Subject: Geometry

Quadrilaterals

Geometry.Quadrilaterals History

Hide minor edits - Show changes to markup

August 07, 2011 by matthew_suan -
Added lines 1-17:

(:title Quadrilaterals:) (:table border=1 cellpadding=5 cellspacing=0:) (:cellnr bgcolor=#d4d7ba colspan=14 align=center:) The Breakdown (:cellnr:)

(:tableend:)

Changed line 28 from:
to:

Changed line 39 from:
to:

Changed line 42 from:
to:

Changed line 45 from:
to:

Changed line 52 from:
to:

Changed line 59 from:
to:

Changed line 71 from:
to:

Changed line 74 from:
to:

Changed line 81 from:
to:

Changed line 86 from:
to:

August 07, 2011 by matthew_suan -
Changed lines 71-72 from:

In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y.

to:

In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y. These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Changed lines 101-107 from:

PIC – FIG 9 Question 2.

to:
Quadrilaterals
Figure 9

Example #2

Changed lines 107-116 from:

These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Solutions are below!

Answers

Question 2.

to:

(:toggle id=box2 show='Show Answer and Solution' init=hide button=1:)

Added line 126:
August 07, 2011 by matthew_suan -
Changed lines 81-84 from:

Question 1.

The statement that ABCD is isosceles has been given to us. From this we can deduce two truths:

to:

The statement that ABCD is isosceles has been given to us. From this we can deduce two truths:

Added line 87:
August 07, 2011 by matthew_suan -
Changed line 73 from:
Quadrilaterals
Figure 8
to:
Quadrilaterals
Figure 8
August 07, 2011 by matthew_suan -
Changed lines 63-64 from:

In the above diagram \angle{a} proportional to \angle{b}.

to:

In the above diagram \angle{a} \propto \angle{b}.

Changed lines 68-69 from:

Thus, in the diagram used above, if , then .

to:

Thus, in the diagram used above, if \angle{A} \propto \angle{B}, then \angle{A} \propto \angle{B} .

Changed lines 73-74 from:

PIC – FIG 8 Question 1.

to:
Quadrilaterals
Figure 8

Example #1

Changed lines 78-93 from:

PIC – FIG 9 Question 2. As before, work out the angle values for each corner.

These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Solutions are below!

Answers

to:

(:toggle id=box3 show='Show Answer and Solution' init=hide button=1:)

Changed line 85 from:

1. also = 5x + 60°.

to:

1. \angle{C}= 5x + 60°.

Added lines 99-115:

PIC – FIG 9 Question 2. As before, work out the angle values for each corner.

These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Solutions are below!

Answers

August 07, 2011 by matthew_suan -
Changed lines 63-64 from:

In the above diagram \angle{a} = \angle{b}.

to:

In the above diagram \angle{a} proportional to \angle{b}.

Changed lines 67-68 from:

Thus, in the diagram used above, if , then .'''

to:

Thus, in the diagram used above, if , then .

August 07, 2011 by matthew_suan -
Changed lines 59-60 from:

'''The base angles of an isosceles trapezoid are congruent. In the above diagram .'''

to:

The base angles of an isosceles trapezoid are congruent.

Changed lines 61-62 from:
Quadrilaterals
Figure 6: Other form of trapezoids.
to:
Quadrilaterals
Figure 7: Other form of trapezoid.

In the above diagram \angle{a} = \angle{b}.

Changed line 66 from:

'''Any trapezoid with congruent base angles is an isosceles trapezoid.

to:

Any trapezoid with congruent base angles is an isosceles trapezoid.

Changed line 70 from:

In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y.

to:

In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y.

August 07, 2011 by matthew_suan -
Changed line 59 from:

'''The base angles of an isosceles trapezoid are congruent.

to:

'''The base angles of an isosceles trapezoid are congruent.

Changed line 65 from:

'''Any trapezoid with congruent base angles is an isosceles trapezoid.

to:

'''Any trapezoid with congruent base angles is an isosceles trapezoid.

August 07, 2011 by matthew_suan -
Changed line 53 from:

The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, \angle{C} and angle{D} .

to:

The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, \angle{C} and \angle{D} .

August 07, 2011 by matthew_suan -
Changed line 43 from:

Trapezoids

to:

What are trapezoids?

Changed lines 45-47 from:

In the following diagram showing Trapezoid Y, and are the legs, and are the bases, and is the median.

PIC – FIG 6 – TRAP Y

to:

In the following diagram showing Trapezoid Y, \bar{AC} and \bar{BD} are the legs, \bar{AB} and \bar{CD} are the bases, and \bar{EF} is the median.

