c. n. colon geometry st. barnabas hs. introduction isosceles triangles can be seen throughout our...
TRANSCRIPT
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Ch 9-6 The Converse of the Isosceles Triangle Theorem
C. N. Colon
Geometry
St. Barnabas HS
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IntroductionIsosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles triangles are a distinct classification of triangles with unique characteristics and parts that have specific names. In this lesson, we will explore the qualities of isosceles triangles.
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• Isosceles triangles have at least two congruent sides, called legs.
• The angle created by the intersection of the legs is called the vertex angle.
• Opposite the vertex angle is the base of the isosceles triangle.
• Each of the remaining angles is referred to as a base angle. The intersection of one leg and the base of the isosceles triangle creates a base angle.
Key Concepts
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Key Concepts
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Theorem
Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the congruent sides are congruent.
Key Concepts
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• If the Isosceles Triangle Theorem is reversed, then that statement is also true.
• This is known as the Converse of the Isosceles Triangle Theorem.
Key Concepts
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Theorem
Converse of the Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Key Concepts
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• If the vertex angle of an isosceles triangle is bisected, the bisector is perpendicular to the base, creating two right triangles.
• In the diagram that follows, D is the midpoint of .
Key Concepts
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• Equilateral triangles are a special type of
isosceles triangle, for which each side of the
triangle is congruent.
• If all sides of a triangle are congruent, then all
angles have the same measure.
Key Concepts
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Theorem
If a triangle is equilateral then it is equiangular, or has equal angles.
Key Concepts
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• Each angle of an equilateral triangle measures 60˚
• Conversely, if a triangle has equal angles, it is equilateral.
Key Concepts
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Key Concepts, continuedTheorem
If a triangle is equiangular, then it is equilateral.
Key Concepts
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• These theorems and properties can be used to solve many triangle problems.
Key Concepts
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Common Errors/Misconceptions
• incorrectly identifying parts of isosceles triangles
• not identifying equilateral triangles as having the same properties of isosceles triangles
• incorrectly setting up and solving equations to find unknown measures of triangles
• misidentifying or leaving out theorems, postulates, or definitions when writing proofs
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YOU TRY
Determine whether with vertices A (–4, 5), B (–1, –4), and C (5, 2) is an isosceles triangle. If it is isosceles, name a pair of congruent angles.
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Use the distance formula to calculate the length of each side.
Calculate the length of .
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Substitute A (–4, 5) and B (–1, –4) for (x1, y1) and (x2, y2).
Simplify.
Calculate the length of
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Calculate the length of
Substitute (–1, –4) and (5, 2) for (x1, y1) and (x2, y2).
Simplify.
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Calculate the length of .
Substitute (–4, 5) and (5, 2) for (x1, y1) and (x2, y2).
Simplify.
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Determine if the triangle is isosceles.
A triangle with at least two congruent sides is an isosceles triangle.
, so is isosceles.
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Identify congruent angles. If two sides of a triangle are congruent, then the angles opposite the sides are congruent.
✔
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Geogebra is a graphing program that can be used to illustrate the properties of isosceles triangles.
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Find the values of x and y.
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Make observations about the figure.
The triangle in the diagram has three congruent sides.
A triangle with three congruent sides is equilateral.
Equilateral triangles are also equiangular.
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The measure of each angle of an equilateral triangle is 60˚.
An exterior angle is also included in the diagram.
The measure of an exterior angle is the supplement of the adjacent interior angle.
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Determine the value of x.
The measure of each angle of an equilateral triangle is 60˚.
Create and solve an equation for x using this information.
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The value of x is 9.
Equation
Solve for x.
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Determine the value of y.
The exterior angle is the supplement to the interior angle.
The interior angle is 60˚ by the properties of equilateral triangles.
The sum of the measures of an exterior angle and interior angle pair equals 180.
Create and solve an equation for y using this information.
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The value of y is 13.
Equation
Simplify.
Solve for y.
✔
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Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite to these angles are congruent.
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Converse of Isosceles Triangle Theorem PROOF
If two angles of a triangle are congruent,then the sides opposite to these angles are congruent.
