This document discusses triangle inequalities and properties related to triangle side lengths and angle measures. It provides examples of:
1) Determining the order of angles from smallest to largest given side lengths
2) Determining the order of side lengths from smallest to largest given angle measures
3) Applying the triangle inequality theorem which states the sum of any two side lengths must be greater than the third side length.
This powerpoint presentation is an introduction for the topic TRIANGLE CONGRUENCE. This topic is in Grade 8 Mathematics. I hope that you will learn something from this sides.
This powerpoint presentation is an introduction for the topic TRIANGLE CONGRUENCE. This topic is in Grade 8 Mathematics. I hope that you will learn something from this sides.
Identify isosceles and equilateral triangles by side length and angle measure
Use the Isosceles Triangle Theorem to solve problems
Use the Equilateral Triangle Corollary to solve problems
* Classify triangles by sides and by angles
* Find the measures of missing angles of right and equiangular triangles
* Find the measures of missing remote interior and exterior angles
* Name polygons based on their number of sides
* Classify polygons based on concave/convex and equilateral/equiangular/regular
* Calculate and use the measures of interior and exterior angles of polygons
Classify triangles by sides and by angles
Find the measures of missing angles of right and equiangular triangles
Find the measures of missing remote interior and exterior angles
Identify a midsegment of a triangle and use it to solve problems.
Analyze the relationship between the angles of a triangle and the lengths of the sides
Determine allowable lengths for sides of triangles
Classify triangles by angles and by sides
Analyze the relationship between the angles and sides of a triangle
Determine allowable lengths for sides of triangles
324 Chapter 5 Relationships Within TrianglesObjective To.docxgilbertkpeters11344
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
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lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
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C T I V I T I
E S T
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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
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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.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
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* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
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* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
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* Graph piecewise-defined functions
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* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
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* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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Sectors of the Indian Economy - Class 10 Study Notes pdf
2.5.5 Triangle Inequalities
1. Triangle Inequalities
The student is able to (I can):
• Analyze the relationship between the angles of a triangle
and the lengths of the sidesand the lengths of the sides
• Determine allowable lengths for sides of triangles
2. If two sides of a triangle are not congruent,
then the larger angle is opposite the longer
side.
If two angles of a triangle are not
congruent, then the longer side is opposite
the larger angle.
AT > AC → m∠C > m∠T
m∠C > m∠T → AT > AC
A
C
T
m∠C > m∠T → AT > AC
3. Example: Given the side lengths, put the
angles in order from smallest to
largest.
A
19 16
∠P is across from 16, ∠N is across from
19, and ∠A is across from 31, so it would
be: ∠P, ∠N, and ∠A
P N
31
4. Example: Given the angle measures, put
the side lengths in order from
smallest to largest.
E
First, we have to calculate m∠E:
m∠E = 180- (70+30) = 80°
So the sides would be:
T N
70° 30°
TE EN TN< <
5. Triangle Inequality Theorem
The sum of any two side lengths of a
triangle is greater than the third side
length.
Example:
1. Which set of lengths forms a triangle?
4, 5, 10 7, 9, 124, 5, 10 7, 9, 12
4 + 5 < 10 7 + 9 > 12
6. Note: To find a range of possible third
sides given two sides, subtract for
the lower bound and add for the
upper bound.
Examples:
2. What is a possible third side for a
triangle with sides 8 and 14?triangle with sides 8 and 14?
14 — 8 = 6 lower bound
14 + 8 = 22 upper bound
The third side can be between 6 and 22.
7. 3. What is the range of values for the
third side of a triangle with sides 11 and
19?
19 — 11 = 8 lower bound
19 + 11 = 30 upper bound
8 < x < 308 < x < 30