Quadrilaterals
Figure 6: A diagram of Trapeziod Y.
Changed lines 49-60 from:

Trapezoid Y is an isosceles trapezoid because its legs are congruent ( = ). A trapezoid without congruent legs is simply called a trapezoid. An isosceles trapezoid is symmetrical when cut at a right angle to the midpoint of the median. The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, and .

Let’s look at some basic principles. Principle 1: The base angles of an isosceles trapezoid are congruent. In the above diagram .

PIC – FIG 7 other trap

to:

Trapezoid Y is an isosceles trapezoid because its legs are congruent (\bar{AC} = \bar{BD}). A trapezoid without congruent legs is simply called a trapezoid.

An isosceles trapezoid is symmetrical when cut at a right angle to the midpoint of the median.

The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, \angle{C} and angle{D} .

Principles related to trapezoids

Let’s look at some basic principles pertaining trapezoids.

Principle 1:

'''The base angles of an isosceles trapezoid are congruent. In the above diagram .'''

Changed lines 62-66 from:

Principle 2: Any trapezoid with congruent base angles is an isosceles trapezoid. Thus, in the diagram used above, if , then .

Applying Algebra to Trapezoid Figures

to:
Quadrilaterals
Figure 6: Other form of trapezoids.

Principle 2:

'''Any trapezoid with congruent base angles is an isosceles trapezoid. Thus, in the diagram used above, if , then .'''

Applying Algebra to Trapezoid Figures

August 07, 2011 by matthew_suan -
Changed line 41 from:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
to:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
August 07, 2011 by matthew_suan -
Changed line 41 from:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
to:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
August 07, 2011 by matthew_suan -
Changed line 41 from:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
to:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
August 07, 2011 by matthew_suan -
Changed lines 26-29 from:

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

to:

Principle 1:

Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2:

If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

Changed lines 34-37 from:
Quadrilaterals
Figure 4: Since quadrilaterals have sum of its angles not more than 360 degrees, angles a and b must add up to 180.

Principle 3: Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

to:
Quadrilaterals
Figure 4: Since quadrilaterals have sum of its angles not more than 360 degrees, angles a and b must add up to 180.

Principle 3:

Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

Changed line 41 from:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
to:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.
August 07, 2011 by matthew_suan -
Changed lines 32-41 from:

PIC – FIG 4

Principle 3: Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

to:
Quadrilaterals
Figure 4: Since quadrilaterals have sum of its angles not more than 360 degrees, angles a and b must add up to 180.

Principle 3: Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

Changed lines 38-46 from:

PIC 3 – FIG 5 – 1 quad with extending sides

Trapezoids

to:
Quadrilaterals
Figure 4: Quadrilaterals with extending sides.

Trapezoids

August 07, 2011 by matthew_suan -
Changed lines 26-28 from:

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

to:

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

August 07, 2011 by matthew_suan -
Changed lines 26-28 from:

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

to:

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

August 07, 2011 by matthew_suan -
Changed lines 10-11 from:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
to:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
Changed line 16 from:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
to:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
August 07, 2011 by matthew_suan -
Changed lines 6-7 from:
Quadrilaterals
Figure 1: Examples of quadrilaterals
to:
Quadrilaterals
Figure 1: Examples of quadrilaterals
Changed lines 10-11 from:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
to:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
Changed line 16 from:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
to:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
August 07, 2011 by matthew_suan -
Changed lines 6-7 from:
Quadrilaterals
Figure 1: Examples of quadrilaterals
to:
Quadrilaterals
Figure 1: Examples of quadrilaterals
Changed lines 10-11 from:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
to:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
Changed line 16 from:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
to:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
August 07, 2011 by matthew_suan -
Changed lines 6-7 from:
Quadrilaterals
Figure 1: Examples of quadrilaterals
to:
Quadrilaterals
Figure 1: Examples of quadrilaterals
Changed lines 10-11 from:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
to:
Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals
Changed lines 16-17 from:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
to:
Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same
Deleted line 136:
August 07, 2011 by matthew_suan -
August 07, 2011 by matthew_suan -
August 07, 2011 by matthew_suan -
Added lines 1-139:

What are quadrilaterals?

A quadrilateral is a four-sided, two-dimensional enclosed figure (quad means four, lateral means side). Squares and rectangles are quadrilaterals, but quadrilaterals are not always square or rectangular. The sides do not have to be the same length, and can meet at any angle.

This section will look at angle relationships within quadrilaterals. Here are a few shapes that meet the requirements:

Quadrilaterals
Figure 1: Examples of quadrilaterals

Here are a few that don’t. None of the following shapes are quadrilaterals. If you don’t understand why, slow down and have a think about each one. The first sentence in this article ought to help you out.

