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1 Study Island Copyright 2014 Edmentum - All rights reserved. Generation Date: 04/01/2014 Generated By: Cheryl Shelton Title: 10 th Grade Geometry Theorems 1. Given: g h Prove: 1 and 2 are supplementary 1. g h 1. Given m 1 = m 3 3. Definition of congruent angles 4. 2 and 3 are supplementary 5. m 3 + m 2 = m 1 + m 2 = Linear Pair Postulate 5. Definition of supplementary angles 6. Substitution property of equality

2 7. 1 and 2 are supplementary 7. Definition of supplementary angles A. Alternate Exterior Angles Theorem B. Alternate Interior Angles Theorem C. Definition of congruent angles D. Definition of linear pair 2. Proving the alternate exterior angle theorem. Given: f g and line d is a transversal. Prove: Line f is parallel to line g. Line d is a transversal given If parallel lines are cut by a transversal, then corresponding angles are congruent transitive property of congruence Which of the following statements and reasons completes the proof? A. 1 4 since vertical angles are congruent. B. 5 4 since alternate interior angles are congruent. C. 5 8 since vertical angles are congruent. D. 4 8 since corresponding angles are congruent.

3 3. In the triangle below, m F = 28, m FHG = 70, and m E = 82. Determine the missing reason to prove HG DE. m F = 28, m FHG = 70, m E = 82 given m F + m FHG + m HGF = m HGF = 180 substitution sum of the interior angles of a triangle equal 180 m HGF = 82 property of subtraction m HGF = m E transitive property of equality HG DE A. B. C. D. If two corresponding angles are congruent, then the third angle is opposite a set of parallel lines. If two triangles have one or more sets of congruent angles, then the lines containing the bases are parallel. If three angles of one triangle are congruent to three angles of another triangle, then the two triangles are congruent and the bases are parallel. If two lines are cut by a transversal to form congruent corresponding angles, then the two lines are parallel.

4 4. Given: Line f is tangent to circle C at point D, 1 2 Prove: CD f 1. Line f is tangent to circle C at point D, and 2 are a linear pair 1. Given 2. Definition of linear pair 3. 1 and 2 are supplementary 3. Linear Pair Postulate 4. m 1 + m 2 = m 1 = m 2 6. m 1 + m 1 = (m 1) = m 1 = is a right angle 4. Definition of supplementary angles 5. Definition of congruent angles 6. Substitution property of equality 7. Distributive property of equality 8. Division property of equality 9. Definition of right angle

5 10. CD f 10. A. Definition of perpendicular lines and segments B. Definition of vertical angles C. Definition of parallel lines and segments D. Definition of congruent angles 5. Given: 2 and 3 are a linear pair, 3 and 4 are a linear pair Prove: and 3 are a linear pair, 3 and 4 are a linear pair 2. 2 and 3 are supplementary, 3 and 4 are supplementary Given Congruent Supplements Theorem A. Transitive Property B. Congruent Complements Theorem

6 C. Right Angle Congruence Theorem D. Linear Pair Postulate 6. Proving the same-side interior angle theorem. Given: f g and line d is a transversal. Prove: m 3 + m 5 = Line f is parallel to line g. Line d is a transversal. given 2. m 3 + m 1 = 180 linear pair property m 3 + m 5 = 180 substitution property Which of the following statements and reasons completes the proof? A. m 5 + m 7 = 180 since they form a linear pair. B. m 3 = m 7 since corresponding angles are congruent. C. m 1 = m 4 since vertical angles are congruent. D. m 1 = m 5 since corresponding angles are congruent. 7.

