TechMathI - 3.6 - The Triangle Inequality Theorem
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The document discusses the triangle inequality theorem and applying it to determine if triangles can be constructed with given side lengths. It also provides examples of writing inequalities to find the maximum possible length of the third side of a triangle given the lengths of two other sides.
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This document discusses the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It provides examples of using the theorem to determine if a triangle can be formed based on given side lengths and to prove relationships between distances. The final example uses the theorem to prove that the total distance between three towns by traveling between two directly is greater than the direct distance between the first and last town.
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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This ensures that the three sides can form a triangle. Some key applications of the triangle inequality include determining if a set of lengths can form a triangle, solving triangle inequality equations, and relating side lengths to angle measures. For example, given a triangle with sides of 6 and 12 units, the third side must be greater than 6 units and less than 18 units to satisfy the triangle inequality.
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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It can be written as inequalities comparing the different side lengths, such as AB + AC > BC. The converse is also true - the side opposite the largest angle will be the longest side. Examples demonstrate using the triangle inequality theorem to determine if a set of lengths could form a triangle or find the possible range of values for the third side of a triangle given two side lengths.
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The document discusses various theorems and properties related to triangles, including:
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This document discusses the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It provides examples of using the theorem to determine if a triangle can be formed based on given side lengths and to prove relationships between distances. The final example uses the theorem to prove that the total distance between three towns by traveling between two directly is greater than the direct distance between the first and last town.
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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This ensures that the three sides can form a triangle. Some key applications of the triangle inequality include determining if a set of lengths can form a triangle, solving triangle inequality equations, and relating side lengths to angle measures. For example, given a triangle with sides of 6 and 12 units, the third side must be greater than 6 units and less than 18 units to satisfy the triangle inequality.
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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It can be written as inequalities comparing the different side lengths, such as AB + AC > BC. The converse is also true - the side opposite the largest angle will be the longest side. Examples demonstrate using the triangle inequality theorem to determine if a set of lengths could form a triangle or find the possible range of values for the third side of a triangle given two side lengths.
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324 Chapter 5 Relationships Within TrianglesObjective To.docx
324 Chapter 5 Relationships Within Triangles
Objective To use inequalities involving angles and sides of triangles
In the Solve It, you explored triangles formed by various lengths of board. You may have
noticed that changing the angle formed by two sides of the sandbox changes the length
of the third side.
Essential Understanding Th e angles and sides of a triangle have special
relationships that involve inequalities.
Property Comparison Property of Inequality
If a 5 b 1 c and c . 0, then a . b.
For a neighborhood improvement project, you
volunteer to help build a new sandbox at
the town playground. You have two boards
that will make up two sides of the
triangular sandbox. One is 5 ft long and the
other is 8 ft long. Boards come in the
lengths shown. Which boards can you use
for the third side of the sandbox? Explain.
Inequalities in
One Triangle
5-6
t
t
tt
o
lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
T
A
C T I V I T I
E S T
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Proof of the Comparison Property of Inequality
Given: a 5 b 1 c, c . 0
Prove: a . b
Statements Reasons
1) c . 0 1) Given
2) b 1 c . b 1 0 2) Addition Property of Inequality
3) b 1 c . b 3) Identity Property of Addition
4) a 5 b 1 c 4) Given
5) a . b 5) Substitution
Proof
hsm11gmse_NA_0506.indd 324 3/6/09 11:56:15 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
Lesson 5-6 Inequalities in One Triangle 325
Th e Comparison Property of Inequality allows you to prove the following corollary to
the Triangle Exterior Angle Th eorem (Th eorem 3-11).
Proof of the Corollary
Given: /1 is an exterior angle of the triangle.
Prove: m/1 . m/2 and m/1 . m/3.
Proof: By the Triangle Exterior Angle Th eorem, m/1 5 m/2 1 m/3. Since
m/2 . 0 and m/3 . 0, you can apply the Comparison Property of
Inequality and conclude that m/1 . m/2 and m/1 . m/3.
