geometry section 4-6 1112

Section 4-6Isosceles and Equilateral Triangles

Wednesday, February 8, 2012

Essential Questions

❖ How do you use properties of isosceles triangles?

❖ How do you use properties of equilateral triangles?


Vocabulary1. Legs of an Isosceles Triangle:

2. Vertex Angle:

3. Base Angles:


Vocabulary1. Legs of an Isosceles Triangle: The two congruent sides

of an isosceles triangle

2. Vertex Angle:

3. Base Angles:




2. Vertex Angle: The included angle between the legs of an isosceles triangle

3. Base Angles:




2. Vertex Angle: The included angle between the legs of an isosceles triangle

3. Base Angles: The angles formed between each leg and the base of an isosceles triangle


Theorems and CorollariesTheorem 4.10 - Isosceles Triangle Theorem:

Theorem 4.11 - Converse of Isosceles Triangle Theorem:

Corollary 4.3 - Equilateral Triangles:



Theorems and CorollariesTheorem 4.10 - Isosceles Triangle Theorem: If two sides

of a triangle are congruent, then the angles opposite those sides are congruent

Theorem 4.11 - Converse of Isosceles Triangle Theorem:






Theorem 4.11 - Converse of Isosceles Triangle Theorem:If two angles of a triangle are congruent, then the sides opposite those angles are congruent.







Corollary 4.3 - Equilateral Triangles: A triangle is equilateral IFF it is equiangular






Corollary 4.3 - Equilateral Triangles: A triangle is equilateral IFF it is equiangular

Corollary 4.4 - Equilateral Triangles: Each angle of an equilateral triangle measures 60°


Example 1a. Name two unmarked congruent angles.

b. Name two unmarked congruentsegments


Example 2Find each measure.

a.

b. PR



180 - 60a.

b. PR



180 - 60 = 120a.

b. PR



180 - 60 = 120 120 ÷ 2a.

b. PR



180 - 60 = 120 120 ÷ 2 = 60a.

b. PR



180 - 60 = 120 120 ÷ 2 = 60= 60°

a.

b. PR



180 - 60 = 120 120 ÷ 2 = 60= 60°

a.

b. PR

Since all three angles will be 60°, this is an equilateral triangle, so PR = 5 cm.


Example 3Find the value of each variable.



6y + 3 = 8y − 5



6y + 3 = 8y − 5− 6y − 6y



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?

4x − 8 = 60



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?

4x − 8 = 60+ 8 + 8



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?

4x − 8 = 60+ 8 + 84x = 68



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?

4x − 8 = 60+ 8 + 84x = 68

44



6y + 3 = 8y − 5− 6y − 6y

3 = 2y − 5+ 5 + 5

8 = 2y22

y = 4

4x − 8 = 4x − 8− 4x − 4x+ 8 + 8

0 = 0Now what?

4x − 8 = 60+ 8 + 84x = 68

44x = 17


Check Your Understanding

Check out p. 287 #1-8 and see if you have an idea of what to do with these problems


Problem Set


Problem Set

p. 287 #9-31 odd (skip 27), 47, 56, 61

“We have, I fear, confused power with greatness.”- Stewart L. Udall


geometry section 4-6 1112

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