7 Given: Point F is the center of circle F Prove: 2 (m 1 + m 3) = m 1. Point F is the center of circle F 1. Given 2. FG, FJ, and FH are radii of circle F 3. Definition of radius 2. FG FJ FH 3. Definition of radius , m 1 = m 2, m 3 = m 4 6. m 5 = (m 1 + m 2) m 6 = (m 3 + m 4) 7. m 5 = (m 1 + m 1) m 6 = (m 3 + m 3) 8. m 5 = (m 1) m 6 = (m 3) 9. m 7 = (m 5 + m 6) 10. m 7 = (180-2 (m 1)) - (180-2 (m 3)) 4. Definition of isosceles triangle 5. Definition of congruent angles Substitution property of equality 8. Distributive property of equality 9. Sum of angles about a point 10. Substitution property of equality 11. m 7 = 2 (m 1 + m 3) 11. Simplification 12. m = 2 (m 1 + m 3) 12. Definition of central angle

8 A. Definition of complementary angles B. Congruent Supplements Theorem C. Linear Pair Postulate D. Triangle Sum Theorem 8. Given isosceles trapezoid ABCD with DF CE and BAF CEB, determine the missing reason to prove AFD CEB. ABCD is an isosceles trapezoid given C D The base angles of an isosceles trapezoid are congruent. DF CE given AB DC definition of isosceles trapezoid AFD BAF BAF CEB given AFD CEB transitive property AFD CEB ASA A. alternate interior angles conjecture B. vertical angle conjecture C. corresponding angles conjecture D. alternate exterior angles conjecture 9.

9 Given: g h Prove: g h 1. Given m 2 = m 3 3. Definition of congruent angles Vertical Angles Theorem 5. m 1 = m 2 5. Definition of congruent angles 6. m 1 = m 3 6. Transitive property of equality Definition of congruent angles A. Definition of complementary angles B. Consecutive Interior Angles Theorem C. Corresponding Angles Postulate D. Definition of parallel lines 10. Given point D is on the bisector of CAB and AD is an altitude of CAB, determine the missing reason to prove CAD BAD. AD AD reflexive property AD is an altitude of CAB given AD CB definition of altitude CDA BDA perpendicular angles are congruent point D is on the bisector of CAB given

10 DAC DAB CAD BAD ASA A. definition of isosceles triangle B. definition of angle bisector C. reflexive property D. definition of an altitude 11. Given: 1 and 2 are complementary, 3 and 4 are complementary, 1 4 Prove: and 2 are complements, 3 and 4 are complements, m 1 + m 2 = 90 m 3 + m 4 = Given 2. Definition of complementary angles 3. m 1 + m 2 = m 3 + m 4 3. Transitive property of equality 4. m 1 = m m 1 + m 2 = m 3 + m 1 6. m 2 = m Substitution property of equality 6. Subtraction property of equality 7. Definition of congruent angles

11 A. Definition of congruent angles B. Transitive Property of Angle Congruence C. Definition of complementary angles D. Right Angle Congruence Theorem 12. Proving the vertical angle theorem. Given: 1 and 2 are vertical angles. Prove: and 2 are vertical angles given 2. m 1 + m 3 = 180 m 2 + m 3 = 180 linear pair property 3. m 1 + m 3 = m 2 + m 3 substitution property 4. m 1 = m 2 A. subtraction property B. reflexive property C. addition property D. transitive property 13. Given: 1 and 2 are supplementary, 2 and 3 are supplementary Prove: 1 3

12 1. 1 and 2 are supplements, 2 and 3 are supplements 2. m 1 + m 2 = 180 m 2 + m 3 = m 1 + m 2 = m 2 + m 3 4. m 1 = m Given Transitive property of equality 4. Subtraction property of equality 5. Definition of congruent angles A. Definition of supplementary angles B. Definition of complementary angles C. Transitive property of equality D. Substitution property of equality 14. Given: 1 2, 1 and 2 are a linear pair Prove: p q 1. 1 and 2 are a linear pair 1. Given 2. 1 and 2 are 2. Linear Pair Postulate

13 supplementary 3. m 1 + m 2 = Definition of supplementary angles 4. Given 5. m 1 = m 2 6. m 1 + m 1 = Definition of congruent angles 6. Substitution property of equality 7. 2 (m 1) = Distributive property 8. m 1 = Division property of equality 9. 1 is a right angle 9. Definition of a right angle 10. p q 10. A. Definition of parallel lines B. Definition of perpendicular lines C. Linear Pair Postulate D. Congruent Supplements Theorem 15. Given m M = 23 and M P, determine the missing reason to prove MNP is obtuse. m M = 23 and M P given m P = 23 transitive property of equality m M + m N + m P = m N + 23 = 180 substitution m N = 134 addition property of equality N is obtuse definition of obtuse angle