Applying the Corollary
Use the fi gure at the right. Why is ml2 S ml3?
In nACD, CB > CD, so by the Isosceles Triangle Th eorem,
m/1 5 m/2. /1 is an exterior angle of nABD, so by the
Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3.
Th en m/2 . m/3 by substitution.
1. Why is m/5 . m/C?
You can use the corollary to Th eorem 3-11 to prove the following theorem.
Corollary Corollary to the Triangle Exterior Angle Theorem
Corollary
Th e measure of an exterior
angle of a triangle is greater
than the measure of each of
its remote interior angles.
If . . .
/1 is an exterior angle
Then . . .
m/1 . m/2 and
m/1 . m/3
2 1
3
Proof
3
4
1
25
A
CD
B
Theorem 5-10
Theorem
If two sides of a triangle are
not congruent, then the
larger angle lies opposite
the longer side.
If . . .
XZ . XY
Then . . .
m/Y . m/Z
You will prove Theorem 5-10 in Exercise 40.
X
Y
Z
G
U
I
m
C
Th
G
How do you identify
an exterior angle?
An exterior angle
must be form.
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This document discusses how to calculate the perimeter of rectangles and squares. It provides examples of calculating perimeters given the length and width of rectangles, or given the side length of squares. Formulas are provided: For rectangles, the perimeter is 2 * length + 2 * width. For squares, the perimeter is 4 * the side length. An example word problem is also worked out calculating the perimeter of a rectangular plot of land and the cost of fencing wire around it.
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324 Chapter 5 Relationships Within TrianglesObjective To.docx
324 Chapter 5 Relationships Within Triangles
Objective To use inequalities involving angles and sides of triangles
In the Solve It, you explored triangles formed by various lengths of board. You may have
noticed that changing the angle formed by two sides of the sandbox changes the length
of the third side.
Essential Understanding Th e angles and sides of a triangle have special
relationships that involve inequalities.
Property Comparison Property of Inequality
If a 5 b 1 c and c . 0, then a . b.
For a neighborhood improvement project, you
volunteer to help build a new sandbox at
the town playground. You have two boards
that will make up two sides of the
triangular sandbox. One is 5 ft long and the
other is 8 ft long. Boards come in the
lengths shown. Which boards can you use
for the third side of the sandbox? Explain.
Inequalities in
One Triangle
5-6
t
t
tt
o
lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
T
A
C T I V I T I
E S T
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Proof of the Comparison Property of Inequality
Given: a 5 b 1 c, c . 0
Prove: a . b
Statements Reasons
1) c . 0 1) Given
2) b 1 c . b 1 0 2) Addition Property of Inequality
3) b 1 c . b 3) Identity Property of Addition
4) a 5 b 1 c 4) Given
5) a . b 5) Substitution
Proof
hsm11gmse_NA_0506.indd 324 3/6/09 11:56:15 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
Lesson 5-6 Inequalities in One Triangle 325
Th e Comparison Property of Inequality allows you to prove the following corollary to
the Triangle Exterior Angle Th eorem (Th eorem 3-11).
Proof of the Corollary
Given: /1 is an exterior angle of the triangle.
Prove: m/1 . m/2 and m/1 . m/3.
Proof: By the Triangle Exterior Angle Th eorem, m/1 5 m/2 1 m/3. Since
m/2 . 0 and m/3 . 0, you can apply the Comparison Property of
Inequality and conclude that m/1 . m/2 and m/1 . m/3.
Applying the Corollary
Use the fi gure at the right. Why is ml2 S ml3?
In nACD, CB > CD, so by the Isosceles Triangle Th eorem,
m/1 5 m/2. /1 is an exterior angle of nABD, so by the
Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3.
Th en m/2 . m/3 by substitution.
