Review Begin at the word Tomorrow Every Time

Review: 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. homecoming! 2. Move to the alternate interior angle. 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.

Triangles & Angle Theorems Days 107 & 108 Learning Target: Students can prove geometric theorems involving lines, angles, triangles and parallelograms.

4. 1 Classifying Triangles Triangle – A figure formed when three noncollinear points are connected by segments. The sides are DE, EF, and DF. E Angle Side The vertices are D, E, and F. The angles are D, E, F. Vertex D F

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

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

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

Fill in the table Acute Scalene Isosceles Equilateral Obtuse Right

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 D F

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. leg The congruent sides are called legs. base angle The side opposite the vertex is the base.

Base Angles Theorem (4. 9) If two sides of a triangle are congruent, then the angles opposite them are congruent. If , then

Converse of Base Angles Theorem (4. 10) If two angles of a triangle are congruent, then the sides opposite them are congruent. If , then

EXAMPLE 1 Apply the Base Angles Theorem Find the measures of the angles. Q SOLUTION 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

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

EXAMPLE 4 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. P )° 0 2 + (12 x Q (20 x-4)° Plugging back in, SOLUTION R Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12 x + 20 = 20 x – 4 20 = 8 x – 4 24 = 8 x 3=x And since there must be 180 degrees in the triangle,

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

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

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

HYPOTENUSE LEG

Interior Angles Exterior Angles

Triangle Sum Theorem (4. 1) The measures of the three interior angles in a triangle add up to be 180º. x° y° x + y + z = 180° z°

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

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

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

Find the value of each variable. (6 x – 7 55° 43° )° (4 0 + y) ° 28° x = 22º y = 57º

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

Exterior Angle Theorem (4. 3) 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. 76 81 72 43 x 38 x 148

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. A m A + m B = 90 2 x° 2 x 3 x° C m A = 2 x = 2(18) = 36 + 3 x = 90 5 x = 90 x = 18 B m B = 3 x = 3(18) = 54