Name: _________________ Date: __________ Ch 2 Review
1. Use inductive reasoning to test the conjecture. Decide if the conjecture seems true or false.
𝐸𝑣𝑒𝑟𝑦 𝑜𝑑𝑑 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑞𝑢𝑎𝑟𝑒𝑠.
a. Write a few whole numbers here: ___________________________________________
b. Test conjecture: _________________________________________________________
c. Can you find a counterexample? ____________________________________________
2. Use inductive reasoning to test the conjecture. Decide if the conjecture seems true or false.
𝑇ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑠 𝑙𝑎𝑟𝑔𝑒𝑟 𝑡ℎ𝑎𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟.
3. What is inductive reasoning?
4. You would use _________________ reasoning to find the next two terms in the sequence.
400, 200, 100, 50, 25, _____, _____. How would you find the next two numbers?
5. Using 𝑝 = 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 ℎ𝑎𝑠 𝑎 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 90° and 𝑞 = 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒, write the
symbolic statement.
a. 𝑝 → 𝑞
b. ~𝑞
c. ~𝑝 → ~𝑞
6. To write a biconditional statement, you first take the ________________, then remove the
_______ and replace the __________ with _________________.
7. Use the following information to complete the chart below. Be sure to include the symbolic
notation as well. 𝐼𝑓 𝑡ℎ𝑒 𝑠𝑘𝑦 𝑖𝑠 𝑐𝑙𝑒𝑎𝑟, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑐𝑎𝑛 𝑠𝑒𝑒 𝑡ℎ𝑒 𝑠𝑡𝑎𝑟𝑠 𝑎𝑡 𝑛𝑖𝑔ℎ𝑡.
𝑝
𝑞
Symbols
Statement
Conditional
Converse
Inverse
Contrapositive
8.
Using 𝑝 = 𝑎 𝑠𝑡𝑢𝑑𝑒𝑛𝑡 𝑖𝑠 𝑜𝑛 𝑡ℎ𝑒 ℎ𝑜𝑛𝑜𝑟 𝑟𝑜𝑙𝑙 and 𝑞 = 𝑡ℎ𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡 ℎ𝑎𝑠 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑎 90 𝑎𝑣𝑒𝑟𝑎𝑔𝑒,
write the symbolic statements below.
a. 𝑝 → 𝑞
b. ~𝑝
c. ~𝑞 → ~𝑝
d. 𝑞 → 𝑝
e. 𝑝 represents the ___________________ and 𝑞 represents the ____________________.
9. Use the statement to write a compound statement for the conjecture. Then determine the
validity of the statement. 𝑝: (−5)(−2) = 10 and 𝑞: 7 − 4 > 3
a. 𝑝 ⋀ 𝑞
b. 𝑝 𝑎𝑛𝑑 ~𝑞
c. 𝑝 ⋁ 𝑞
10. Write a valid argument using the Law of Syllogism for the following statements.
If today is Monday, then I will watch TV.
Symbols: _________
If I watch TV, then I will make popcorn.
Symbols: _________
(Conclusion) _____________________
Symbols: _________
11. Write a valid conclusion for the statements 𝐼𝑓 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 60°,
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑎𝑛 𝑎𝑐𝑢𝑡𝑒 𝑎𝑛𝑔𝑙𝑒. 𝑇ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 60°. __________________________.
What law did you use? _____________________________ What is the symbolic notation of
this law?
12. 𝐼𝑓 𝑡ℎ𝑒 𝑠𝑢𝑛 𝑐𝑜𝑚𝑒𝑠 𝑜𝑢𝑡, 𝑆𝑢𝑧𝑦 𝑤𝑖𝑙𝑙 𝑔𝑜 𝑡𝑜 𝑡ℎ𝑒 𝑏𝑒𝑎𝑐ℎ. 𝑆𝑢𝑧𝑦 𝑤𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑏𝑒𝑎𝑐ℎ. Is this valid? ______
Why or why not? ______________________________________________________________
13. 𝐼𝑓 𝑖𝑡 𝑟𝑎𝑖𝑛𝑠 𝑡𝑜𝑑𝑎𝑦, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 ℎ𝑎𝑣𝑒 𝑎 𝑝𝑖𝑐𝑛𝑖𝑐. 𝐼𝐹 𝑤𝑒 𝑑𝑜 𝑛𝑜𝑡 ℎ𝑎𝑣𝑒 𝑎 𝑝𝑖𝑐𝑛𝑖𝑐, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡
𝑠𝑒𝑒 𝑜𝑢𝑟 𝑓𝑟𝑖𝑒𝑛𝑑𝑠. Is this valid? _______ Why or why not? _____________________________
What if you were given "𝑤𝑒 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 𝑠𝑒𝑒 𝑜𝑢𝑟 𝑓𝑟𝑖𝑒𝑛𝑑𝑠" for the conclusion. Would this be
valid?______ Why or why not? __________________________________________________
14. Fill in the reasons for each mini proof.
a) 𝑥 + 7 = 23
𝑥 = 16
c) 𝑥 − 24 = 50
𝑥 = 74
b)