14 MNP is obtuse definition of obtuse triangle A. The square of the length of the third side of a triangle is equal to the sum of the squares of the other two sides. B. The sum of the interior angles of a triangle equals 180. C. Correpsonding angles sum to 180. D. Adjacent angles in a triangle are congruent. 16. Given: ABC with exterior 4 Prove: m 4 = m 1 + m 2 1. ABC with exterior 4 Given 2. m 4 + m 3 = 180 Linear Pair 3. m 1 + m 2 + m 3 = 180 Triangle Sum Theorem 4. Substitution Property 5. m 4 = m 1 + m 2 Subtraction Property A. m 4 + m 3 = m 1 + m 2 + m 4 B. m 4 + m 3 = m 1 + m 2 + m 3 C. m 1 + m 3 + m 4 = 180 D. m 1 + m 2 + m 4 = 180

15 17. Given m FAH = m CBD + m CDB, determine the missing statement in the proof below. m FAH = m CBD + m CDB m DCE = m CBD + m CDB m FAH = m DCE given The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. transitive property of equality converse of the alternate exterior angle conjecture A. B. m FAH = m DCB C. CBD is isosceles D. 18. Proving the congruent supplements theorem. Given: 1 3, 1 and 2 are supplementary, and 3 and 4 are supplementary. Prove: 2 4

16 1. m 1 + m 2 = 180 m 3 + m 4 = m 1 + m 2 = m 3 + m 4 definition of supplementary angles substitution property m 1 = m 3 given 4. m 1 + m 2 = m 1 + m 4 5. m 2 = m subtraction property A. addition property B. substitution property C. reflexive property D. symmetric property 19. Given: CB Prove: m BCA + m CAB + m ABC = CB Given 2. m DAC + m CAB + m BAE = 180 Angle Addition

17 3. m DAC = m BCA m BAE = m ABC m BCA + m CAB + m ABC = 180 Substitution Property Alternate Exterior Angles; A. Alternate Exterior Angles B. C. D. Alternate Exterior Angles; Alternate Interior Angles Alternate Interior Angles; Alternate Exterior Angles Alternate Interior Angles; Alternate Interior Angles 20. Given: 3 and 4 are a linear pair, 1 and 3 are supplementary Prove: g h 1. 3 and 4 are a linear pair, 1. Given

18 1 and 3 are supplementary 2. 3 and 4 are supplementary 2. Linear Pair Postulate 3. m 1 + m 3 = 180 m 3 + m 4 = m 1 + m 3 = m 3 + m 4 5. m 1 = m g h 3. Definition of supplementary angles 4. Transitive property of equality 5. Subtraction property of equality 6. Definition of congruent angles 7. A. Alternate Interior Angles Theorem B. Alternate Exterior Angles Theorem C. Linear Pair Postulate D. Corresponding Angles Postulate 21. In the triangles below, NP RP, and PM PQ. Determine the missing reason to prove that MNP QRP. Statement NP RP, PM PQ Reason given m NPM = m RPQ MNP QRP SAS

19 A. An angle is congruent to itself. B. Alternate exterior angles are congruent. C. Vertical angles are congruent. D. Alternate interior angles are congruent. 22. Proving the alternate interior angle theorem. Given: f g and line d is a transversal. Prove: Line f is parallel to line g. Line d is a transversal given If parallel lines are cut by a transversal, then corresponding angles are congruent transitive property of congruence Which of the following statements and reasons completes the proof? A. B. C. D. 1 8 since alternate exterior angles are congruent. 5 8 since vertical angles are congruent. 1 4 since vertical angles are congruent. 4 8 since corresponding angles are congruent.

20 23. In scalene triangle WXY below, WYA BAY. Determine the missing reason to prove AXB ~ WXY. WYA BAY given AB WY If two parallel lines are cut by a transversal, then alternate interior angles are congruent. XWY XYW XAB XBA X AXB ~ X WXY reflexive property If three angles of one triangle are congruent to three angles of another triangle, then the two triangles are similar. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. A. B. If two angles are congruent to the same angle, then they are congruent. C. If two parallel lines are cut by a transversal, then corresponding angles are congruent. D. If two angles are vertical angles, then they are congruent.

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