
by SSS=SSS. The corresponding angles of the congruent triangles are equal:
∠1=∠2=∠3=∠4,
∠5=∠6=∠7=∠8
and
∠9=∠10=∠11=∠12.
∠9 and ∠10 are supplementary in addition to being equal, hence ∠9=∠10=∠11=∠12=90∘. We have proven the following theorem:
The diagonals of a rhombus are perpendicular and bisect the angles. See Figure 3.2.3.

Find w, x, y, and z:

Solution
ABCD is a rhombus since it is a parallelogram all of whose sides equal 6. According to Theorem 3.2.1, the diagonals are perpendicular and bisect the angles. Therefore w∘=40∘ since AC bisects ∠BAD. ∠AED=90∘ so x∘=180∘−(90∘+40∘)=180∘−130∘=50∘ (the sum of the angles of ∠AED is 180∘ ). Finally y∘=w∘=40∘ (compare with Figure 3.2.3) and z∘=x∘=50∘.
Answer
w=40,x=50,y=40,z=50.
Figure 3.2.4 shows rhombus ABCD of Example 3.2.1 with all its angles identified.

A rectangle is a parallelogram in which all the angles are right angles (Figure 3.2.5). It has all the properties of a parallelogram plus some additional ones as well. It is not actually necessary to be told that all the angles are right angles:


A parallelogram with just one right angle must be a rectangle.
In Figure 3.2.6 if ∠A is a right angle then all the other angles must be right angles too.
- Proof
-
In Figure 3.2.6, ∠C=∠A=90∘ because the opposite angles of a parallelogram are equal (Theorem 3.2.1, section 3.1). ∠B=90∘ and ∠D=90∘ because the successive angles of a parallelogram are supplementary (Theorem 3.2.2, section 3.2).
Find x and y:

Solution
Theorem 3.2.2, ABCD is a rectangle. x∘=40∘ because alternate interior angles of parallel lines AB and CD must be equal. Since the figure is a rectangle ∠BCD=90∘ and y∘=90∘−x∘=90∘−40∘=50∘.
Answer: x=40,y=50
Let us draw the diagonals of rectangle ABCD (Figure 3.2.7).

We will show △ABC≅△BAD. AB=BA because of identity. ∠A=∠B=90∘. BC=AD because the opposite sides of a parallelogram are equal. Then △ABC≅△BAD by SAS=SAS. Therefore diagonal AC= diagonal BD because they are corresponding sides of congruent triangles. We have proven:
The diagonals of a rectangle are equal. In Figure 3.2.7, AC=BD.
Find w, x, y, z, AC and BD:

Solution
x=3 because the diagonals of a parallelogram bisect each other. So AC=3+3=6. BD=AC=6 since the diagonals of a rectangle are equal (Theorem 3.2.3). Therefore y=z=3 since diagonal BD is bisected by diagonal AC.
Answer: x=y=z=3 and AC=BD=6.
Find x, y, and z:

Solution
x∘=35∘, because alternate interior angles of parallel lines are equal. y∘=x∘=35∘ because they are base angles of isosceles triangle ABE so (AE= BE\) because the diagonals of a rectangle are equal and bisect each other). z∘=180∘−(x∘+y∘)=180∘−(35∘+35∘)=180∘−70∘=110∘. Figure 3.2.8 shows rectangle ABCD with all the angles identified.
Answer: x=y=z=3,AC=BD=6.

The Square
A square is a rectangle with all its sides equal. It is therefore also a rhombus. So it has all the properties of the rectangle and all the properties of the rhombus.


