Geometry Tutor Worksheet 20 Geometric Proofs
Geometry Tutor Worksheet 20 Geometric Proofs
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Geometry Tutor - Worksheet 20 – Geometric Proofs 1. Fill in the reasons for the proof below. Given: 5𝑥 + 1 = 21 Prove: 𝑥 = 4 Statements 1) 5𝑥 + 1 = 21
Reasons 1)
2) 5𝑥 = 20 3) 𝑥 = 4
2) 3)
2. Fill in the reasons for the proof below. 1
Given: 𝑥 − 5 = 10 2
Prove: 𝑥 = 30 Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1)
2) 2 ( 𝑥 − 5) = 20
2)
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) 4)
2
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3. Fill in the reasons for the proof below. Given: 5(𝑥 + 3) = −4 Prove: 𝑥 = −
19 5
Statements 1) 5(𝑥 + 3) = −4
Reasons 1)
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) 3) 4)
5
4. Fill in the missing statements and reasons for the proof below. Given:
Prove: 𝑥 = 50 Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1)
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6)
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) 6) Division Property 3
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5. Fill in the missing statements and reasons for the proof below. Given: ⃗⃗⃗⃗⃗ 𝐴𝐵 bisects ∠𝑅𝐴𝑁
Prove: 𝑥 = 75 Statements 1)
Reasons 1)
2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 4) 5) −𝑥 = −75 6)
2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) 6)
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6. Fill in the missing statements and reasons for the proof below. Given: 𝑚∠1 = 𝑚∠3
Prove: ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵 Statements 1)
Reasons 1)
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) 3) 4) Angle Addition Postulate 5) 6)
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7. Fill in the missing statements and reasons for the proof below. Given: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷
̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ Prove: 𝐴𝐶 Statements 1)
Reasons 1)
2) 𝐴𝐵 = 𝐶𝐷 3) 4) ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 6)
2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6)
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8. Fill in the statements and reasons for the proof below. (Hint: This proof is similar to question 4.) Given: ∠𝐶𝐷𝐸 and ∠𝐸𝐷𝐹 are supplementary.
Prove: 𝑥 = 40 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
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9. Fill in the missing statements and reasons for the proof below. Given: 𝑋𝑌 = 42
Prove: 𝑥 = 5 Statements 1)
Reasons 1)
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 4) 5) 6) 7)
2) Segment Addition Postulate 3) 4) 5) 6) 7)
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10. Fill in the missing statements for the proof below. (Hint: This proof involves vertical angles.) Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3 Statements 1)
Reasons 1) Given
2) 3) 4) 5)
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
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11. Fill in the missing statements and reasons for the proof below. Given: ∠1 ≅ ∠3
Prove: ∠6 ≅ ∠4 Statements 1)
Reasons 1)
2) ∠3 ≅ ∠6 3) 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) 3) Transitive Property of Congruence 4) 5)
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12. Fill in the missing statements and reasons for the proof below. (Hint: This proof is similar to question 9.) Given: 𝑄𝑆 = 42
Prove: 𝑥 = 13 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
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Answers - Geometry Tutor - Worksheet 20 – Geometric Proofs 1. Fill in the reasons for the proof below. Given: 5𝑥 + 1 = 21 Prove: 𝑥 = 4 Statements 1) 5𝑥 + 1 = 21
Reasons 1)
2) 5𝑥 = 20 3) 𝑥 = 4
2) 3)
In the first line, Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by subtracting 1 from both sides of the equation, so Reason #2 is Subtraction Property. In the third line, the both sides of the last equation were divided by 5, so Reason #3 is Division Property. Answer: Statements 1) 5𝑥 + 1 = 21
Reasons 1) Given
2) 5𝑥 = 20 3) 𝑥 = 4
2) Subtraction Property 3) Division Property
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2. Fill in the reasons for the proof below. 1
Given: 𝑥 − 5 = 10 2
Prove: 𝑥 = 30 Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1)
2) 2 ( 𝑥 − 5) = 20
2)
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) 4)
2
Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by multiplying by 2 on both sides of the equation, so Reason #2 is Multiplication Property. In the third line, the left side of the last equation is changed by distributing the 2, so Reason #3 is Distributive Property. In the fourth line, the last equation was changed by adding 10 to both sides, so Reason #4 is Addition Property. Answer: Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1) Given
2) 2 ( 𝑥 − 5) = 20
2) Multiplication Property
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) Distributive Property 4) Addition Property
2
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3. Fill in the reasons for the proof below. Given: 5(𝑥 + 3) = −4 Prove: 𝑥 = −
19 5
Statements 1) 5(𝑥 + 3) = −4
Reasons 1)
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) 3) 4)
5
Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by distributing 5 on the left side of the equation, so Reason #2 is Distributive Property. In the third line, the last equation is changed by subtracting 15 from both sides, so Reason #3 is Subtraction Property. In the fourth line, the last equation was changed by dividing both sides by 5, so Reason #4 is Division Property. Answer: Statements 1) 5(𝑥 + 3) = −4
Reasons 1) Given
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) Distributive Property 3) Subtraction Property 4) Division Property
5
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4. Fill in the missing statements and reasons for the proof below. Given: ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary.
