1. Move to the consecutive interior angle. Warm-up Begin at the word “Tomorrow”. Every Time you move, write down the word(s) upon which you land. is show spirit 1. Move to the consecutive interior angle. 2. Move to the alternate interior angle. homecoming! Tomorrow 3. Move to the corresponding angle. 4. Move to the alternate exterior. 5. Move to the exterior linear pair. because your 6. Move to the alternate exterior angle. it 7. Move to the vertical angle.

4.1 Triangles & Angles January 22nd -23rd

4.1 Classifying Triangles Triangle – A figure formed when three noncollinear points are connected by segments. The sides are DE, EF, and DF. The vertices are D, E, and F. The angles are D,  E,  F. Angle E Side Vertex F D

Triangles Classified by Angles Acute Obtuse Right 17º 50º 120º 60º 30° 70º 43º 60º All acute angles One obtuse angle One right angle

Triangles Classified by Sides Isosceles Equilateral Scalene no sides congruent all sides congruent at least two sides congruent

Classify each triangle by its angles and by its sides. 60° A B C 45° E F G

Try These:  ABC has angles that measure 110, 50, and 20. Classify the triangle by its angles.  RST has sides that measure 3 feet, 4 feet, and 5 feet. Classify the triangle by its sides.

Opposite Side- opposite the vertex ex. DF is opposite E. Adjacent Sides- share a vertex ex. The sides DE & EF are adjacent to E. Opposite Side- opposite the vertex ex. DF is opposite E. E F D

Parts of Isosceles Triangles The angle formed by the congruent sides is called the vertex angle. The two angles formed by the base and one of the congruent sides are called base angles. The congruent sides are called legs. leg leg base angle base angle The side opposite the vertex is the base.

Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. If , then

Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent. If , then

Apply the Base Angles Theorem EXAMPLE 1 Apply the Base Angles Theorem Find the measures of the angles. SOLUTION Q P Since a triangle has 180°, 180 – 30 = 150° for the other two angles. Since the opposite sides are congruent, angles Q and P must be congruent. 150/2 = 75° each. (30)° R

Apply the Base Angles Theorem EXAMPLE 2 Apply the Base Angles Theorem Find the measures of the angles. Q P (48)° R

Apply the Base Angles Theorem EXAMPLE 3 Apply the Base Angles Theorem Find the measures of the angles. Q P (62)° R

EXAMPLE 4 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. P SOLUTION (12x+20)° Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4 20 = 8x – 4 24 = 8x 3 = x (20x-4)° Q R Plugging back in, And since there must be 180 degrees in the triangle,

Apply the Base Angles Theorem EXAMPLE 5 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. Q P (11x+8)° (5x+50)° R

Apply the Base Angles Theorem EXAMPLE 6 Apply the Base Angles Theorem Find the value of x. Then find the length of the labeled sides. SOLUTION Q P (80)° (80)° Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40 4x = 40 x = 10 3x+40 7x Plugging back in, QR = 7(10)= 70 PR = 3(10) + 40 = 70 R

Apply the Base Angles Theorem EXAMPLE 7 Apply the Base Angles Theorem Find the value of x. Then find the length of the labeled sides. P (50)° 5x+3 (50)° R Q 10x – 2

Right Triangles HYPOTENUSE LEG LEG

Interior Angles Exterior Angles

x + y + z = 180° Triangle Sum Theorem The measures of the three interior angles in a triangle add up to be 180º. x + y + z = 180° x° y° z°

m T = 59º m R + m S + m T = 180º 54º + 67º + m T = 180º 54° 54º + 67º + m T = 180º 121º + m T = 180º 67° S T m T = 59º

y = 40º m  D + m DCE + m E = 180º 55º + 85º + y = 180º B 55º + 85º + y = 180º y° 140º + y = 180º C x° 85° y = 40º 55° D A

Find the value of each variable. x° 43° x° 57° x = 50º

Find the value of each variable. 55° (6x – 7)° 43° (40 + y)° 28° x = 22º y = 57º

Find the value of each variable. 50° 53° x° 50° 62° x = 65º

Exterior Angle Theorem The measure of the exterior angle is equal to the sum of two nonadjacent interior angles 1 m1+m2 =m3 2 3

Ex. 1: Find x. B. A. 72 43 148 76 x x 38 81

Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. x + y = 90º x° y°

Find mA and mB in right triangle ABC. mA + m B = 90 2x + 3x = 90 2x° 5x = 90 x = 18 3x° C B mA = 2x mB = 3x = 2(18) = 3(18) = 54 = 36

Homework: Practice WS