Draw , the bisector of the vertex angle .
Since is the angle bisector,
By the Reflexive Property,
It is given that .Therefore, by AAS congruence, .Since corresponding parts of congruent trianglesare congruent,
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p. 361# 4-20 (mo4)
HOMEWORK
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C. N. ColόnGeometry
Ch 9-7Proving Right Triangles Congruent by Hypotenuse Leg(HL)
Reference: SIMON PEREZ.
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CONGRUENT TRIANGLES
Corresponding Parts of Congruent Triangles are Congruent
ABC KLM by CPCTC
B C
A
L M
K
We would have to prove that all six pairs of corresponding parts are congruent!
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ABC KLM by SSS
B C
A
L M
K
We only had to prove that three pairs of corresponding parts are congruent!
CONGRUENT TRIANGLES by Side-Side-Side
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ABC KLM by SAS
B C
A
L M
K
We only had to prove that three pairs of corresponding parts are congruent!
CONGRUENT TRIANGLES by Side-Angle-Side
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ABC KLM by ASA
B C
A
L M
K
We only had to prove that three pairs of corresponding parts are congruent!
CONGRUENT TRIANGLES by Angle-Side-Angle
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ABC KLM by AAS
B C
A
L M
K
We only had to prove that three pairs of corresponding parts are congruent!
CONGRUENT TRIANGLES by Angle-Angle-Side
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RST JKL by SSS RST JKL by SAS
RST JKL by ASA RST JKL by AAS
SUMMARY: CONGRUENCE THEOREMS IN TRIANGLES
R
J
S
K
T
L R
J
S
K
T
L
R
J
S
K
T
LR
J
S
K
T
L
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List the parts that are missing to be marked as congruent for both triangles to be congruent by AAS:
RST JKL by AAS
S T
R
K
L
J
If
SRT KJL
STR KLJ
Missing partRS KJ
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42
Missing parts
ABC KLM by CPCTC
B C
AL
M
K
List the parts that are missing to be marked as congruent for both triangles to be congruent by CPCTC:
IF THEN
AB KL
AC KMBC LM BAC LKM
ABC KLMACB KML
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List the parts that are missing to be marked as congruent for both triangles to be congruent by SAS:
RST JKL by SAS
S T
R
K
L
J
SR KJ
RT JLIf Missing part SRT KJL
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List the parts that are missing to be marked as congruent for both triangles to be congruent by SSS:
RST JKL by SSS
S T
R
K
L
J
SR KJRT JLIfTS LK Missing part
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List the parts that are missing to be marked as congruent for both triangles to be congruent by ASA:
RST JKL by ASA
S T
R
K
L
J
RT JLIf
SRT KJL
Missing partSTR KLJ
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S T
R
K L
J
RST JKL by LL
RIGHT TRIANGLES CONGRUENT by LEG - LEG
We only had to prove that two pairs of corresponding parts are congruent!
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S T
R
K L
J RST JKL by L A
RIGHT TRIANGLES CONGRUENT by LEG-ANGLE
We only had to prove that two pairs of corresponding parts are congruent!
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S T
R
K L
J RST JKL by H L
RIGHT TRIANGLES CONGRUENT BY
HYPOTENUSE - LEG
We only had to prove that two pairs of corresponding parts are congruent!
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S T
R
K L
J RST JKL by HA
RIGHT TRIANGLES CONGRUENT byHYPOTENUSE - ANGLE
We only had to prove that two pairs of corresponding parts are congruent!
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RST JKL by HL
RST JKL by LL RST JKL by LA
RST JKL by HA
SUMMARY: CONGRUENCE THEOREMS IN RIGHT TRIANGLES
R
J
S
K
T
LR
J
S
K
T
L
R
J
S
K
T
L
R
J
S
K
T
L
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ST
R
K L
J
List the parts that are missing to be marked as congruent for both triangles to be congruent by LA
RST JKL by LAIf
STR KLJ
Missing partRS KJ
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p. 366 # 4, 12 and 14
HOMEWORK