session 5 warm-up begin at the word “tomorrow”. every time you move, write down the word(s) upon...
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Session 5 Warm-upBegin at the word
“Tomorrow”. Every Time you move, write down the
word(s) upon which you land.
Tomorrow
it
is
homecoming!
because
spirit
your
show
1. Move to the consecutive interior angle.2. Move to the alternate interior angle.3. Move to the corresponding angle.4. Move to the alternate exterior.5. Move to the exterior linear pair.
7. Move to the vertical angle.6. Move to the alternate exterior angle.
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Session 5 Daily Check
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CCGPS Analytic GeometryDay 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
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4.1 Triangles & Angles4.1 Triangles & Angles
August 11, 2014August 11, 2014
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4.1 Classifying Triangles
Triangle – A figure formed when three noncollinear points are connected by segments.
E
DF
Angle
SideVertex
The sides are DE, EF, and DF.The vertices are D, E, and F.The angles are D, E, F.
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Triangles Classified by AnglesAcute Obtuse Right
60º
50º
70º
All acute anglesOne obtuse angle
One right angle
120º
43º
17º
30°
60º
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Triangles Classified by Sides
Scalene Isosceles Equilateral
no sidescongruent at least two
sides congruent
all sidescongruent
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Classify each triangle by its angles and by its sides.
60°
60° 60°A B
C
45°
45°
E
F G
EFG is a right
isosceles triangle.
ABC is an acute
equilateral triangle
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Fill in the tableAcute Obtuse Right
Scalene
Isosceles
Equilateral
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Try These:1. ABC has angles that
measure 110, 50, and 20. Classify the triangle by its angles.
2. RST has sides that measure 3 feet, 4 feet, and 5 feet. Classify the triangle by its sides.
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Adjacent Sides- share a vertex ex. The sides DE & EF are adjacent to <E.
E
D F
Opposite Side- opposite the vertex ex. DF is opposite .<E.
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Parts of Isosceles Triangles The angle formed by the congruent sides is called the vertex angle.
leg leg The congruent sides are called legs.
The side opposite the vertex is the base.
base anglebase angle
The two angles formed by the base and one of the congruent sides are called base angles.
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Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
If , thenACAB CB
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Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
If , thenCB ACAB
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EXAMPLE 1 Apply the Base Angles Theorem
P
R
Q
(30)°
Find the measures of the angles.SOLUTION
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.
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EXAMPLE 2 Apply the Base Angles Theorem
P
R
Q(48)°
Find the measures of the angles.
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EXAMPLE 3 Apply the Base Angles Theorem
P
R
Q(62)°
Find the measures of the angles.
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EXAMPLE 4 Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
RQ(20x-4)°
(12x+20)° SOLUTION
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
Plugging back in,
And since there must be 180 degrees in the triangle,
564)3(20
5620)3(12
Rm
Pm
685656180Qm
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EXAMPLE 5 Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
R
Q(11x+8)° (5x+50)°
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EXAMPLE 6 Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
R
Q(80)° (80)°
SOLUTION
Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40
4x = 40
x = 10
7x 3x+40
Plugging back in,
QR = 7(10)= 70PR = 3(10) + 40 = 70
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EXAMPLE 7 Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
RQ
(50)°
(50)°
10x – 2
5x+3
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LEG
LEG
HYPOTENUSE
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Interior Angles Exterior Angles
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Triangle Sum TheoremTriangle Sum TheoremThe measures of the three interior angles
in a triangle add up to be 180º.
x°
y° z°
x + y + z = 180°
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54°
67°
R
S T
m R + m S + m T = 180º 54º + 67º + m T = 180º
121º + m T = 180º
m T = 59º
Find in RST.m T
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85° x°55°
y°
A
B
C
D
E m D + m DCE + m E = 180º55º + 85º + y = 180º
140º + y = 180º
y = 40º
Find the value of each variable in DCE
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Find the value of each variable.
x = 50º
x°
x° 43°
57°
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Find the value of each variable.
x = 22º
(6x – 7)°43°55°
28°
(40 + y)°
y = 57º
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Find the value of each variable.
x = 65º
62°
50°
50°
53°
x°
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The measure of the exterior angle is equal to the sum of two nonadjacent interior angles
1
2 3
m1+m2 =m3
Exterior Angle TheoremExterior Angle Theorem
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x
43
3881
148
72
x76
Ex. 1: Find x.
A. B.
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Corollary to the Triangle Sum TheoremCorollary to the Triangle Sum Theorem
The acute angles of a right triangle arecomplementary.
x°
y°
x + y = 90º
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Find mA and mB in right triangle ABC.
A
BC
2x°
3x°
mA + m B = 90
2x + 3x = 90
5x = 90x = 18
mA = 2x
= 2(18)
= 36
mB = 3x
= 3(18)
= 54
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