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Tools of Geometry I
Tools of Geometry II
Reasoning & Proof I
Reasoning & Proof II
Parallel and Perpendicular Lines
100
How many planes are shown in Figure 1-1
Eight (8)
100
Name the vertex of each given angle using Figure 1-2. a. ∠1 b. ∠4
a. B b. E
100
Make a conjecture based on the given information. Draw a picture to illustrate your conjecture. Given: Lines J and K are parallel.
Sample Ans: Then they are perpendicular to the same line.
100
Identify the hypothesis and conclusion of each statement. a. If no sides of a triangle are equal, then it is scalene. b. If 6-x=11, then x=-5.
a. Hypothesis: If no sides of a triangle are equal Conclusion: then it is scalene. b. Hypothesis: If 6-x=11 Conclusion: then x=-5
100
Define transversal.
A line the intersects two or more lines at different points.
200
The intersection of plane ABD and plane DJK in Figure 1-1
Line ED
200
Use Figure 1-2. Write another name for each angle. a. ∠3 b. ∠DEF c. ∠2
a. ∠DCG OR ∠GCD b. ∠4 c. ∠BCG OR ∠GCB
200
Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. Given: r is a rational number. Conjecture: r is a whole number.
False. Sample Ans: 1/2 is a rational number, but not a whole one.
200
Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. 1. (a) If it rains, then the field will be muddy. (b) If the field is muddy, then the game will be cancelled. 2. (a) If you read a book, then you enjoy reading. (b) If you are in the 10th grade, then you passed the 9th.
1. Valid 2. No conclusion
200
Give two different examples of a transversal that you experience in everyday real life.
Anything will do.
300
Use Figure 1-1. What is the intersection of plane ACD and plane EDJ? (is it a point or a line). Explain.
A line. Planes intersect at a line. (Only lines intersect at points.)
300
Using figure 1-3, answer the following. a. Name two acute vertical angles & obtuse vertical angles. b. Name a pair of complementary adjacent angles and a pair of supplementary adjacent angles.
a. ∠BCG and ∠FGE & ∠BGF and ∠CGE b. ∠BEC and ∠CED & ∠CBE and ∠EBA
300
Use the following statements to write a compound statement for each conjunction and disjunction and find its truth value. p: (-3)^2 = 9, q: A robin is a fish, r: An acute angle measures less than 90 degrees 1. p and q 2. p or ~r
1. (-3)^2 = 9 and a robin is a fish. FALSE 2. (-3)^2 = 9 or an acute angle does not measure less than 90 degrees. TRUE
300
Determine if statement (3) follows from statements (1) & (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, state invalid. A. (1)If it snows outside, you will wear your winter coat. (2) It is snowing outside. (3) You will wear your winter coat. B. (1) Two complementary angles are both acute. (2) ∠1 and ∠2 are acute angles. (3) ∠1 and ∠2 are complementary.
A. Law of Detachment B. Invalid
300
Using the Figure 3-1, find the measure of each angle given that m∠5=72 and m∠9=102 a. ∠1 b. ∠4 c. ∠7
a. m∠1= 102 b. m∠4= 102 c. m∠7=108
400
What is the value of the variable BC if B is between A and C? a. AB = 4x, BC = 5x; AB = 16 b. AB = 17, BC = 3m; AC = 32 c. AB = 5n+5, BC = 2n; AC=54
a. BC = 20 b. BC = 15 c. BC = 14
400
Using Figure 1-3, if m∠BCG=4a+5, m∠FGE=6a-15, find m∠BGF.
m∠BGF=135
400
Complete the truth table from Figure 2-1.
Ans. 2nd Column q: T, F, T, F 3rd Column ~q: F, T, F, T 4th Column p∨~q: T, T, F, T
400
Justify each statement with a property of equality or a property or congruence. 1. Segment XD is congruent to itself. 2. If JK≅XY and XY≅LM, then JK≅LM.
1. Reflexive 2. Transitive
400
Define the following: a. Exterior and Interior Angles b. Consecutive Interior Angles c. Alternate Exterior and Interior Angles d. Corresponding Angles
a. Exterior angles are those that lie on the far outer parts of the lines being intersected by a transversal. Interior angles are those that lie within the lines being intersected by a transversal. b. Consecutive Interior angles are those that are on the same side of the transversal and are within the lines being intersected by a transversal. c. Alternate Exterior angles are those that lie on the outer parts of the lines being cut by the transversal and are on opposite sides of the transversal. Alternate Interior angles are those that lie on the inside of the lines being cut by a transversal and are on opposite sides of the transversal. d. Corresponding Angles are those that lie on the same side of the transversal and respectively lie on the same position upon the lines that are being intersected.
500
Solve the following problems. 1. Use the Pythagorean Theorem or Distance Formula to find the distance between each pair of points. a. A(0,0) , B(-3,4) b. T(-1,3) , N(0,2) 2. Find the coordinates of the midpoint of a segment having the endpoints A(2.8,-3.4) and D(1.2,5.6).
1a. 5 units 1b. √2 units 2.(2, 1.1)
500
Answer the following a. The measure of ∠A is nine less than the measure of ∠B. If ∠A and ∠B for a linear pare, what are their measures? b. The measure of an angle's complement is 17 more than the measure of the angle. Find the measurement of the angle and its complement.
a. ∠A = 85.5 degrees, ∠B = 94.5 degrees b. The measurement of the angle is 36.5 degrees, which makes its complement 53.5 degrees.
500
Write the converse, inverse, and contrapositive of the following conditional statement. Determine whether the conditional is true or false. If false, find a counterexample. If two angles are congruent angles, then they have the same measure.
Converse: If two angles have the same measure, then they are congruent angles. Inverse: If two angles are not congruent angles, then they do not have the same measure. Contrapositive: If two angles do not have the same measure, then they are not congruent angles. The conditional is true.
500
Justify the following using a Two-Column proof or using a Paragraph Proof. Given: JK≅KL, HJ≅GH, and KL≅HJ Prove: GH≅JK
Proof: (Paragraph) It is given that JK≅KL and KL≅HJ. Thus, JK≅HJ by the Transitive Property. It is also given that HJ≅GH. By the Transitive Property, JK≅GH. Therefore, GH≅JK by the Symmetric Property. Proof: (Two-Column) Statements Reasons 1. JK≅KL, KL≅HJ 1. Given 2. JK≅HJ 2. Transitive Property 3. HJ≅GH 3. Given 4. JK≅GH 4. Transitive Property 5. GH≅JK 5. Symmetric Property
500
Find the values for x and y for Figure 3-2.
x = 10 y = 12