1. Why is m/5 . m/C?
You can use the corollary to Th eorem 3-11 to prove the following theorem.
Corollary Corollary to the Triangle Exterior Angle Theorem
Corollary
Th e measure of an exterior
angle of a triangle is greater
than the measure of each of
its remote interior angles.
If . . .
/1 is an exterior angle
Then . . .
m/1 . m/2 and
m/1 . m/3
2 1
3
Proof
3
4
1
25
A
CD
B
Theorem 5-10
Theorem
If two sides of a triangle are
not congruent, then the
larger angle lies opposite
the longer side.
If . . .
XZ . XY
Then . . .
m/Y . m/Z
You will prove Theorem 5-10 in Exercise 40.
X
Y
Z
G
U
I
m
C
Th
G
How do you identify
an exterior angle?
An exterior angle
must be form.
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This document provides information on calculating the area and perimeter of basic shapes. It defines key terms like area, perimeter, dimensions, and circumference. Formulas are provided for calculating the perimeter of rectangles, triangles, circles, parallelograms, trapezoids, and the area of triangles, rectangles, parallelograms, trapezoids, and circles. Examples are given for calculating perimeter and area of various shapes. Practice problems are included for applying the formulas.
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This document discusses how to calculate the perimeter of rectangles and squares. It provides examples of calculating perimeters given the length and width of rectangles, or given the side length of squares. Formulas are provided: For rectangles, the perimeter is 2 * length + 2 * width. For squares, the perimeter is 4 * the side length. An example word problem is also worked out calculating the perimeter of a rectangular plot of land and the cost of fencing wire around it.
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This document discusses how to calculate the perimeter of rectangles and squares. It provides examples of calculating perimeters given the length and width of rectangles, or given the side length of squares. Formulas are provided: For rectangles, the perimeter is 2 * length + 2 * width. For squares, the perimeter is 4 * the side length. An example word problem is also worked out calculating the perimeter of a rectangular plot of land and the cost of fencing wire around it.
Similar to TechMathI - 3.6 - The Triangle Inequality Theorem (20)
324 Chapter 5 Relationships Within TrianglesObjective To.docx
324 Chapter 5 Relationships Within TrianglesObjective To.docx
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The document contains instructions for five math centers focused on polynomial operations: adding, subtracting, distributing, and multiplying polynomials. Students are directed to copy problems from each center and show their work. Solution banks with ordered terms are provided for students to write their answers.
TechMathI - Point slope form
The document provides examples and explanations of finding the rate of change and linear equations from tables of data points and given slopes and points. It includes examples of using the point-slope formula to write linear equations from a given point and slope, as well as examples of writing linear equations from two given points. Students are prompted to write linear equations for several examples.
TechMathI - Standard form
The document discusses finding the x-intercept and y-intercept of linear equations by setting either x or y equal to 0 and solving for the other variable. It provides examples of finding the intercepts of equations and graphing the resulting point coordinates. Students are asked to practice finding and graphing intercepts for several equations.
TechMathI - Slope intercept form
This document discusses linear equations and slope-intercept form. It provides examples of:
1) Finding the slope and y-intercept of linear equations in slope-intercept form like y = 4x + 2.
2) Writing equations of lines given the slope and y-intercept, such as an equation with m = 3 and y-intercept of 7.
3) Finding the slope and y-intercept from two points on a line, then writing the equation in slope-intercept form.
TechMathI - Rate of change and slope
The document contains information about rates of change, slope, and lines. It includes examples of calculating rates of change from tables of data and slope from ordered pairs using the slope formula. It discusses horizontal and vertical lines having slopes of 0 and undefined, and positive and negative slopes indicating the direction a line slants.
TechMathI - Ch6 Test
This document contains a 36 question multiple choice test on various math concepts including:
- Calculating unit rates
- Solving proportions
- Finding percentages of quantities
- Calculating percent increases and decreases
The questions require skills like setting up and solving proportions, unit conversion, percent calculation, and percent change.
