TRIA
NGLE S
UM
PROPE
RTIE
S
4. 1
TO CLARIFY*******
A triangle is a polygon with three sides.
A triangle with vertices A, B, and C is called triangle ABC
TRIANGLES ON A PLANE
We can find the side lengths
√(-1 – 0)2 + (2 – 0)2 = √5 ≈ 2.2
√(6 – 0)2 + (3 – 0)2 = √45 ≈ 6.7
√(6 – -1)2 + (3 – 2)2 = √50 ≈ 7.1
This is a scalene triangle
We can also determine if it is a right triangle. (hint, look for perpendicular angles)
Slope of OP = (2-0)/(-1-0) = -2
Slope of OQ = (3-0)/(6-0) = ½
The lines are perpendicular and form a right angle so this is a right scalene triangle
TRY IT OUT
Triangle ABC has the vertices A(0,0), B(3,3) and C (-3,3). Classify it by its sides. Then determine if it is a right triangle.
EXTENDING SIDES
When you extend the sides of a polygon there are new angles formed.
The original angles (on the inside) are called interior* angles.
The new angles formed are called exterior* angles.
TRIANGLE SUM THEOREM
4.1: The sum of the measures of the interior angles of a triangle is 180°
PROVE IT
Given: Triangle ABC
Prove: m<1 + m<2 + m<3 = 180°
Statements Reasons
1. Draw BD parallel to AC
2. M<4 + m<2 + m<5
3. <1 c= <4, <3 c= <54. m<1 = m<4, m<3 =
m<5
5. m<1 + m<2 + m<3 = 180°
1. Parallel Postulate2. Addition Angle
Postulate and def of a straight <
3. Alternate Interior Angles
4. Definition of congruent angles
5. Substitution property
a
b
c1
2
3
4 5
EXTERIOR ANGLE THEOREM
4.2: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles
APPLY THEOREM 4.2
Find m<JKM
Step 1: Write an equation
Step 2: Plug in x
(2x – 5) = 70 + x
2(75) -5 = 145
COROLLARY*
A corollary to a theorem is a statement that can be proved easily by using the theorem.
Corollary to the triangle sum theorem: The acute angles of a right triangle are complementary
APPLY
CONGRUENCE
4. 2
CONGRUENT FIGURES
Two figures are congruent if they have exactly the same size and shape.
All of the parts of one figure are congruent to the corresponding parts* of the other figure.
USE PROPERTIES OF CONGRUENT FIGURESDEFG c= SPQR
Find x
Find y
8
10
THIRD ANGLES THEOREM
4.3: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
PROPERTIES OF CONGRUENT TRIANGLES THEOREMReflexive property: ABC is congruent to ABC
Symmetric property: if ABC is congruent to DEF then DEF is congruent to ABC
Transitive Property: If ABC is congruent to DEF and DEF is congruent to JKL,
then ABC is congruent to JKL
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