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Some Basic Facts of Triangles
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments.
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown.
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles* All triangles are rigid.
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles* All triangles are rigid.Each side of a triangle ties down the other two sides so they can't move.
Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles* All triangles are rigid.Each side of a triangle ties down the other two sides so they can't move. Shapes with 4 or more sides may be squashed as shown.
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Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles* All triangles are rigid.Each side of a triangle ties down the other two sides so they can't move. Shapes with 4 or more sides may be squashed as shown.
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Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles
* All triangles are flat, i.e. every triangle lies in a plane.
* All triangles are rigid.Each side of a triangle ties down the other two sides so they can't move. Shapes with 4 or more sides may be squashed as shown.
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Some Basic Facts of TrianglesA triangle is a three–sided polygon, formed by three line segments. The standard way of labeling a triangle is to label a side and it's opposite angle with the same letter as shown. Basic Facts of Triangles
* All triangles are flat, i.e. every triangle lies in a plane. A shape made from 4 or more sides might be bent and protrudes into space as shown.
* All triangles are rigid.Each side of a triangle ties down the other two sides so they can't move. Shapes with 4 or more sides may be squashed as shown.
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Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.* (The Triangle Inequality)This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
2315
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
2315
less than 38
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
2315
less than 38but must be more than 23 – 15 = 8.
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
2315
less than 38but must be more than 23 – 15 = 8. 23
15
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
2315
less than 38but must be more than 23 – 15 = 8.
more than 8
2315
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesTwo sides of any triangle can’t be too short such that they can't meet as shown.
Example A.
a. The lengths of two sides of a triangle are 15 and 23. The length of the third side must be between what values?
The third side must be less than 15 + 23 = 38,
* (The Triangle Inequality) Let x, y, and z be the lengths of the sides of a triangle. The sum of the lengths of any two sides is more than the length of third side, i.e. x + y > z, x + z > y, and y + z > x.
Hence the length of the third side must be between 8 and 38.
2315
less than 38but must be more than 23 – 15 = 8.
more than 8
2315
This translates into an inequality about the lengths of the sides.
Some Basic Facts of TrianglesSince triangles are flat, we can joint the three angles as shown.
Some Basic Facts of Triangles
a
b
c
Since triangles are flat, we can joint the three angles as shown.
Some Basic Facts of Triangles
a
b
c
a
Since triangles are flat, we can joint the three angles as shown.
Some Basic Facts of Triangles
a
b
c
a c c
Since triangles are flat, we can joint the three angles as shown.
Some Basic Facts of TrianglesSince triangles are flat, we can joint the three angles as shown.
a
b
c
a c c c
Some Basic Facts of TrianglesSince triangles are flat, we can joint the three angles as shown.So the angles of any triangle may be joint into a straight line.
a
b
c
a c c c
Some Basic Facts of TrianglesSince triangles are flat, we can joint the three angles as shown.So the angles of any triangle may be joint into a straight line.
a
b
c
a c c c
* The sum of all three angles is 1800.
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
Fact About Similar Triangles
Fact About Similar Triangles
congruent triangles (the same)
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
Fact About Similar Triangles
A triangle that has two equal sides is called an isosceles triangle.
congruent triangles (the same)
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
Fact About Similar Triangles
A triangle that has two equal sides is called an isosceles triangle.
congruent triangles (the same)
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
isosceles triangles
Fact About Similar Triangles
A triangle that has two equal sides is called an isosceles triangle.
congruent triangles (the same)
* An isosceles trianglemay be cut into two congruent right triangles that are the mirror images of each other.
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
isosceles triangles
Fact About Similar Triangles
A triangle that has two equal sides is called an isosceles triangle.
congruent triangles (the same)
* An isosceles trianglemay be cut into two congruent right triangles that are the mirror images of each other.
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
isosceles triangles
Fact About Similar Triangles
A triangle that has two equal sides is called an isosceles triangle.
congruent triangles (the same)
* An isosceles trianglemay be cut into two congruent right triangles that are the mirror images of each other.
Two triangles that are exactly the same so they may be stacked on top of each other perfectly are said to be congruent.
isosceles triangles
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross–Multiplication for Solving Proportional Equations
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If
Cross–Multiplication for Solving Proportional Equations
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a.
2 3
(x + 2) (x – 5) = b.
2 (x + 1)
x= c. 3
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x
2 3
(x + 2) (x – 5) = b.
