1. Session 5 Warm-up

2. Session 5 Daily Check

3. CCGPS Analytic Geometry Day 5 (8-13-13)
UNIT QUESTION: How do I prove geometric theorems involving lines, angles, triangles and parallelograms?
Standards: MCC9-12.G.SRT.1-5, MCC9-12.A.CO.6-13
Today’s Question:
If the legs of an isosceles triangle are congruent, what do we know about the angles opposite them?
Standard: MCC9-12.G.CO.10

4. 4.1 Triangles & Angles
August 13, 2013

5. 4.1 Classifying Triangles
Triangle – A figure formed when three noncollinear points are connected by segments.
The sides are DE, EF, and DF.
The vertices are D, E, and F.
The angles are D,  E,  F.
Angle E
Side
Vertex F D

6. Triangles Classified by Angles
Acute
Obtuse
Right
17º
50º
120º
60º
30°
70º
43º
60º
All acute angles
One obtuse angle
One right angle

7. Triangles Classified by Sides
Isosceles
Equilateral
Scalene
no sides
congruent
all sides
congruent
at least two
sides congruent

8. Classify each triangle by its angles and by its sides.
60° A B C
45° E F G

9. Opposite Side- opposite the vertex ex. DF is opposite ∠ E.
Adjacent Sides- share a vertex ex. The sides DE & EF are adjacent to ∠E.
Opposite Side- opposite the vertex ex. DF is opposite ∠ E. E F D

10. Parts of Isosceles Triangles
The angle formed by the congruent sides is called the vertex angle.
The two angles formed by the base and one of the congruent sides are called base angles.
The congruent sides are called legs.
leg
leg
base angle
base angle
The side opposite the vertex is the base.

11. Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
If , then

12. Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
If , then

13. Apply the Base Angles Theorem
EXAMPLE 1
Apply the Base Angles Theorem
Find the measures of the angles. Q P
(30)° R

14. Apply the Base Angles Theorem
EXAMPLE 1
Apply the Base Angles Theorem
Find the measures of the angles.
SOLUTION Q P
Since a triangle has 180°, 180 – 30 = 150° for the other two angles.
Since the opposite sides are congruent, angles Q and P must be congruent.
150/2 = 75° each.
(30)° R

15. Apply the Base Angles Theorem
EXAMPLE 2
Apply the Base Angles Theorem
Find the measures of the angles. Q P
(48)° R

16. Apply the Base Angles Theorem
EXAMPLE 3
Apply the Base Angles Theorem
Find the measures of the angles. Q P
(62)° R

17. EXAMPLE 4
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle. P
(12x+20)°
(20x-4)° Q R

18. Apply the Base Angles Theorem
EXAMPLE 4
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle. P
SOLUTION
(12x+20)°
Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4
20 = 8x – 4
24 = 8x
3 = x
(20x-4)° Q R
Plugging back in,
And since there must be 180 degrees in the triangle,

19. Apply the Base Angles Theorem
EXAMPLE 5
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle. Q P
(11x+8)°
(5x+50)° R

20. Apply the Base Angles Theorem
EXAMPLE 6
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
SOLUTION Q P
(80)°
(80)°
Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40
4x = 40
x = 10
3x+40 7x
Plugging back in,
QR = 7(10)= 70
PR = 3(10) + 40 = 70 R

21. Apply the Base Angles Theorem
EXAMPLE 7
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides. P
(50)°
5x+3
(50)° R Q
10x – 2

22. Right Triangles
HYPOTENUSE
LEG
LEG

23. Interior Angles
Exterior Angles

24. x + y + z = 180° Triangle Sum Theorem
The measures of the three interior angles in a triangle add up to be 180º.
x + y + z = 180° x° y° z°

25. m T = 59º m R + m S + m T = 180º 54º + 67º + m T = 180º
54°
54º º + m T = 180º
121º + m T = 180º
67° S T
m T = 59º

26. y = 40º m  D + m DCE + m E = 180º 55º + 85º + y = 180ºB
55º + 85º + y = 180º y°
140º + y = 180º C x°
85°
y = 40º
55° D A

27. Find the value of each variable.x°
43° x°
57°
x = 50º

28. Find the value of each variable.
55°
(6x – 7)°
43°
(40 + y)°
28°
x = 22º
y = 57º

29. Find the value of each variable.
50°
53° x°
50°
62°
x = 65º

30. Exterior Angle Theorem
The measure of the exterior angle is equal to the sum of two nonadjacent interior angles 1
m1+m2 =m3 2 3

31. Ex. 1: Find x. B. A. 72 43
148 76 x x 38 81

32. Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are
complementary.
x + y = 90º x° y°

33. Find mA and mB in the right triangle ABC.
mA + m B = 90
2x x = 90
2x°
5x = 90
x = 18
3x° C B
mA = 2x
mB = 3x
= 2(18)
= 3(18)
= 54
= 36

34. Homework:
Practice WS