TRIANGLE SUM PROPERTIES 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° StatementsReasons 1.Draw BD parallel to AC 2.M<4 + m<2 + m<5 1.<1 c= <4, <3 c= <5 2.m<1 = m<4, m<3 = m<5 3.m<1 + m<2 + m<3 = 180° 1.Parallel Postulate 2.Addition Angle Postulate and def of a straight < 3.Alternate Interior Angles 4.Definition of congruent angles 5.Substitution property a b c

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 FIGURES DEFG 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 THEOREM Reflexive 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