Download 4.1 Triangles and Angles

Triangles and Angles
Ch 4
Lesson 1
Triangles Classified: done based
on two things
• They can be
classified by angles.
• Acute (less than 90)
• Right (equal to 90)
• Obtuse (greater than
90)
• Equiangular (all
angles are congruent)
• They can be
classified by sides
• Equilateral (all sides
are equal)
• Isosceles (two sides
are equal)
• Scalene (no sides are
equal)
Parts of a Triangle
• All triangles have three legs. The longest
leg is called the hypotenuse.
Parts of a triangle
• Sides
• Angles
– Point A is a vertex
– Interior angles
– Exterior angles
–
are adjacent
sides
Types of Triangles
• Acute Scalene
– No equal sides
– No equal angles
– All angles less than 90
• Right Isosceles
– Two sides are equal
– One right angle
Theorem 4.1 (Triangle Sum)
• The sum of the measure of the interior
angles of a triangle is 180 degrees.
• x + y + z = 180
Example #1
• Prove that the sum of angles inside a
triangle is 180 degrees
• Given= Triangle ABC
• Prove= m<1 + m<2 + m<3 = 180 degrees
• Draw a line BD parallel to AC then proof
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Solution:
Two column proof
• Statement
–
–
–
–
–
–
–
Draw BD II AC
m<4+m<2+m<5 = 180
<4≅<1
m<4≅m<1
<5≅<3
m<5=m<3
m<1+m<2+m<3=180
• Reason
–
–
–
–
–
–
–
Parallel lines
Def of a straight line
Interior alternate <s
Def of Congruent <s
Interior alternate <s
Def of Congruent <s
Substitution prop.
Theorem 4.2 Exterior Angle
Theorem
• The measure of the exterior angle is equal
is equal to the sum of the two nonadjacent
interior angles.
• m<1= m<A +m<B
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Example #2
• Find x
• 2x+10 = x + 65
2x+10 = x + 65
-10
-10
2x
= x + 55
2x
= x + 55
-x
-x
x = 55
Corollary
• It is a statement that can be easily proven
by using a theorem
• The two acute angles of the right angle
triangle are complementary (=90)
• m<A + m<B = 90