Foundations for Geometry



Tools of Geometry—Project Problems—Set 1

From Module 1

Refer to the figure for Exercises 1 and 2.

[pic]

1. Which represents the name of the ray whose endpoint is K and that passes through R?

2. In the diagram, how many different rays have endpoint R?

Refer to the figure for Exercises 3 and 4.

[pic]

3. What is MP?

4. What is LP?

5. An angle whose measure is 70° is what type of angle?

6. [pic] bisects (FGH, m(FGJ ( (7x ( 9)°, and m(HGJ ( (2x ( 36)°. What is m(FGH?

7. An angle measuring 22° is bisected. What is the measure of the angles that are formed?

A 11° C 33°

B 22° D 44°

8. What is the next letter in the sequence?

D, H, L, P, . . .

9. Which is the counterexample that proves the conjecture false?

“If two rays have the same endpoint, then they are opposite rays.”

A [pic] C [pic]

B [pic] D [pic]

10. Identify the hypothesis of the conditional statement “Two angles are complementary if the sum of their measures is 90 degrees.”

11. Which conditional statement has the same truth value as this statement?

“The sum of two odd numbers is even.”

A If two even numbers are added, then their sum is even.

B If an even and odd number are added, then their sum is even.

C If two even numbers are multiplied, then their product is odd.

D If two odd numbers are multiplied, then their product is even.

Tools of Geometry--Project Problems—Set 1

Use the figure for Exercises 12–15.

[pic]

12. Name a line.

13. Name a segment on line n.

14. Name a ray with endpoint A.

15. Name the intersection of [pic] and [pic]

16. Z is in the interior of (WXY.

If m(WXZ ( 110°, and m(ZXY ( 20°, what is m(WXY?

17. (A and (B are complementary.

m(A ( 29(. Find m(B.

18. Find the next item in the pattern.

2, 5, 8, 11, 14, . . .

19. Show that the conjecture is false by finding a counterexample. When the letters i and e appear next to each other in a word, the letter i always comes before the letter e.

20. Identify the hypothesis and conclusion of the statement “If it is raining, then there are clouds in the sky.”

21. Given: If Lewis earns a scholarship, he can go to college. Lewis earns a scholarship.

Conjecture: Lewis can go to college. Determine whether the conjecture is

valid by the Law of Detachment.

From Module 2

1. Consider the related biconditional statement for the conditional statement “If Shelly lives in Texas, then she lives in the United States.”

Which of the following statements is true about the related biconditional statement?

A The biconditional is true because the conditional is true.

B The biconditional is false because the conditional and its converse are false.

C The biconditional is true because the conditional and its converse are true.

D The biconditional is false because the converse of the conditional is false.

2. If r + 14 ’ −9, what property

justifies r ’ −23?

3. If [pic] ’ 8, what property justifies

x − 1 ’ 16?

A

4. If 5 ’ 2k, what property justifies 2k ’ 5?

5. Completes the statement:

If 6x ’ 5 and d ’ 6x, then ______ by

the Transitive Property of Equality.

6. Completes the statement:

If [pic] then ______ by the Symmetric Property of Congruence.

7. Given: L bisects [pic]; M bisects [pic]

Prove: KL ’ MN

[pic]

Proof:

Since L bisects [pic] and M bisects [pic] by definition of bisect, [pic] and [pic] Then, by the      ?     , [pic] Finally,

KL ’ MN by the definition of congruent segments.

8. Given: ∠1 ” ∠4

Prove: ∠2 ” ∠3

Proof:

|Statements |Reasons |

|1. ∠1 ” ∠4 |1. Given |

|2. ∠1 and ∠2 are supp., |2. Lin. Pairs |

|and ∠3 and ∠4 are |Thm. |

|supp. | |

|3. ∠2 ” ∠3 |3.      ?      |

What goes in the blank to complete the proof?

9. For the following statement, what is the conclusion?

If ABCD is a rhombus, then it is a parallelogram.

10. Use the Reflexive Property of Congruence to complete the statement “∠A ” ________.”

11. Complete the sentence “A _________ is any statement that you can prove.”

12. What is the reason for Step 2?

|Statements |Reasons |

|1. ∠1 ” ∠2 and |1. Given |

|∠2 ” ∠3. | |

|2. m∠1 ’ m∠2 and m∠2 ’ m∠3. |2.       ?       |

|3. m∠1 ’ m∠3 |3. Trans. Prop. of ’ |

|4. ∠1 ” ∠3 |4. Def. of ” [pic] |

13. The box is part of a flowchart proof. Identify the statement.

[pic]

14. Write True or False. A paragraph proof is less formal than a two-column proof, so you do not need to include every step.

From Module 3

Refer to the figure for Exercises 1–3.

[pic]

1. Which segment is perpendicular to [pic]?

2. Which segment is parallel to [pic]?

3. Which segment is NOT skew to [pic]?

Refer to the figure for Exercises 4–5.

[pic]

4. Name 4 pairs of angles are corresponding angles.

5. Name 2 pairs of angles are alternate exterior angles.

6. What is the value of x?

[pic]

7. Name two pairs of same side interior angles.

[pic]

8. Why is m ( n?

9. Identify a pair of parallel segments.

[pic]

10. Write True or False. Parallel lines intersect.

11. How many angles are formed by two lines and a transversal?

12. What is the name given to the angle pair ∠3 and ∠5?

[pic]

13. If parallel lines are intersected by a transversal, how many pairs of corresponding angles are there?

14. Complete the sentence. If a transversal intersects parallel lines and an obtuse angle is formed, all the obtuse angles are ________.

15. Given r || s. What is the measure of ∠1?

[pic]

16. Write True or False. You can use the measures of the angles formed by two lines and a transversal to determine whether the two lines are parallel.

17. If ∠2 ” ∠8, then r || s by which theorem?

[pic]

18. If two coplanar lines are cut by a transversal so that right angles are formed, how many different angle measures are there?

19. Name the shortest segment from

C to [pic]

[pic]

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