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Mathematics LibreTexts

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.

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Figure 3.2.1: A rhombus.
clipboard_e30ed209bad8de53bbaaefec6503bf0cd.png
Figure 3.2.2: A rhombus with diagonals.

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

ADECDECBEABE

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:

Theorem 3.2.1

The diagonals of a rhombus are perpendicular and bisect the angles. See Figure 3.2.3.

clipboard_e77169a8f72d68d5163bef0dc944d0fe8.png
Figure 3.2.3: The diagonals of a rhombus are perpendicular and bisect the angles.
Example 3.2.1

Find w, x, y, and z:

clipboard_ee2c102aa1c0fd84f2c25e6021ec36fba.png

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)=180130=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.

clipboard_eee8503d3c3dfa063e0987b134801aa7d.png
Figure 3.2.4: The rhombus of Example 3.2.1 with all 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:

clipboard_e3d4c7aa97fcc7b8b1285ca590d70a11b.png
Figure 3.2.5: A rectangle.
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Figure 3.2.6: A parallelogram with just one right angle must also be a rectangle.
Theorem 3.2.2

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).

Example 3.2.2

Find x and y:

clipboard_e5f54077a211435a82c7bece4f6a023be.png

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=90x=9040=50.

Answer: x=40,y=50

Let us draw the diagonals of rectangle ABCD (Figure 3.2.7).

clipboard_e5f2dc56c6da94b3935964309b2e0c91b.png
Figure 3.2.7: Rectangle wIth diagonals drawn.

We will show ABCBAD. AB=BA because of identity. A=B=90. BC=AD because the opposite sides of a parallelogram are equal. Then ABCBAD by SAS=SAS. Therefore diagonal AC= diagonal BD because they are corresponding sides of congruent triangles. We have proven:

Theorem 3.2.3

The diagonals of a rectangle are equal. In Figure 3.2.7, AC=BD.

Example 3.2.3

Find w, x, y, z, AC and BD:

clipboard_e03d55530fa44a029018f73948fd4f1f3.png

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.

Example 3.2.4

Find x, y, and z:

clipboard_ec8bb2ab0895e960dc5e621f25d29ea59.png

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)=18070=110. Figure 3.2.8 shows rectangle ABCD with all the angles identified.

Answer: x=y=z=3,AC=BD=6.

clipboard_e02e8b9c827f9970febe626ca9a23429d.png
Figure 3.2.8: The rectangle of Example 3.2.4 with all the angles identified.

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.

clipboard_e397d73d938173a5f11cf67a3dd4d4a99.png
Figure 3.2.9: A square.
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Figure 3.2.10: A trapezoid.

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:

Theorem 3.2.4

The base angles of an isosceles trapezoid are equal. In Figure 3.2.11, A=B and C=D

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Figure 3.2.11: An isosceles trapezoid
Example 3.2.5

Find x,y, and z:

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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=180x=18055=125 and z=18055=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:

Theorem 3.2.5

The diagonals of an isosecles trapezoid are equal.

In Figure 3.2.13, AC=BD

屏幕快照 2020-11-11 下午4.42.41.png
Figure 3.2.13. The diagonals AC and BD are equal.
Proof

BC=AD, given, ABC=BAD because they are the base angles of isosceles trapezoid ABCD (Theorem 3.2.4). AB=BA, identity. Therefore ABCBAD by SAS=SAS. So AC=BD because they are corresponding sides of the congruent triangles.

Example 3.2.6

Find x if AC=2x and BD=3x:

屏幕快照 2020-11-11 下午4.46.11.png

Solution

By Theorem 3.2.5,

AC=BD2x=3x(x)2x=(3x)(x)2=3xx2x23x+2=0(x1)(x2)=0

x1=0x=1 x2=0x=2

Check, x=1:

屏幕快照 2020-11-11 下午4.54.21.png

Check, x=2:

屏幕快照 2020-11-11 下午4.55.03.png

Answer: x=1 or x=2.

SUMMARY

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THE PARALLELOGRAM

A quadrilateral in which the opposite sides are parallel.

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THE RHOMBUS

A parallelogram in which all of the sides are equal.

屏幕快照 2020-11-16 下午3.15.42.png

THE RECTANGLE

A parallelogram in which all of the angles are equal to 90.

屏幕快照 2020-11-16 下午3.16.37.png

THE SQUARE

A parallelogram which is both a rhombus and a rectangle.

屏幕快照 2020-11-16 下午3.23.32.png

THE TRAPEZOID

A quadrilateral with just one pair of parallel sides.

屏幕快照 2020-11-16 下午3.24.43.png

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:

Screen Shot 2020-11-16 at 3.39.32 PM.png

2. Find w,x,y, and z:

Screen Shot 2020-11-16 at 3.39.53 PM.png

3. Find x and y:

Screen Shot 2020-11-16 at 3.40.08 PM.png

4. Find x and y:

Screen Shot 2020-11-16 at 3.40.26 PM.png

5. Find x,y,z,AC and BD:

Screen Shot 2020-11-16 at 3.40.53 PM.png

6. Find x,y, and z:

Screen Shot 2020-11-16 at 3.42.15 PM.png

7. Find x,y, and z:

Screen Shot 2020-11-16 at 3.41.57 PM.png

8. Find x,y, and z:

Screen Shot 2020-11-16 at 3.42.45 PM.png

9. Find x if AC=3x and BD=4x1:

Screen Shot 2020-11-16 at 3.43.05 PM.png

10. Find x and y:

Screen Shot 2020-11-16 at 3.43.21 PM.png

11. Find x,y, and z:

Screen Shot 2020-11-16 at 3.43.41 PM.png

12. Find x,y, and z:

Screen Shot 2020-11-16 at 3.44.03 PM.png

13. Find x,y, and z:

Screen Shot 2020-11-16 at 3.44.23 PM.png

14. Find x,y, and z:

Screen Shot 2020-11-16 at 3.44.44 PM.png

15. Find x and y:

Screen Shot 2020-11-16 at 3.45.08 PM.png

16. Find x,y,A,B,C, and D:

Screen Shot 2020-11-16 at 3.45.25 PM.png

17. Find w,x,y, and z:

Screen Shot 2020-11-16 at 3.45.44 PM.png

18. Find x,y, and z:

Screen Shot 2020-11-16 at 3.46.00 PM.png

19. Find x if AC=x213 and BD=2x+2:

Screen Shot 2020-11-16 at 3.46.21 PM.png

20. Find x, AC and BD:

Screen Shot 2020-11-16 at 3.46.41 PM.png


This page titled 3.2: Other Quadrilaterals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Henry Africk (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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