A trapezoid is a quadrilateral with two and only two sides parallel. The parallel sides are called bases and the other two sides are called legs. In Figure 3.2.8 AB and CD are the bases and AD and BC are the legs. ∠A and ∠B are a pair of base angles. ∠C and ∠D are another pair of base angles.
An isosceles trapezoid is a trapezoid in which the legs are equal. In Figure 3.2.8, ABCD is an isosceles trapezoid with AD=BC. An isosceles trapezoid has the following property:
The base angles of an isosceles trapezoid are equal. In Figure 3.2.11, ∠A=∠B and ∠C=∠D

Find x,y, and z:

Solution
x∘=55∘ because ∠A and ∠B, the base angles of isosceles trapezoid ABCD, are equal. Now the interior angles of parallel lines on the same side of the transversal are supplementary (Theorem 3 section 1.4). Therefore y∘=180∘−x∘=180∘−55∘=125∘ and z∘=180∘−55∘=125∘.
Answer: x=22, y=z=125.
Proof of Theorem 3.2.4: Draw DE parallel to CB as in Figure 3.2.12. ∠1=∠B because corresponding angles of parallel lines are equal, DE=BC because they are the opposite sides of parallelogram BCDE. Therefore AD=DE. So △ADE is isosceles and its base angles, ∠A and ∠1, are equal. We have proven A=∠1=∠B. To prove ∠C=∠D, observe that they are both supplements of ∠A=∠B (Theorem 3.2.3, section 1.4).
The isosceles trapezoid has one additional property:
The diagonals of an isosecles trapezoid are equal.
In Figure 3.2.13, AC=BD

- Proof
-
BC=AD, given, ∠ABC=∠BAD because they are the base angles of isosceles trapezoid ABCD (Theorem 3.2.4). AB=BA, identity. Therefore △ABC≅△BAD by SAS=SAS. So AC=BD because they are corresponding sides of the congruent triangles.
Find x if AC=2x and BD=3−x:
Solution
By Theorem 3.2.5,
AC=BD2x=3−x(x)2x=(3−x)(x)2=3x−x2x2−3x+2=0(x−1)(x−2)=0
x−1=0x=1 x−2=0x=2
Check, x=1:
Check, x=2:
Answer: x=1 or x=2.
SUMMARY
THE PARALLELOGRAM
A quadrilateral in which the opposite sides are parallel.
THE RHOMBUS
A parallelogram in which all of the sides are equal.
THE RECTANGLE
A parallelogram in which all of the angles are equal to 90∘.
THE SQUARE
A parallelogram which is both a rhombus and a rectangle.
THE TRAPEZOID
A quadrilateral with just one pair of parallel sides.
THE ISOSCELES TRAPEZOID
A trapezoid in which the non-parallel sides are equal.
PROPERTIES OF QUADRILATERALS
Opposite sides are parallel | Opposite sides are equal | Opposite angles | Diagonals bisect each other | Diagonals are equal | Diagonals are perpendicular | Diagonals bisect the angles | All sides are equal | All angles are equal | |
---|---|---|---|---|---|---|---|---|---|
Parallelogram | YES | YES | YES | YES | - | - | - | - | - |
Rhombus | YES | YES | YES | YES | - | YES | YES | YES | - |
Rectangle | YES | YES | YES | YES | YES | - | - | - | YES |
Trapezoid | * | - | - | - | - | - | - | - | - |
Isosceles Trapezoid | * | * | - | - | YES | - | - | - | - |
*One pair only.
Problems
For each of the following state any theorems used in obtaining your answer.
1. Find w,x,y, and z:
2. Find w,x,y, and z:
3. Find x and y:
4. Find x and y:
5. Find x,y,z,AC and BD:
6. Find x,y, and z:
7. Find x,y, and z:
8. Find x,y, and z:
9. Find x if AC=3x and BD=4x−1:
10. Find x and y:
11. Find x,y, and z:
12. Find x,y, and z:
13. Find x,y, and z:
14. Find x,y, and z:
15. Find x and y:
16. Find x,y,∠A,∠B,∠C, and ∠D:
17. Find w,x,y, and z:
18. Find x,y, and z:
19. Find x if AC=x2−13 and BD=2x+2:
20. Find x, AC and BD: