Session 3Session 3 Daily CheckDaily Check1. Name the type of angles. (1 point each)a) b)

1 2

21

2. Solve for x. (4 points each)a)

b) and are complementary.

10x-104x+22

BA7 1A x 4 1B x

Homework ReviewHomework Review

CCGPS Analytic GeometryDay 3 (8-9-13)

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:Which angles are congruent to each other when parallel lines are cut by a transversal?Standard: MCC9-12.A.CO.9

Parallel Lines and Parallel Lines and TransversalsTransversals

Parallel Lines – Two lines are parallel if and only if they are in the same plane and do not intersect.

AB

CD

AB CD

Parallel Planes – Planes that do not intersect.

Skew Lines – two lines that are NOT in the same plane and do NOT intersect

Ex 1Ex 1: Name all the parts of the prism shown : Name all the parts of the prism shown below. Assume segments that look parallel are below. Assume segments that look parallel are parallel.parallel.

A

B C

D

EF

G

1. A plane parallel to plane AFE. Plane BGD

2. All segments that intersect GB.

AB, FG, DG, BC

3. All segments parallel to FE. GD, BC

4. All segments skew to ED. BG, FA, BC

Transversal – A line, line segment, or ray that intersects two or more lines at different points.

a

b

t Line t is a transversal.

Special AnglesSpecial Angles

12

34

56

7 8

Interior Angles – lie between the two lines (3, 4, 5, and 6)

Alternate Interior Angles – are on opposite sides of the transversal. (3 & 6 AND 4 and 5)

Consecutive Interior Angles – are on the same side of the transversal. (3 & 5 AND 4 & 6)

65

43

More Special AnglesMore Special Angles

21

7 8

Exterior Angles – lie outside the two lines (1, 2, 7, and 8)

Alternate Exterior Angles – are on opposite sides of the transversal (1& 8 AND 2 & 7)

Ex. 2Ex. 2: Identify each pair of angles as alternate : Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, alternate exterior, consecutive interior, or vertical.interior, or vertical.

1

2

3

45

67

8

a. 1 and 2

b. 6 and 7

c. 3 and 4

d. 3 and 8

Alt. Ext. Angles

Vertical Angles

Alt. Int. Angles

Consec. Int. Angles

Alternate Interior Angles Alternate Interior Angles TheoremTheorem

If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.

21

34 56

78

2 6

3 7

Alternate Exterior Angles Alternate Exterior Angles TheoremTheorem

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles are congruent.

21

34 56

78

1 5

4 8

Consecutive Interior Angles Consecutive Interior Angles TheoremTheorem

If two parallel lines are cut by a transversal, then each pair of consecutive interior angles are supplementary.

21

34 56

78

m2 + m3 = 180°

m6 + m7 = 180°

Ex. 3Ex. 3 In the figure, p In the figure, pq. If mq. If m5 = 28°, find 5 = 28°, find the measure of each angle.the measure of each angle.

213 4

5 67 8

a. m8 =

b. m1 =

c. m2 =

d. m3 =

e. m4 =

28°

28°

152°

152°

28°

p

q

Ex. 4Ex. 4 In the figure, s In the figure, st. Find the t. Find the mmCBG.CBG.

S

t

3x -5

4x -29

A

B

CD

E

F

G3x – 5 = 4x - 29-5 = x - 29

24 = x

Step 1: Solve for x.

Step 2: mCBG = mABE = 3x -5.3x-5 = 3(24) – 5 = 72-5 = 67°

Ex: 5Ex: 5 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

10 and 6

Ex: 6Ex: 6 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

13 and 14

Ex: 7Ex: 7 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

6 and 14

Ex: 8Ex: 8 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

5 and 1

Ex: 9Ex: 9 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

15 and 12

Ex: 10Ex: 10 Identify each pair of angles as: Identify each pair of angles as: alt. alt. interior, alt. exterior, consecutive interior, alt. exterior, consecutive interior, or vertical.interior, or vertical.

116

215

1110

129

138

67

314

45

16 and 2