TechMathI - Ch6 Review
This document contains 30 math word problems testing foundations concepts like rates, proportions, percentages, and percent change. The problems can be solved by setting up and solving proportions, finding unit rates, calculating percentages, or determining percent change between two amounts. Key information includes rates, costs, percentages, initial and final amounts, and measurements involved in rate and percent problems.
TechMathI - Ch6 Quiz
This document contains a 15 question quiz on math concepts like proportions, ratios, rates, similar triangles, and unit conversions. The questions require setting up and solving proportions, finding unit rates, calculating rates per hour, and using similar triangles to determine unknown lengths.
TechMathI - 6.2 - Similarity examples
Similar figures are two-dimensional shapes that have the same shape but can differ in size. To be similar, the corresponding sides of the figures must be proportional, meaning they differ by a constant scale factor. Similar figures have the same angle measures but different side lengths. Problems involving similar figures can be solved by identifying the scale factor between corresponding sides and using proportional reasoning.
TechMathI - 6.2 - Similarity
This document contains 4 math word problems involving similar triangles and scaling factors to determine unknown side lengths or heights. It also provides examples of finding the height of objects like trees using the similarity of triangles formed by shadows of objects of known height.
TechMathI - 6.1 - Ratio and Proportion
1. The document provides examples of ratios, proportions, rates, unit rates, and examples of setting up and solving proportions using cross multiplication.
2. It includes multi-step proportion examples involving fractions with variables in both the numerator and denominator.
3. Additional practice problems are provided for readers to try setting up and solving different types of proportions.
TechMathI - Ch5 Test
This document contains a 33 question test on polygons and their properties. The test covers identifying polygons by name and type, finding missing side lengths and angle measures of polygons, calculating sums of interior and exterior angles, determining which statements about polygon properties are true or false, and identifying polygons based on given properties. The test includes multiple choice, true/false, and short answer questions.
TechMathI - Ch5 Vocab Quiz
This document is a vocabulary quiz that tests knowledge of polygon terms. It contains 17 multiple choice questions that ask the learner to match polygon names like triangle, square, pentagon, and hexagon with their definitions relating to number of sides, side properties, angles, and geometric shapes. The word bank at the bottom provides additional polygon shape names as potential answers.
TechMathI - 5.6 - Trapezoids
This document discusses properties and definitions related to trapezoids. It defines a trapezoid as a quadrilateral with one pair of parallel sides, and defines an isosceles trapezoid as one where the non-parallel sides are congruent. It states that the diagonals of an isosceles trapezoid are congruent and that the median of any trapezoid is parallel to the bases and has a length that is half the sum of the base lengths. It includes examples of finding missing side lengths using the median property.
TechMathI - 5.5 - Special Parallelograms
1. The document discusses different types of quadrilaterals including trapezoids, parallelograms, rectangles, squares, and rhombuses.
2. It provides steps to identify these shapes based on their properties like the number of parallel sides and whether the angles are right angles or the sides are congruent.
3. Tests are described to determine if a quadrilateral is a parallelogram based on properties like opposite sides being parallel and congruent.
TechMathI - 5.4 - Parallelograms
This document provides examples and practice problems for identifying angle measures in parallelograms. It gives the side lengths of parallelogram ABCD and asks to find the three other angle measures. It also provides a practice problem to find missing angle measures for an additional parallelogram.
TechMathI - 5.3 - Properties of Quadrilaterals
This document provides information about different types of quadrilaterals (four-sided polygons) through a foldable activity. It defines and lists properties of parallelograms, rectangles, rhombuses, squares, and trapezoids. Key quadrilaterals are distinguished by their parallel sides, right angles, and congruent sides or angles. The foldable allows students to draw and label examples of each quadrilateral type.
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- 9. 80 miles 100 miles What is the longest that the side can possibly be? 62 miles 91 miles
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