Prove: 𝑥 = 50 Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1)
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6)
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) 6) Division Property
Statement #1 is the given information, so Reason #1 is Given. In the third line, Reason #3 is Substitution Property so the first equation was changed by substituting the algebraic expressions from the figure into the equation. In the fifth line, the last equation is changed by subtracting 30 from both sides, so Reason #5 is Subtraction Property. In the sixth line, Reason #6 is Division Property, so the last equation was changed by dividing both sides by 3. Answer: Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1) Given
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) (2𝑥 + 30) + 𝑥 = 180 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6) 𝑥 = 50
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property
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5. Fill in the missing statements and reasons for the proof below. Given: ⃗⃗⃗⃗⃗ 𝐴𝐵 bisects ∠𝑅𝐴𝑁
Prove: 𝑥 = 75 Statements 1)
Reasons 1)
2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 4) 5) −𝑥 = −75 6)
2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) 6)
Statement #1 is the given information and Reason #1 is Given. In the third line, since Reason #3 is Definition of Congruent ∠s, Statement #3 says that the measures of the angles are equal. Reason #4 is Substitution Property, so the algebraic equation that matches Statement #3 is Statement #4. In the fifth line, the last equation is changed by subtracting 2𝑥 from both sides, so Reason #5 is Subtraction Property. The sixth line is the last line, so Statement #6 is what we are trying to prove and Reason #6 is Division Property. Answer: Statements ⃗⃗⃗⃗⃗ bisects ∠𝑅𝐴𝑁 1) 𝐴𝐵 2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 𝑚∠𝑅𝐴𝐵 = 𝑚∠𝐵𝐴𝑁 4) 𝑥 = 2𝑥 − 75 5) −𝑥 = −75 6) 𝑥 = 75
Reasons 1) Given 2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) Subtraction Property 6) Division Property
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6. Fill in the missing statements and reasons for the proof below. Given: 𝑚∠1 = 𝑚∠3
Prove: ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵 Statements 1)
Reasons 1)
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) 3) 4) Angle Addition Postulate 5) 6)
Statement #1 is the given information and Reason #1 is Given. Statement #2 says something is equal to itself, so Reason #2 is Reflexive Property. Since Statement #3 adds a value to both sides, Reason #3 is Addition Property. Use Line 4 to build Line 5. Statement #5 sets the two measures equal to each other, so 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 and Reason #5 is Substitution Property. Statement #6 is what we are trying to prove and Reason #6 is Definition of Congruent Angles, since we said their measures are equal. Answer: Statements 1) 𝑚∠1 = 𝑚∠3
Reasons 1) Given
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) Reflexive Property 3) Addition Property 4) Angle Addition Postulate 5) Substitution Property 6) Definition of Congruent ∠s 17
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7. Fill in the missing statements and reasons for the proof below. Given: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷
̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ Prove: 𝐴𝐶 Statements 1)
Reasons 1)
2) 𝐴𝐵 = 𝐶𝐷 3) 4) ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 6)
2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6)
In the first line, Statement #1 is the given information and Reason #1 is Given. Since Reason #3 is Reflexive Property, Statement #3 states something is equal to itself. Use Line #5 to build Statement #4. Statement #5 uses two segments added together. Statement #6 is what we are trying to prove and Reason #6 is Substitution Property, since we just said the segments are equal. Answer: Statements 1) ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷 2) 𝐴𝐵 = 𝐶𝐷 3) 𝐵𝐶 = 𝐵𝐶 4) 𝐴𝐵 + 𝐵𝐶 = 𝐶𝐷 + 𝐵𝐶 ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 ̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ 6) 𝐴𝐶
