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

Session 5 Daily Check

CCGPS Analytic Geometry Day 5 ( ) UNIT QUESTION: How do I prove geometric theorems involving lines, angles, triangles and parallelograms? Standards: MCC9-12.G.SRT.1-5, MCC9-12.A.CO.6-13 Today’s Question: If the legs of an isosceles triangle are congruent, what do we know about the angles opposite them? Standard: MCC9-12.G.CO.10

4.1 Triangles & Angles August 11, 2014

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

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

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

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

Fill in the table AcuteObtuseRight Scalene Isosceles Equilateral

Try These: 1.  ABC has angles that measure 110, 50, and 20. Classify the triangle by its angles. 2.  RST has sides that measure 3 feet, 4 feet, and 5 feet. Classify the triangle by its sides.

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

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

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

EXAMPLE 1 Apply the Base Angles Theorem P R Q (30)° Find the measures of the angles. SOLUTION 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.

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

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

EXAMPLE 4 Apply the Base Angles Theorem Find the value of x. Then find the measure of each angle. P RQ (20x-4)° (12x+20)° SOLUTION 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 Plugging back in, 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. P R Q (11x+8)°(5x+50)°

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

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

LEG HYPOTENUSE

Interior AnglesExterior Angles

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

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

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

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

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

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

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

x x76 Ex. 1: Find x. A. B.

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

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