GEOMETRY R Unit 2: Angles and Parallel Lines

Day Classwork Homework

Friday 9/16

Unit 1 Test

Monday 9/19

Angle Relationships HW 2.1

Tuesday 9/20

Angle Relationships with Transversals HW 2.2

Wednesday 9/21

Proving Lines are Parallel and Constructing Parallel Lines

HW 2.3

Thursday 9/22

Interior and Exterior Angle Theorems

HW 2.4

Friday 9/23

More Unknown Angle Problems with Justifications

Unit 2 Quiz 1

HW 2.5

Monday 9/26

Unknown Angle Proofs

HW 2.6

Tuesday

9/27

Proofs with Auxiliary Lines

HW 2.7

Wednesday

9/28

Review

Unit 2 Quiz 2

Review Packet

Thursday 10/29

Review Study!

Friday 9/30

Unit 2 Test

Angles [1]

Vocabulary Term Example

Adjacent Angles - Two angles in the same plane that share a common vertex and a common ray, but share

no common interior points.

Linear Pair - Two adjacent angles with non-common

sides that are opposite rays.

Complementary Angles – Two angles with measures

that have a sum of 90 degrees.

Supplementary Angles – Two angles with measures

that have a sum of 180 degrees.

Angle Addition Postulate: If two adjacent angles

combine to form a whole angle, the measure of the whole angle equals the sum of the measures of the two

adjacent angles.

Supplement Theorem – Two angles that form a linear

pair are supplementary angles.

Examples: 1. Find the measure of ∠𝑎.

2. Find the measure of ∠𝑓.

𝐴

𝐵

3. Find the measure of ∠𝑐. 4. Find the value of x.

5. Two angles are complementary. One angle is six less than twice the other angle. Find both angles.

Vertical Angles - Two non-adjacent angles formed by

two intersecting lines.

Vertical Angles Theorem - Vertical angles are

congruent.

Examples:

1. If 21 4m x x and 2 36m x , find 3m .

1 2 3

y

2x + 1

3x + 10

y — 389

x + 10

2. Find the value of x and y. Find the values of the missing variables.

3.

4.

5.

3x + 6

x - 11x - 10y

6.

7.

8.

ANGLES AND PARALLEL LINES [2]

Vocabulary Examples Transversal - A line that intersects two or

more coplanar lines.

Corresponding Angles - Non-adjacent angles that lie on the same side of the transversal,

one exterior and one interior.

Alternate Interior Angles - Non-adjacent interior angles that lie on opposite sides of a

transversal.

Alternate Exterior Angles - Non-adjacent exterior angles that lie on opposite sides

of a transversal.

Theorems Words Diagram

Corresponding Angles Postulate

If 2 parallel lines are cut by a

transversal, then corresponding angles are congruent.

Alternate Interior Angles Theorem

If 2 parallel lines are cut by a

transversal, then alternate interior angles are congruent.

Consecutive Interior Angles Theorem

If 2 parallel lines are cut by a transversal, then consecutive

interior angles are supplementary

Alternate Exterior Angles Theorem

If 2 parallel lines are cut by a

transversal, then alternate exterior angles are congruent.

Finding Values of Variables The special relationships between the angles formed by two parallel lines and a transversal can be used to find unknown values.

1. If m∠4 = 2x – 17 and m∠1 = 85, find x.

2. Find y if m∠3 = 4y + 30 and m∠7 = 7y + 6.

3. If m∠1 = 4x + 7 and m∠5 = 5x – 13, find x.

4. Find y if m∠7 = 68 and m∠6 = 3y – 2.

An _________________________________is sometimes useful when solving for unknown angles. In this figure, we can use the auxiliary line to find the measure of ∠𝑊. What is the measure of ∠𝑊?

Exercises Find the unknown angles that are labeled. Justify each step.

1.

2.

3.

4.

5.

6.

7.

8.

9.

Find the unknown (labeled) angles. Give reasons for your solutions.

1.

2.

3.

4.

PROVING LINES ARE PARALLEL [3]

Examples

1. Given the following information, is it possible to prove that any of the lines shown are parallel? Justify your reasoning.

a. ∠2 ≅ ∠8 c. ∠1 ≅ ∠15

b. ∠3 ≅ ∠11 d. ∠8 ≅ ∠6 e. ∠12 ≅ ∠14

Words Diagram

Converse of the Corresponding Angles

Postulate

If 2 lines are cut by a transversal such that corresponding angles

are congruent, then the lines are parallel.

Converse of the

Alternate Interior Angles Theorem

If 2 lines are cut by a transversal

such that alternate interior angles are congruent, then the lines are

parallel.

Converse of the

Consecutive Interior Angles Theorem

If 2 lines are cut by a transversal

such that consecutive interior angles are supplementary, then

the lines are parallel.

Converse of the

Alternate Exterior Angles Theorem

If 2 lines are cut by a transversal

such that alternate exterior angles are congruent, then the lines are

parallel.

Converse of the Perpendicular

Transversal Theorem

If 2 lines are perpendicular to the

same line, then they are parallel to each other.

2. Find m∠MRQ so that a||b. Show your work.

3. Use the converse of the corresponding angles postulate to construct a line parallel to 𝐴𝐵 ⃡ , passing through point C.