4𝑥 = 28
𝑥=7
d) 2𝑥 + 5 = 25
2𝑥 = 20
𝑥 = 10
15. Complete the proofs below.
𝑥+3
a. Given: 𝐺𝑀 = 28
Prove: 𝑥 = 11
𝐺
2𝑥 − 8
𝐸
𝑀
𝐺𝑀 = 28
𝐺𝐸 = 𝑥 + 3, 𝐸𝑀 = 2𝑥 − 8
𝐺𝐸 + 𝐸𝑀 = 𝐺𝑀
𝑥 + 3 + 2𝑥 − 8 = 28
3𝑥 − 5 = 28
3𝑥 = 33
𝑥 = 11
b. Given: 𝐴𝐵 ≅ 𝐵𝐶, 𝐵𝐶 ≅ 𝐶𝐷
Prove: 3 = 𝑥
2𝑥 + 1
𝐴
4𝑥 − 5
𝐵
𝐶
𝐷
𝐴𝐵 ≅ 𝐵𝐶, 𝐵𝐶 ≅ 𝐶𝐷
𝐴𝐵 ≅ 𝐶𝐷
𝐴𝐵 = 𝐶𝐷
2𝑥 + 1 = 4𝑥 − 5
1 = 2𝑥 − 5
6 = 2𝑥
3=𝑥
c. Given: ∠1 and ∠2 are vertical angles
Prove: 𝑚∠1 = 𝑚∠2
𝑙
1
2
𝑚
∠1 and ∠2 are vertical angles
∠1 ≅ ∠2
𝑚∠1 = 𝑚∠2
16. Name
a.
b.
c.
d.
e.
the property for each.
𝑥=𝑥
𝑎 = 𝑏, 𝑏 = 𝑐, ∴ 𝑎 = 𝑐
𝐴𝐵 = 𝐶𝐷, 𝐴𝐵 − 3 = 𝐶𝐷 − 3
𝑚∠1 = 𝑚∠3, 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2
𝑥
= 10, 𝑥 = 20
2
f. 𝑚∠1 = 30°, 𝑚∠1 + 𝑚∠2 = 90°, 30° + 𝑚∠2 = 90°
g. If point 𝑄 is between ∠𝐴𝐵𝐶, then 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐴𝐵𝑄 + 𝑚∠𝑄𝐵𝐶
h. 𝑥 + 𝑦 = 3, 3 = 𝑥 + 𝑦
17. Write a 2 column proof to solve −2(𝑥 + 2) + 4 = 12.
−2(𝑥 + 2) + 4 = 12
−2(𝑥 + 2) = 8
−2𝑥 − 4 = 8
−2𝑥 = 12
𝑥 = −6
18. Solve for 𝑥.
4𝑥 + 5
105
19. For each statement below, determine if the statement is valid. Explain.
a. ∠𝑃𝑆𝑇 ≅ ∠𝑈𝑊𝑉
𝑇
𝑃
b. ∠𝑅𝑆𝑈 ≅ ∠𝑃𝑆𝑇
𝑉
𝑈
c. 𝑅𝑇 is parallel to 𝑉𝑋
𝑆
d. 𝑅𝑇 and 𝑃𝑍 are perpendicular
e. 𝑉𝑋 and 𝑃𝑍 intersect at 𝑊.
𝑍
𝑊
𝑅
𝑋
20. Given: ∠1 ≅ ∠2
Prove: ∠3 ≅ ∠4
1
∠1 ≅ ∠2
∠3 ≅ ∠1
∠2 ≅ ∠4
∠3 ≅ ∠4
4
3
2
21. Which of the following statements cannot be assumed from the diagram?
𝑭
𝐸
𝐷
𝐵
a. 𝐸, 𝐷, and 𝐶 are collinear
⃡ and ⃡𝐸𝐶 is 𝐷
b. The intersection of 𝐵𝐷
⃡𝐸𝐶
c. ⃡𝐵𝐷
d. 𝐸𝐶
𝑮
𝐴
𝐶
plane 𝐺
Chapter 1 Review material…some of the material from chapter 1 will be on the chapter 2
test!
22. Define the following terms:
a. Supplementary angles
b. Complementary angles
c. Linear pair
d. Vertical angles
e. Midpoint formula
f. Distance formula
23. Given that ∠𝐴𝐵𝐷 is a straight angle, find 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝐶𝐵𝐷.
𝐶
(2𝑥 + 9)°
𝐴
(10𝑥 + 15)°
𝐵
𝐷
24. Given that ∠𝑋𝑌𝑍 and ∠𝐿𝑀𝑁 are complementary angles, find 𝑚∠𝑋𝑌𝑍 and 𝑚∠𝐿𝑀𝑁.
𝑍
(2𝑥 + 6)°
𝑁
𝑀
(5𝑥 + 21)°
𝑌
𝐿
𝑋
25. Determine if 𝐴𝐵 ≅ 𝐶𝐷. You must use the distance formula. Then find the midpoint of 𝐴𝐵 and
𝐶𝐷.
𝐴𝐵: 𝐴(1,1), 𝐵(5,3)
𝐶𝐷: 𝐶(−3, −2), 𝐷(−1,2)