TRIA

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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