Reasons 1) Given 2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6) Substitution Property
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8. Fill in the statements and reasons for the proof below. (Hint: This proof is similar to question 4.) Given:
Prove: 𝑥 = 40 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
Statement #1 is the given information, and Reason #1 is Given. In the second line, Statement #2 is the equation using the definition of supplementary ∠s, so Reason #2 is Definition of Supplementary ∠s. In the third line, Substitute algebraic expressions from the figure into the equation, so Reason #3 is Substitution. In Line #4 remove the parentheses using the Distributive Property. In the fifth line, subtract 20 from both sides, so Reason #5 is Subtraction Property. In the sixth line, the Statement is what we are proving and Reason #6 is Division Property, because the last equation was changed by dividing both sides by 4. Answer: Statements 1) ∠𝐶𝐷𝐸 and ∠𝐸𝐷𝐹 are supplementary
Reasons 1) Given
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 𝑥 + (3𝑥 + 20) = 180 4) 4𝑥 + 20 = 180 5) 4𝑥 = 160 6) 𝑥 = 40
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property 19
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9. Fill in the missing statements and reasons for the proof below. Given: 𝑋𝑌 = 42
Prove: 𝑥 = 5 Statements 1)
Reasons 1)
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 4) 5) 6) 7)
2) Segment Addition Postulate 3) 4) 5) 6) 7)
Statement #1 is the given information, and Reason #1 is Given. In the third line, Statement #3 is the equation using the algebraic expressions in the figure, so Reason #3 is Substitution Property. In Line #4 remove the parentheses using the Distributive Property. In the fifth line, combine like terms on the left side, so Reason #5 is Combine Like Terms. Line #6 results from subtracting 12 from both sides of the equation. The seventh line, the Statement is what we are proving and Reason #7 is Division Property, because the last equation was changed by dividing both sides by 6. Answer: Statements 1) 𝑋𝑌 = 42
Reasons 1) Given
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 3(𝑥 + 4) + 3𝑛 = 42 4) 3𝑥 + 12 + 3𝑥 = 42 5) 6𝑥 + 12 = 42 6) 6𝑥 = 30 7) 𝑥 = 5
2) Segment Addition Postulate 3) Substitution Property 4) Distributive Property 5) Combine Like Terms 6) Subtraction Property 7) Division Property 20
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10. Fill in the missing statements for the proof below. (Hint: This proof involves vertical angles.) Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3 Statements 1)
Reasons 1) Given
2) 3) 4) 5)
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
Statement #1 is the given information, because Reason #1 is Given. In the second line, Statement #2 says ∠4 ≅ ∠2 because Reason #2 says vertical angles are congruent. This statement must use information already given in previous statements. In the third line, Statement #3 is ∠1 ≅ ∠2, using information in the previous statements and because Reason #3 is Transitive Property of Congruence. In Line #4 Statement #4 is ∠1 ≅ ∠3 because Reason #4 says vertical angles are congruent. This statement must also use information already given in previous statements. In the fifth line, the Statement is what we are proving and Reason #5 is Transitive Property of Congruence. Answer: Statements 1) ∠1 ≅ ∠4
Reasons 1) Given
2) ∠4 ≅ ∠2 3) ∠1 ≅ ∠2 4) ∠1 ≅ ∠3 5) ∠2 ≅ ∠3
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence 21
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11. Fill in the missing statements and reasons for the proof below. Given: ∠1 ≅ ∠3
Prove: ∠6 ≅ ∠4 Statements 1)
Reasons 1)
2) ∠3 ≅ ∠6 3) 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) 3) Transitive Property of Congruence 4) 5)
Statement #1 is the given information, because Reason #1 is Given. In the second line, Statement #2 says ∠3 ≅ ∠6 so Reason #2 says vertical angles are congruent. This statement must use information already given in previous statements. In the third line, Statement #3 is ∠1 ≅ ∠6, using information in the previous statements and because Reason #3 is Transitive Property of Congruence. In Line #4 Statement #4 is ∠1 ≅ ∠4 so Reason #4 says vertical angles are congruent. In the fifth line, the Statement is what we are proving and Reason #5 is Transitive Property of Congruence. Answer: Statements 1) ∠1 ≅ ∠3
Reasons 1) Given
2) ∠3 ≅ ∠6 3) ∠1 ≅ ∠6 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
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12. Fill in the missing statements and reasons for the proof below. (Hint: This proof is similar to question 9.) Given: 𝑄𝑆 = 42
Prove: 𝑥 = 13 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
Statement #1 is the given information, and Reason #1 is Given. In Line #2, using the figure, write a statement adding two segments to make a larger segment, so Reason #2 is Segment Addition Postulate. In the third line, Statement #3 is the equation you can write using the algebraic expressions in the figure, so Reason #3 is Substitution Property. In Line #4 remove the parentheses using the Distributive Property. Line #5 results from subtracting 3 from both sides of the equation. The sixth line, the Statement is what we are proving and Reason #6 is Division Property, because the last equation was changed by dividing both sides by 3. Answer: Statements 1) 𝑄𝑆 = 42
Reasons 1) Given
2) 𝑄𝑅 + 𝑅𝑆 = 𝑄𝑆 3) (𝑥 + 3) + 2𝑥 = 42 4) 3𝑥 + 3 = 42 5) 3𝑥 = 39 6) 𝑥 = 13
2) Segment Addition Postulate 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property 23
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Geometry Tutor - Worksheet 20 – Geometric Proofs 1. Fill in the reasons for the proof below. Given: 5𝑥 + 1 = 21 Prove: 𝑥 = 4 Statements 1) 5𝑥 + 1 = 21
Reasons 1)
2) 5𝑥 = 20 3) 𝑥 = 4
2) 3)
2. Fill in the reasons for the proof below. 1
Given: 𝑥 − 5 = 10 2
Prove: 𝑥 = 30 Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1)
2) 2 ( 𝑥 − 5) = 20
2)
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) 4)
2
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3. Fill in the reasons for the proof below. Given: 5(𝑥 + 3) = −4 Prove: 𝑥 = −
19 5
Statements 1) 5(𝑥 + 3) = −4
Reasons 1)
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) 3) 4)
5
4. Fill in the missing statements and reasons for the proof below. Given:
Prove: 𝑥 = 50 Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1)
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6)
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) 6) Division Property 3
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5. Fill in the missing statements and reasons for the proof below. Given: ⃗⃗⃗⃗⃗ 𝐴𝐵 bisects ∠𝑅𝐴𝑁
Prove: 𝑥 = 75 Statements 1)
Reasons 1)
2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 4) 5) −𝑥 = −75 6)
2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) 6)
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6. Fill in the missing statements and reasons for the proof below. Given: 𝑚∠1 = 𝑚∠3
Prove: ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵 Statements 1)
Reasons 1)
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) 3) 4) Angle Addition Postulate 5) 6)
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7. Fill in the missing statements and reasons for the proof below. Given: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷
̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ Prove: 𝐴𝐶 Statements 1)
Reasons 1)
2) 𝐴𝐵 = 𝐶𝐷 3) 4) ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 6)
2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6)
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8. Fill in the statements and reasons for the proof below. (Hint: This proof is similar to question 4.) Given: ∠𝐶𝐷𝐸 and ∠𝐸𝐷𝐹 are supplementary.
Prove: 𝑥 = 40 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
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9. Fill in the missing statements and reasons for the proof below. Given: 𝑋𝑌 = 42
Prove: 𝑥 = 5 Statements 1)
Reasons 1)
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 4) 5) 6) 7)
2) Segment Addition Postulate 3) 4) 5) 6) 7)
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10. Fill in the missing statements for the proof below. (Hint: This proof involves vertical angles.) Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3 Statements 1)
Reasons 1) Given
2) 3) 4) 5)
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
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11. Fill in the missing statements and reasons for the proof below. Given: ∠1 ≅ ∠3
Prove: ∠6 ≅ ∠4 Statements 1)
Reasons 1)
2) ∠3 ≅ ∠6 3) 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) 3) Transitive Property of Congruence 4) 5)
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12. Fill in the missing statements and reasons for the proof below. (Hint: This proof is similar to question 9.) Given: 𝑄𝑆 = 42
Prove: 𝑥 = 13 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
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Answers - Geometry Tutor - Worksheet 20 – Geometric Proofs 1. Fill in the reasons for the proof below. Given: 5𝑥 + 1 = 21 Prove: 𝑥 = 4 Statements 1) 5𝑥 + 1 = 21
Reasons 1)
2) 5𝑥 = 20 3) 𝑥 = 4
2) 3)
In the first line, Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by subtracting 1 from both sides of the equation, so Reason #2 is Subtraction Property. In the third line, the both sides of the last equation were divided by 5, so Reason #3 is Division Property. Answer: Statements 1) 5𝑥 + 1 = 21
Reasons 1) Given
2) 5𝑥 = 20 3) 𝑥 = 4
2) Subtraction Property 3) Division Property
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2. Fill in the reasons for the proof below. 1
Given: 𝑥 − 5 = 10 2
Prove: 𝑥 = 30 Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1)
2) 2 ( 𝑥 − 5) = 20
2)
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) 4)
2
Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by multiplying by 2 on both sides of the equation, so Reason #2 is Multiplication Property. In the third line, the left side of the last equation is changed by distributing the 2, so Reason #3 is Distributive Property. In the fourth line, the last equation was changed by adding 10 to both sides, so Reason #4 is Addition Property. Answer: Statements 1 1) 𝑥 − 5 = 10 2
1
Reasons 1) Given
2) 2 ( 𝑥 − 5) = 20
2) Multiplication Property
3) 𝑥 − 10 = 20 4) 𝑥 = 30
3) Distributive Property 4) Addition Property
2
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3. Fill in the reasons for the proof below. Given: 5(𝑥 + 3) = −4 Prove: 𝑥 = −
19 5
Statements 1) 5(𝑥 + 3) = −4
Reasons 1)
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) 3) 4)
5
Statement #1 is the given information, so Reason #1 is Given. In the second line, the given equation was changed by distributing 5 on the left side of the equation, so Reason #2 is Distributive Property. In the third line, the last equation is changed by subtracting 15 from both sides, so Reason #3 is Subtraction Property. In the fourth line, the last equation was changed by dividing both sides by 5, so Reason #4 is Division Property. Answer: Statements 1) 5(𝑥 + 3) = −4
Reasons 1) Given
2) 5𝑥 + 15 = −4 3) 5𝑥 = −19 19 4) 𝑥 = −
2) Distributive Property 3) Subtraction Property 4) Division Property
5
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4. Fill in the missing statements and reasons for the proof below. Given: ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary.
Prove: 𝑥 = 50 Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1)
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6)
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) 6) Division Property
Statement #1 is the given information, so Reason #1 is Given. In the third line, Reason #3 is Substitution Property so the first equation was changed by substituting the algebraic expressions from the figure into the equation. In the fifth line, the last equation is changed by subtracting 30 from both sides, so Reason #5 is Subtraction Property. In the sixth line, Reason #6 is Division Property, so the last equation was changed by dividing both sides by 3. Answer: Statements 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary
Reasons 1) Given
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) (2𝑥 + 30) + 𝑥 = 180 4) 3𝑥 + 30 = 180 5) 3𝑥 = 150 6) 𝑥 = 50
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property