Quadrilaterals
Figure 2: Examples of shapes that are not quadrilaterals

Sum of all angles in a quadrilateral

What do you think the angles of different quadrilaterals add up to? Why? Do you think quadrilaterals all have different angle sums, or does it depend on the length of their sides? (An angle sum is the total value of all interior angles in a shape.) In the following diagram (see Figure 3), does A have the same angle sum as B?

Quadrilaterals
Figure 3: Totally different quadrilaterals A and B but their angle sums are just the same

Think about cutting a square or rectangle in half diagonally…

If you already know that the angles of any and every triangle add to 180°, it should make sense to you that the angles of a quadrilateral sum to 360° (two times 180°). This is easy to do with square and rectangular quadrilaterals, but harder when the sides are not all equal.

Principles related to quadrilaterals

Later on in this article we’ll have a look at trapezoids. For now, let’s examine some principles related to quadrilaterals of a general nature.

Principle 1: Interior angles of any quadrilateral will always add to exactly 360°.

Principle 2: If two opposite angles in a quadrilateral are right angles, the other two opposite angles are supplementary (they add to 180°).

In the following diagram, angles a and b must add to 180° for the total enclosure to equal 360°.

PIC – FIG 4

Principle 3: Quadrilateral edges drawn to extend past corners will create supplementary interior/exterior angles.

In the following diagram, angles a and c are supplementary, and angles b and d are supplementary.

PIC 3 – FIG 5 – 1 quad with extending sides

Trapezoids

A trapezoid is a quadrilateral that has exactly two parallel sides. In other words, it has two parallel sides and two non-parallel sides. The parallel sides are called bases, the non-parallel sides connecting the bases are called legs. The median is the finite line segment connecting the two midpoints of the legs. In the following diagram showing Trapezoid Y, and are the legs, and are the bases, and is the median.

PIC – FIG 6 – TRAP Y

Trapezoid Y is an isosceles trapezoid because its legs are congruent ( = ). A trapezoid without congruent legs is simply called a trapezoid. An isosceles trapezoid is symmetrical when cut at a right angle to the midpoint of the median. The base angles of a trapezoid are the angles at the ends of its longer base, in Trapezoid Y’s case, and .

Let’s look at some basic principles. Principle 1: The base angles of an isosceles trapezoid are congruent. In the above diagram .

PIC – FIG 7 other trap

Principle 2: Any trapezoid with congruent base angles is an isosceles trapezoid. Thus, in the diagram used above, if , then .

Applying Algebra to Trapezoid Figures In the following diagram, use your knowledge of geometry to find values for all four angles of each trapezoid by first finding values for x and y.

PIC – FIG 8 Question 1. ABCD is an isosceles trapezoid.

PIC – FIG 9 Question 2. As before, work out the angle values for each corner.

These shouldn’t be too hard for you to solve, just use a little logic and you’ll be fine.

Solutions are below!

Answers Please note that a lower case x means ‘letter x’ which replaces a number, while upper case X denotes multiplication. Question 1.

The statement that ABCD is isosceles has been given to us. From this we can deduce two truths: 1. also = 5x + 60°. 2. 5x = 2x + y. Therefore 5x – y = 2x. Therefore y = 3x. Working out the values for x and y is not too difficult from here. Because and are parallel, we know that = 180°. In other words 10x + 60 = 180°. Therefore 10x = 120°. (Deduct 60° from 180°.) Therefore x = 120°/10 x = 12°. Finding values for angles is as simple as substituting this value into x in the diagram. = 5x = 5 X 12 = 60° = 5x + 60 = (5 X 12) + 60 = 120° Because ABCD is isosceles, we know and .

Question 2.

Solving the second diagram is a little more interesting. The fact that alone tells us that ABCD is not an isosceles trapezoid, so we’ll have to work out at least three angles in order to deduce the fourth. We know that because and are parallel, we know that = 180°, and that = 180°. We’ll work with and first. To put this mathematically, 12x – 15 + 15 = 180°. Simplified, 12x = 180°. Therefore x = 180/12 = 15°. = 5 X 15 = 75°. = 7 X 15 = 105°. (A possibility to consider: These two values are equal to 180, therefore the lines are parallel, ABCD is indeed a trapezoid. If 180°, and are not parallel… and ABCD is not a trapezoid.) = 9 X 15 = 135°. There are two possible ways to find a value for . The first is by deductive reasoning: = 360 – ( + + ) = 360 – (75 + 105 + 135) = 45°. To put this in words, must equal 360 minus the sum of the other three angles. The second is by algebra: We know that there are 24 fifteens in 360 ( 360/15 = 24). Because the other three angles sum to 21x, we know = 3x. We’ve been given a value of x + 2y for . From this we can conclude x = y because x + 2y = 3x. = 3x = 3 X 15 = 45°.