Download Section 4.6 – Isosceles, Equilateral, and Right Triangles

Section 4.6 – Isosceles, Equilateral, and Right Triangles
Exercise: Use a piece of patty paper and a ruler to construct an isosceles triangle (half of
the class try obtuse isosceles and the other half try acute isosceles). Use a protractor to
measure all of the angles. What is your conclusion?
Base Angles (of an Isosceles Triangle):
Vertex Angle (of an Isosceles Triangle):
Base Angles Theorem:
Converse of the Base Angles Theorem:
Examples:
1. Use the diagram of ABC to prove the Base Angles Theorem.
Given: AB  AC , AD bisects BAC
Prove: B  C
Statements
Reasons
1. AB  AC , AD bisects BAC
1.
2.
2. Definition of Angle Bisector
3.
3. Reflexive Property
4.
4. SAS
5. B  C
5.
Using the Base Angles Theorem, we can draw in several congruence marks in the diagram
on the left. Using its converse, we can draw in several congruence marks in the diagram on
the right,
Sometimes we will talk about COROLLARIES. These are statements that can be easily
proven using a theorem or a definition.
Corollary to Base Angles Theorem:
If a triangle is equilateral, then it is equiangular.
Corollary to Converse of Base Angles Theorem:
If a triangle is equiangular, then it is equilateral.
Examples: Find the value of x and y.
1.
2.
So far, we have learned four ways to prove triangles are congruent. These are
________________________________________________________________________________________________________.
There is one more: Hypotenuse-Leg (HL).
Hypotenuse-Leg (HL) Congruence Theorem:
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse
and the leg of a second right triangle, then the two triangles are congruent.
(The proof of this is on p. 833 if you want to see! It is pretty tough )
Examples: Is it possible to say the following triangles are congruent? Use any triangle
congruence we have learned.
1.
2.
3.
4.