c. n. colon geometry st. barnabas hs. introduction isosceles triangles can be seen throughout our...
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Ch 9-6 The Converse of the Isosceles Triangle Theorem
C. N. Colon
Geometry
St. Barnabas HS
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.
• 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
Key Concepts
Theorem
Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the congruent sides are congruent.
Key Concepts
• 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
Theorem
Converse of the Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Key Concepts
• 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
• 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
Theorem
If a triangle is equilateral then it is equiangular, or has equal angles.
Key Concepts
• Each angle of an equilateral triangle measures 60˚
• Conversely, if a triangle has equal angles, it is equilateral.
Key Concepts
Key Concepts, continuedTheorem
If a triangle is equiangular, then it is equilateral.
Key Concepts
• These theorems and properties can be used to solve many triangle problems.
Key Concepts
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
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.
Use the distance formula to calculate the length of each side.
Calculate the length of .
Substitute A (–4, 5) and B (–1, –4) for (x1, y1) and (x2, y2).
Simplify.
Calculate the length of
Calculate the length of
Substitute (–1, –4) and (5, 2) for (x1, y1) and (x2, y2).
Simplify.
Calculate the length of .
Substitute (–4, 5) and (5, 2) for (x1, y1) and (x2, y2).
Simplify.
Determine if the triangle is isosceles.
A triangle with at least two congruent sides is an isosceles triangle.
, so is isosceles.
Identify congruent angles. If two sides of a triangle are congruent, then the angles opposite the sides are congruent.
✔
Geogebra is a graphing program that can be used to illustrate the properties of isosceles triangles.
Find the values of x and y.
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.
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.
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.
The value of x is 9.
Equation
Solve for x.
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.
The value of y is 13.
Equation
Simplify.
Solve for y.
✔
Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite to these angles are congruent.
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,
p. 361# 4-20 (mo4)
HOMEWORK
C. N. ColόnGeometry
Ch 9-7Proving Right Triangles Congruent by Hypotenuse Leg(HL)
Reference: SIMON PEREZ.
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!
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
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
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
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
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
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
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
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
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
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
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!
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!
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!
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!
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
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
p. 366 # 4, 12 and 14
HOMEWORK