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5. Fill in the missing statements and reasons for the proof below. Given: ⃗⃗⃗⃗⃗ 𝐴𝐵 bisects ∠𝑅𝐴𝑁
Prove: 𝑥 = 75 Statements 1)
Reasons 1)
2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 4) 5) −𝑥 = −75 6)
2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) 6)
Statement #1 is the given information and Reason #1 is Given. In the third line, since Reason #3 is Definition of Congruent ∠s, Statement #3 says that the measures of the angles are equal. Reason #4 is Substitution Property, so the algebraic equation that matches Statement #3 is Statement #4. In the fifth line, the last equation is changed by subtracting 2𝑥 from both sides, so Reason #5 is Subtraction Property. The sixth line is the last line, so Statement #6 is what we are trying to prove and Reason #6 is Division Property. Answer: Statements ⃗⃗⃗⃗⃗ bisects ∠𝑅𝐴𝑁 1) 𝐴𝐵 2) ∠𝑅𝐴𝐵 ≅ ∠𝐵𝐴𝑁 3) 𝑚∠𝑅𝐴𝐵 = 𝑚∠𝐵𝐴𝑁 4) 𝑥 = 2𝑥 − 75 5) −𝑥 = −75 6) 𝑥 = 75
Reasons 1) Given 2) Definition of ∠ Bisector 3) Definition of congruent ∠s 4) Substitution Property 5) Subtraction Property 6) Division Property
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6. Fill in the missing statements and reasons for the proof below. Given: 𝑚∠1 = 𝑚∠3
Prove: ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵 Statements 1)
Reasons 1)
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) 3) 4) Angle Addition Postulate 5) 6)
Statement #1 is the given information and Reason #1 is Given. Statement #2 says something is equal to itself, so Reason #2 is Reflexive Property. Since Statement #3 adds a value to both sides, Reason #3 is Addition Property. Use Line 4 to build Line 5. Statement #5 sets the two measures equal to each other, so 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 and Reason #5 is Substitution Property. Statement #6 is what we are trying to prove and Reason #6 is Definition of Congruent Angles, since we said their measures are equal. Answer: Statements 1) 𝑚∠1 = 𝑚∠3
Reasons 1) Given
2) 𝑚∠2 = 𝑚∠2 3) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5) 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 6) ∠𝐴𝐸𝐶 ≅ ∠𝐷𝐸𝐵
2) Reflexive Property 3) Addition Property 4) Angle Addition Postulate 5) Substitution Property 6) Definition of Congruent ∠s 17
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7. Fill in the missing statements and reasons for the proof below. Given: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷
̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ Prove: 𝐴𝐶 Statements 1)
Reasons 1)
2) 𝐴𝐵 = 𝐶𝐷 3) 4) ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 6)
2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6)
In the first line, Statement #1 is the given information and Reason #1 is Given. Since Reason #3 is Reflexive Property, Statement #3 states something is equal to itself. Use Line #5 to build Statement #4. Statement #5 uses two segments added together. Statement #6 is what we are trying to prove and Reason #6 is Substitution Property, since we just said the segments are equal. Answer: Statements 1) ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷 2) 𝐴𝐵 = 𝐶𝐷 3) 𝐵𝐶 = 𝐵𝐶 4) 𝐴𝐵 + 𝐵𝐶 = 𝐶𝐷 + 𝐵𝐶 ̅̅̅̅ 5) ̅̅̅̅ 𝐴𝐵 + ̅̅̅̅ 𝐵𝐶 = 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ 𝐶𝐷 + ̅̅̅̅ 𝐵𝐶 = 𝐵𝐷 ̅̅̅̅ ≅ 𝐵𝐷 ̅̅̅̅ 6) 𝐴𝐶
Reasons 1) Given 2) Definition of Congruent Segments 3) Reflexive Property 4) Addition Property 5) Segment Addition Postulate 6) Substitution Property
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8. Fill in the statements and reasons for the proof below. (Hint: This proof is similar to question 4.) Given:
Prove: 𝑥 = 40 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
Statement #1 is the given information, and Reason #1 is Given. In the second line, Statement #2 is the equation using the definition of supplementary ∠s, so Reason #2 is Definition of Supplementary ∠s. In the third line, Substitute algebraic expressions from the figure into the equation, so Reason #3 is Substitution. In Line #4 remove the parentheses using the Distributive Property. In the fifth line, subtract 20 from both sides, so Reason #5 is Subtraction Property. In the sixth line, the Statement is what we are proving and Reason #6 is Division Property, because the last equation was changed by dividing both sides by 4. Answer: Statements 1) ∠𝐶𝐷𝐸 and ∠𝐸𝐷𝐹 are supplementary
Reasons 1) Given
2) 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 3) 𝑥 + (3𝑥 + 20) = 180 4) 4𝑥 + 20 = 180 5) 4𝑥 = 160 6) 𝑥 = 40
2) Definition of Supplementary ∠s 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property 19