2 (x + 1)
x= c. 3
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
5 6
2 (x + 1)
x= c. 3
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)
5 6
2 (x + 1)
x= c. 3
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)2x – 10 = 3x + 6
5 6
2 (x + 1)
x= c. 3
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)2x – 10 = 3x + 6
5 6
2 (x + 1)
x= c. 3
–10 – 6 = 3x – 2x –16 = x
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)2x – 10 = 3x + 6
5 6
2 (x + 1)
x= c.x(x + 1) = 6
3
–10 – 6 = 3x – 2x –16 = x
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
Cross
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)2x – 10 = 3x + 6
5 6
2 (x + 1)
x= c.x(x + 1) = 6
3
x2 + x = 6 x2 + x – 6 = 0
–10 – 6 = 3x – 2x –16 = x
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
Cross
A B
C D ,= If then AD = BC.
Cross–Multiplication for Solving Proportional Equations
Example B. Solve for x.3 x
5 2 = a. Cross
6 = 5x= x
2 3
(x + 2) (x – 5) = b.
2(x – 5) = 3(x + 2)2x – 10 = 3x + 6
5 6
2 (x + 1)
x= c.x(x + 1) = 6
3
x2 + x = 6 x2 + x – 6 = 0
(x + 3) (x – 2) = 0 So the solutions are x = –3 and x = 2.
–10 – 6 = 3x – 2x –16 = x
Two equal ratios A B
C D, =
are said to be in proportion or proportional.
Review on ProportionsA:B = C:D, or two fractions that
Cross
Cross
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional.
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
= C=A B a b c
that is, all three fractions are the same.
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
= C=A B a b c
= C ,A B,
ab
A =a c C ,
B =b c
that is, all three fractions are the same.
etc..
From this triple–equal–ratios, we get the proportions:
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
= C=A B a b c
= C ,A B,
ab
A =a c C ,
B =b c
that is, all three fractions are the same.
Example C. Given that the followingtriangles are similar, what is x?
5 4
3
10
8
etc..
x
From this triple–equal–ratios, we get the proportions:
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
= C=A B a b c
= C ,A B,
ab
A =a c C ,
B =b c
that is, all three fractions are the same.
Example C. Given that the followingtriangles are similar, what is x?
5 4
3
10
8
etc..
x
We must have that = 10 x3 5
From this triple–equal–ratios, we get the proportions:
Fact About Similar TrianglesTwo triangles are similar (the same shape) if the corresponding sides of triangles are proportional. So given the following triangles, they are similar means that
= C=A B a b c
= C ,A B,
ab
A =a c C ,
B =b c
that is, all three fractions are the same.
Example C. Given that the followingtriangles are similar, what is x?
5 4
3
10
8
etc..
x
We must have that = 10 x3 5 = 2 so x = 6.
From this triple–equal–ratios, we get the proportions:
Similar triangles appear in the following situations. Examples of Similar Triangles
Similar triangles appear in the following situations. Examples of Similar Triangles
I. Given that DE is parallel to AB,
Similar triangles appear in the following situations. Examples of Similar Triangles
I. Given that DE is parallel to AB, then ΔABC is similar to ΔDEC.
II. Given a right triangle as shown,
Similar triangles appear in the following situations. Examples of Similar Triangles
I. Given that DE is parallel to AB, then ΔABC is similar to ΔDEC.
A
B
C
II. Given a right triangle as shown, with CD perpendicular to AB,
Similar triangles appear in the following situations. Examples of Similar Triangles
I. Given that DE is parallel to AB, then ΔABC is similar to ΔDEC.
A
B
C
D
II. Given a right triangle as shown, with CD perpendicular to AB, then the trianglesΔABC, ΔACD, and ΔBCDare similar.
Similar triangles appear in the following situations. Examples of Similar Triangles
I. Given that DE is parallel to AB, then ΔABC is similar to ΔDEC.
A
B
C
D
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 =
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 =
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 = 4(x + 5) = 10x
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 = 4(x + 5) = 10x
4x + 20 = 10x
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 = 4(x + 5) = 10x
4x + 20 = 10x20 = 6x
Examples of Similar TrianglesExample D. Find x, assuming the sides are parallel.
6x + 6 5
3 =
By similar triangles,
x + 6 6*53 = = 10
So x = 4
Example E. Find x, assuming the sides are parallel.By similar triangles,
xx + 5 10
4 = 4(x + 5) = 10x
4x + 20 = 10x20 = 6x
x = 20/6 = 10/3
5x
9
6
5
x+8
9x
5x
x+3
6
x
6 4
14 x
6 x–3
14x
6 4
x + 6a. b. c.
d. e. f.
45
xy
z
Ex. (Similar Triangles) Solve for the variables.
1
2 x
y
z
1 2
x
yz
g. h. i.
