1. Isosceles, Equilateral, Right Triangles

2. I. Isosceles Triangle (at least two sides are congruent)
vertex 
Legs: the two congruent sides of the triangle.
Base: the remaining side of the triangle.
leg
leg
Base Angles: the angles at the base.
base ’s
Vertex Angle: the angle opposite the base.
base

3. Base Angles Theorem
If 2 sides of a triangle are congruent, then the angles opposite them are congruent. A
If AB  AC, then B  C B C

4. Ex. 1 Find the measure of F and G.
m F = 65o base angles thm
65o H

5. Converse of Base Angles Theorem
If 2 angles of a triangle are congruent, then the sides opposite them are congruent. A
If B  C , then AB  AC B C

6. Ex. 2 Find the value of x. 4x + 4 = 32 converse of base angles thmQ 32 S

7. ex. 3. The vertex of an isosceles triangle is 60o
ex. 3 The vertex of an isosceles triangle is 60o. What is the measure of the base angles. A
mB = mC = x
Base angles thm
60o
x + x + 60 = 180
∆ sum thm
2x =
2x = 120
x = _120_ 2 x x C B
x = 60
mB = mC = 60o

8. Ex. 4 Find the mB. A
68o xo C B

9. II. Equilateral Triangle (all sides are congruent)
All equilateral triangles are equiangular and all equiangular triangles are equilateral. All 3 sides are congruent and all 3 angles are congruent.

10. Ex. 3 Find the length of each side of the equiangular triangle.
6x + 3 = 7x – 1 equiangular equilateral triangle
6x – 7x = - 1 – 3
-x = -4
x= 4
Determine the length of the sides by substituting x = 4 into the equation.
7(4) – 1
=28 – 1
= 27 units

11. Open your Textbooks Pg 188 #8-14 (even) Pg 189 #18-24 (even)

12. Pythagorean Theorem and the Distance Formula

13. Hypotenuse
(opposite the right angle)
Side c a
Side b
Pythagorean Theorem is written as when a and b represent the lengths of the sides and c represents the length of the hypotenuse.
c2 = a2 + b2

14. Pythagorean Theorem
(hypotenuse)2 = (side)2 + (side)2
c2 = a2 + b2

15. c2 = a2 + b2 c2 = 82 + 62 c2 = 64 + 36 c2 = 100 c = 100 c = 10 cm
Ex. 1 Use Pythagorean Theorem to find the missing side.
8 cm
c2 = a2 + b2
c2 =
6 cm c
c2 =
c2 = 100
c = 100
c = 10 cm

16. Ex. 2 Find the unknown side.
c2 = a2 + b2
132 = x2 + 32
169 = x2 + 9
169 – 9 = x2
160 = x2
x= 160 x
13 cm
3 cm

17. Ex. 3. Find the length of the hypotenuse of the right triangle
Ex. 3 Find the length of the hypotenuse of the right triangle. Do the lengths of the sides of the triangle form a Pythagorean triple?  
c2 = a2 + b2
x2 = x
x2 =
21cm
x2 = 841
x = 841
x = 29 cm
20cm
20, 21, and 29 form a Pythagorean triple, because they are integers that satisfy c2 = a2 + b2

18. Open your Textbooks
Pg. 195 #2, 4, 10, 14
Pg 197 #35, 36

19. Converse of Pythagorean Theorem

20. If c2 < a2 + b2, then ΔABC is an acute triangle.
We can use the Converse of the Pythagorean theorem to determine whether a triangle is an acute, obtuse, and right triangle.
If c2 < a2 + b2, then ΔABC is an acute triangle.
If c2 = a2 + b2, then ΔABC is a right triangle.
If c2 > a2 + b2, then ΔABC is an obtuse triangle.
*Note: c represents the largest side measure. a c b

21. ∆WXY is an acute triangle
Ex1. Classify the triangles as acute, obtuse, or right.
Do the Pythagorean Theorem
c2 = a2 + b2
112 =
121 =
121 < 130 (121 is less than 130)
∆WXY is an acute triangle X
7cm 11cm
9cm W Y

22. ∆ABC is a right triangle
Ex 2. Classify the triangles as acute, obtuse, or right.
c2 = a2 + b2
372 =
1369 =
1369 = 1369
∆ABC is a right triangle A 12 37 c 35 B

23. Ex 3. Classify the triangles as acute, obtuse, or right.
c2 = a2 + b2
352 =
1225 =
1225 > 1044
∆WXY is an obtuse triangle
12cm cm
30cm

24. Distance Formula: the distance d between two points A(x1, y1) and B(x2, y2) is
Ex. 4 Find the distance between T(5, 2) and R(-4, -1) to the nearest tenth (one decimal place).
d = √90

25. Class work
Pg. 195 #5 Pg. 197 #28 Pg. 203 #2, 6, 14, Pg 204 # 18-20, 25