Download Isosceles, Equilateral and Right Triangles

Isosceles, Equilateral and Right
Triangles
November 7, 2012
By the end of this lesson,
you will be able to:
• Use properties of isosceles and equilateral
triangles to show angle measures, side lengths
and prove triangle congruence.
• Use properties of right triangles to show angle
measures, side lengths and prove triangle
congruence.
• Prove triangle congruence using the
Hypotenuse-Leg (HL) Theorem
Draw and label an ISOSCELES
triangle with the following:
•
•
•
•
•
Vertices A, B & C
Legs
Base
Base Angles
Vertex Angle
• Base Angles Theorem:
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
If 𝐴𝐵 ≅ 𝐴𝐶, then 𝐵 ≅ 𝐶
• Base Angles Theorem:
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
• Converse of the Base
Angles Theorem:
If two angles of a triangle are
congruent, then the sides
opposite them are congruent.
If 𝐴𝐵 ≅ 𝐴𝐶, then 𝐵 ≅ 𝐶
If 𝐵 ≅ 𝐶, then 𝐴𝐵 ≅ 𝐴𝐶
• Base Angles Theorem:
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
• Converse of the Base
Angles Theorem:
If two angles of a triangle are
congruent, then the sides
opposite them are congruent.
If 𝐴𝐵 ≅ 𝐴𝐶, then 𝐵 ≅ 𝐶
If 𝐵 ≅ 𝐶, then 𝐴𝐵 ≅ 𝐴𝐶
Both of these statements are TRUE, so we can write this in the form of a
BICONDITIONAL STATEMENT.
Both of these statements are TRUE, so we can write this in the form of a
BICONDITIONAL STATEMENT.
Two angles of a triangle are congruent
if and only if the sides opposite
them are congruent.
An EQUILATERAL triangle is a
special type of triangle.
An EQUILATERAL triangle is a
special type of triangle.
If a triangle is
EQUIANGULAR, then it is
EQUILATERAL.
An EQUILATERAL triangle is a
special type of triangle.
If a triangle is
EQUIANGULAR, then it is
EQUILATERAL.
If a triangle is
EQUILATERAL, then it is
EQUIANGULAR.
So far we have learned about four ways to prove that triangles are congruent.
SSS
(
)
Congruence Postulate
SAS
(
)
Congruence Postulate
(
)
Congruence Postulate
(
)
Congruence Postulate
ASA
AAS
There is ONE MORE WAY to prove triangle congruence. This way is
SPECIAL because it can only be used when
working with
RIGHT
triangles.
There is ONE MORE WAY to prove triangle congruence. This way is
SPECIAL because it can only be used when
working with
RIGHT
triangles.
HYPOTENUSE-LEG (HL) CONGRUENCE THEOREM: If the
hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of a second right
triangle, then the two triangles are congruent.
If 𝐵𝐶 ≅ 𝐸𝐹 and 𝐴𝐶 ≅ 𝐷𝐹, then ∆ABC ≅ ∆DEF