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4.1: Apply Triangle Sum
Properties
Aim: To classify triangles and find
measures of their angles.
What is a triangle?
Classifying Triangles by Sides
Scalene Triangle:
– Has no congruent sides.
abc
Isosceles Triangle:
- Has at least two congruent sides.
ca
Equilateral Triangle:
- Has three congruent sides.
Classifying Triangles by Angles
Acute Triangle:
- Has three acute angles.
Right Triangle:
- Has one right angle.
Obtuse Triangle:
- Has one obtuse angle.
Equiangular Triangle:
- Has three congruent angles.
Graph Triangle ABC
 Graph
A(-5, 4), B(2, 6), and C(4, -1).
Classify the triangle by its sides using the
distance formula.
d
x2  x1    y2  y1 
2
AB  53
BC  53
AC  106
This is an Isosceles
2
d
A
•
x2  x1 2   y2  y1 2
B
•
•
C
How can we check if any of the angles
are right angles?
Using the coordinates A(-5, 4), B(2, 6), and C(4,
-1). Find the slope of each line.
2
AB 
7
7
BC  
2
5
AC 
9
What type of triangle is
this considered to be?
Right Isosceles Triangle
Mini-Activity
 Move
you.
 Draw
your desk to the person next to
1 straight line (use your ID
card for a straight edge).
 Take the Triangle give to you and
tear off each angle.
Theorems

Triangle Sum Theorem:
– The sum of the measures of the interior angles
of a triangle is 180°.

Exterior Angle Theorem:
– The measure of an exterior angle of a triangle
is equal to the sum of the measures of the two
nonadjacent interior angles.

Corollary to the Triangle Sum Theorem:
– The acute angles of a right triangle are
complementary.
4.2: Apply Congruence and Triangles
Date: 11/2
Aim: To identify congruent triangles.
Do Now:Find m JKM.
Find an angle measure
Find m JKM.
STEP 1 Write and solve an equation to find the value
of x.
(2x – 5)° = 70° + x°
x = 75
Apply the Exterior Angle Theorem.
Solve for x.
STEP 2 Substitute 75 for x in 2x – 5 to find m JKM.
2x – 5 = 2 75 – 5 = 145
ANSWER The measure of  JKM is 145°.
Find angle measures from a verbal description
Use the corollary to set up and solve an equation.
x° + 2x° = 90°
x = 30
Corollary to the Triangle Sum Theorem
Solve for x.
ANSWER
So, the measures of the acute angles are 30° and
2(30°) = 60° .
3. Find the measure of
1 in the diagram shown.
ANSWER The measure of  1 in the diagram is 65°.
Third Angle Theorem
 If
two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are
also congruent.
4.2: Apply Congruence and
Triangles
Date: 11/8/10
Aim: To identify congruent
triangles.
Do Now: Take out Homework.
4.3: Prove Triangles Congruent by
SSS
Date: 11/9/10
Aim: To use the side lengths to
prove triangles are congruent.
Postulate
 Side-Side-Side
(SSS) Congruence
Postulate
–If three sides of one triangle are
congruent to three sides of another
triangle, then the two triangles are
congruent.
4.4: Prove Triangles Congruent by
SAS and HL
Date: 11/10/10
Aim: To use sides and angle
lengths to prove congruence.
Do Now:
Side-Angle-Side Congruence Postulate
(SAS)
 If
two sides and the included angle of
one triangle are congruent to two
sides and the included angle of a
second triangle, then the two
triangles are congruent.
4.4: Prove Triangles Congruent by
SAS and HL
Date: 11/12/10
Aim: To use sides and angle
lengths to prove congruence.
Hypotenuse-Leg Congruence Theorem
(HL)
 If
the hypotenuse and a leg of a right
triangle are congruent to the
hypotenuse and a leg of a second
right triangle, then the two triangles
are congruent.
4.5: Prove Triangles Congruent by
ASA and AAS
Date: 11/15/10
Aim: To use two more methods to
prove congruence (ASA and AAS).
Angle-Side-Angle Congruence
Postulate (ASA)
 If
two angles and the included side of
one triangle are congruent to two
angles and the included side of a
second triangle, then the two
triangles are congruent.
Angle-Angle-Side Congruence
Theorem (AAS)
 If
two angles and a non-included side
of one triangle are congruent to the
two angles and the corresponding
non-included side of a second
triangle, then the two triangle are
congruent.
4.6: Use Congruent Triangles
Date: 12/1/09
Aim: To use congruent triangles to
prove corresponding parts
congruent.
4.6: Use Congruent Triangles
Date: 12/2/09
Aim: To use congruent triangles to
prove corresponding parts
congruent.
4.7: Use Isosceles and Equilateral
Triangles
Date: 12/3/09
Aim: To use theorems about
isosceles and equilateral triangles.
Base Angles Theorem
 If
two sides of a triangle are
congruent, then the angels opposite
them are congruent.
Converse of Base Angles Theorem
 If
two angles of a triangle are
congruent, then the sides opposite
them are congruent.
Corollaries
 Corollary
to the Base Angle Theorem:
– If a triangle is equilateral, then it is
equiangular.
 Corollary
to the Converse of the Base
Angles Theorem:
– If a triangle is equiangular, then it is
equilateral.