4. Use the converse of the alternate interior angles theorem to construct a line parallel to 𝐴𝐵 ⃡ , passing through point C.

A B

C

A B

C

ANGLES OF TRIANGLES [4 & 5]

The sum of the interior angles of a triangle is 180 degrees.

The acute angles of a right triangle are complementary.

Each angle of an equilateral triangle measures 60 degrees.

The sum of the interior angles of a quadrilateral is 360 degrees.

Examples 1. The vertex angle of an isosceles triangle exceeds the measure of a base angle by

30. Find the measure of the vertex angle.

2. Classify triangle ABC based on its sides and angles if 𝑚∠A = (9x)°, 𝑚∠B=(3x-6)°, and 𝑚∠C = (11x + 2)°.

Proof of Exterior Angle Theorem:

Theorem Words Example

The Exterior Angle Theorem

An exterior angle of a

triangle equals the sum of its two remote interior

angles

Examples

1. Find x: 2. Find m a and m b in the figure to the right. Justify your

results. In each figure, determine the measures of the unknown (labeled) angles. Give reasons for each step of your calculations.

3.

4.

5.

2x

5x 12x

6.

7.

8.

9.

10.

11.

12.

13.

14.

UNKNOWN ANGLE PROOFS [6]

Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction is called inductive reasoning. These conclusions lack logical certainty. The system of reasoning that uses facts, rules, definitions, or properties to reach logical conclusions is called deductive reasoning.

1. Angles that form a linear pair are supplementary. Use this fact to prove that vertical angles are equal in measure.

2. Given the diagram at the right, prove that 𝑚∠𝑤 + 𝑚∠𝑥 + 𝑚∠𝑧 = 180°.

3. Given the diagram at the right, prove that 𝑚∠𝑤 = 𝑚∠𝑦 + 𝑚∠𝑧.

4. In the diagram at the right, prove that 𝑚∠𝑦 + 𝑚∠𝑧 = 𝑚∠𝑤 + 𝑚∠𝑥.

5. In the figure at the right, 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ ∥ 𝐷𝐸̅̅ ̅̅ .

Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐶𝐷𝐸.

6. In the figure at the right, prove that the sum of the angles

marked by arrows is 900°.

7. In the figure at the right, prove that 𝐷𝐶̅̅ ̅̅ ⊥ 𝐸𝐹̅̅ ̅̅ .

8. In the labeled figure on the right, prove that 𝑚 || 𝑛.

Z

a

b

d

c

9. In the diagram on the right, prove that the sum of the angles marked by arrows is 360°.

10. In the diagram at the right, prove that 𝑚∠𝑎 + 𝑚∠𝑑 − 𝑚∠𝑏 = 180.

MORE UNKNOWN ANGLE PROBLEMS [7]

A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. A theorem is a statement or conjecture that can be proven true by undefined terms,

definitions, and postulates.

1. In the figure on the right, 𝐴𝐵̅̅ ̅̅ ∥ 𝐷𝐸̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ ∥ 𝐸𝐹̅̅ ̅̅ . Prove that B E . (Hint: Extend 𝐵𝐶̅̅ ̅̅ and 𝐸𝐷̅̅ ̅̅ )

2. Given: 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ Prove: 𝑚∠𝑧 = 𝑚∠𝑥 + 𝑚∠𝑦

3. In the figure, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸. Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐶𝐷𝐸.

4. In the figure, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸. Prove that 𝑚∠𝐵 + 𝑚∠𝐷 = 180.

5. In the figure, prove that 𝑚∠𝐴𝐷𝐶 = 𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶.

6. Use the theorems involving parallel lines discussed to prove that the three angles of a triangle sum to 180°. For this proof, you will need to draw an auxiliary line, parallel to one of the triangle’s sides and passing through the vertex opposite that side. Add any necessary labels and write out your proof.

7. A theorem states that in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other of the two parallel lines. Prove this theorem. (a) Construct and label an appropriate figure, (b) state the given information and the theorem to be proved, then (c) list the necessary steps to demonstrate the proof.

8. In the figure, 𝐴𝐵 || 𝐷𝐸 and 𝐵𝐶 || 𝐸𝐹. Prove that ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹.

9. In the figure, 𝐴𝐵 || 𝐶𝐷.

Prove that 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐴 + 𝑚∠𝐶.

MORE EUCLIDEAN PROOFS

1. Given: ∠1 ≅ ∠2 Prove: 𝑙𝑖𝑛𝑒 𝑙 ∥ 𝑙𝑖𝑛𝑒 𝑚

2. Given:𝑊𝑋̅̅ ̅̅ ̅ ∥ 𝑌𝑍̅̅̅̅ , ∠2 ≅ ∠3 Prove: 𝑊𝑌̅̅ ̅̅ ̅ ∥ 𝑋𝑍̅̅ ̅̅

3. Given: ∠1 ≅ ∠6, ∠3 ≅ ∠6 Prove: 𝐷𝐶̅̅ ̅̅ ∥ 𝐴𝐵̅̅ ̅̅

1

4. Given: AE bisects DAC

2 B

Prove: ||AE BC

5. Given: CB bisects ACD ; 1 3

Prove: AB CD

6. Given: 1 2

Prove: 3 4

A

2

B

D

E 1

C

3

2

4

3

m

k

A B

C 2

4

1

D

3