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9. Fill in the missing statements and reasons for the proof below. Given: 𝑋𝑌 = 42
Prove: 𝑥 = 5 Statements 1)
Reasons 1)
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 4) 5) 6) 7)
2) Segment Addition Postulate 3) 4) 5) 6) 7)
Statement #1 is the given information, and Reason #1 is Given. In the third line, Statement #3 is the equation using the algebraic expressions in the figure, so Reason #3 is Substitution Property. In Line #4 remove the parentheses using the Distributive Property. In the fifth line, combine like terms on the left side, so Reason #5 is Combine Like Terms. Line #6 results from subtracting 12 from both sides of the equation. The seventh line, the Statement is what we are proving and Reason #7 is Division Property, because the last equation was changed by dividing both sides by 6. Answer: Statements 1) 𝑋𝑌 = 42
Reasons 1) Given
2) 𝑋𝑍 + 𝑍𝑌 = 𝑋𝑌 3) 3(𝑥 + 4) + 3𝑛 = 42 4) 3𝑥 + 12 + 3𝑥 = 42 5) 6𝑥 + 12 = 42 6) 6𝑥 = 30 7) 𝑥 = 5
2) Segment Addition Postulate 3) Substitution Property 4) Distributive Property 5) Combine Like Terms 6) Subtraction Property 7) Division Property 20
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10. Fill in the missing statements for the proof below. (Hint: This proof involves vertical angles.) Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3 Statements 1)
Reasons 1) Given
2) 3) 4) 5)
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
Statement #1 is the given information, because Reason #1 is Given. In the second line, Statement #2 says ∠4 ≅ ∠2 because Reason #2 says vertical angles are congruent. This statement must use information already given in previous statements. In the third line, Statement #3 is ∠1 ≅ ∠2, using information in the previous statements and because Reason #3 is Transitive Property of Congruence. In Line #4 Statement #4 is ∠1 ≅ ∠3 because Reason #4 says vertical angles are congruent. This statement must also use information already given in previous statements. In the fifth line, the Statement is what we are proving and Reason #5 is Transitive Property of Congruence. Answer: Statements 1) ∠1 ≅ ∠4
Reasons 1) Given
2) ∠4 ≅ ∠2 3) ∠1 ≅ ∠2 4) ∠1 ≅ ∠3 5) ∠2 ≅ ∠3
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence 21
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11. Fill in the missing statements and reasons for the proof below. Given: ∠1 ≅ ∠3
Prove: ∠6 ≅ ∠4 Statements 1)
Reasons 1)
2) ∠3 ≅ ∠6 3) 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) 3) Transitive Property of Congruence 4) 5)
Statement #1 is the given information, because Reason #1 is Given. In the second line, Statement #2 says ∠3 ≅ ∠6 so Reason #2 says vertical angles are congruent. This statement must use information already given in previous statements. In the third line, Statement #3 is ∠1 ≅ ∠6, using information in the previous statements and because Reason #3 is Transitive Property of Congruence. In Line #4 Statement #4 is ∠1 ≅ ∠4 so Reason #4 says vertical angles are congruent. In the fifth line, the Statement is what we are proving and Reason #5 is Transitive Property of Congruence. Answer: Statements 1) ∠1 ≅ ∠3
Reasons 1) Given
2) ∠3 ≅ ∠6 3) ∠1 ≅ ∠6 4) ∠1 ≅ ∠4 5) ∠6 ≅ ∠4
2) Vertical ∠s are ≅ 3) Transitive Property of Congruence 4) Vertical ∠s are ≅ 5) Transitive Property of Congruence
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12. Fill in the missing statements and reasons for the proof below. (Hint: This proof is similar to question 9.) Given: 𝑄𝑆 = 42
Prove: 𝑥 = 13 Statements 1)
Reasons 1)
2) 3) 4) 5) 6)
2) 3) 4) 5) 6)
Statement #1 is the given information, and Reason #1 is Given. In Line #2, using the figure, write a statement adding two segments to make a larger segment, so Reason #2 is Segment Addition Postulate. In the third line, Statement #3 is the equation you can write using the algebraic expressions in the figure, so Reason #3 is Substitution Property. In Line #4 remove the parentheses using the Distributive Property. Line #5 results from subtracting 3 from both sides of the equation. The sixth line, the Statement is what we are proving and Reason #6 is Division Property, because the last equation was changed by dividing both sides by 3. Answer: Statements 1) 𝑄𝑆 = 42
Reasons 1) Given
2) 𝑄𝑅 + 𝑅𝑆 = 𝑄𝑆 3) (𝑥 + 3) + 2𝑥 = 42 4) 3𝑥 + 3 = 42 5) 3𝑥 = 39 6) 𝑥 = 13
2) Segment Addition Postulate 3) Substitution Property 4) Distributive Property 5) Subtraction Property 6) Division Property 23
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