Triangles and Angles Ch 4 Lesson 1 Triangles

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 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 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 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 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 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 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 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 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 Substitution prop.

Theorem 4. 2 Exterior Angle Theorem • The measure of the exterior angle is equal to the sum of the two nonadjacent interior angles. • m<1= m<A +m<B

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = x + 55 -x -x x = 55

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = x + 55 -x -x x = 55

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = x + 55 -x -x x = 55

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = x + 55 -x -x x = 55

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = x + 55 -x -x x = 55

Example #2 • Find x • 2 x+10 = x + 65 -10 2 x = 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