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Long Term Memory Review Algebra 1 Answer Key Week 11

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Geometry: Answer Fundamental

Answer Key

This provides the answers and solutions for the Put Me in, Jitney! do boxes, organized by sections.

Taking the Brunt out of Proofs

  1. Yes
  2. Theorem 8.3: If two angles are complementary to the same angle, and then these two angles are congruent.

?A and ?B are complementary, and ?C and ?B are complementary.

Given: ?A and ?B are complementary, and ?C and ?B are complementary.

Prove: ?A ~= ?C.

Statements Reasons
1. ?A and ?B are complementary, and ?C and ?B are complementary. Given
2. m?A + m?B = 90 , thousand?C + thou?B = 90 Definition of complementary
3. m?A = 90 - m?B, m?C = xc - yard?B Subtraction holding of equality
iv. m?A = m?C Substitution (step iii)
5. ?A ~= ?C Definition of ~=

Proving Segment and Angle Relationships

  1. If E is betwixt D and F, and so DE = DF ? EF.

E is between D and F.

E is between D and F.

Given: Eastward is between D and F

Prove: DE = DF ? EF.

Statements Reasons
1. E is between D and F Given
ii. D, E, and F are collinear points, and Due east is on DF Definition of between
3. DE + EF = DF Segment Addition Postulate
iv. DE = DF ? EF Subtraction holding of equality

two. If ?BD divides ?ABC into two angles, ?ABD and ?DBC, then grand?ABC = thousand?ABC - m?DBC.

?BD divides ?ABC into two angles, ?ABD and ?DBC.

Given: ?BD divides ?ABC into two angles, ?ABD and ?DBC

Prove: m?ABD = thou?ABC - m?DBC.

Statements Reasons
1. ?BD divides ?ABC into two angles, ?ABD and ?DBC Given
2. m?ABD + thou?DBC = m?ABC Angle Improver Postulate
3. k?ABD = m?ABC - m?DBC Subtraction property of equality

3. The angle bisector of an angle is unique.

?ABC with two angle bisectors: ?BD and ?BE.

Given: ?ABC with two angle bisectors: ?BD and ?BE.

Prove: k?DBC = 0.

Statements Reasons
i. ?BD and ?BE bisect ?ABC Given
2. ?ABC ~= ?DBC and ?ABE ~= ?EBC Definition of angel bisector
3. 1000?ABD = m?DBC and m?ABE ~= m?EBC Definition of ~=
four. g?ABD + m?DBE + m?EBC = grand?ABC Angle Add-on Postulate
5. m?ABD + g?DBC = m?ABC and yard?ABE + m?EBC = m?ABC Angle Addition Postulate
6. 2m?ABD = grand?ABC and 2m?EBC = thousand?ABC Commutation (steps 3 and v)
7. m?ABD = m?ABC/ii and m?EBC = m?ABC/2 Algebra
8. yard?ABC/2 + m?DBE + m?ABC/2 = m?ABC Substitution (steps 4 and 7)
9. m?ABC + g?DBE = g?ABC Algebra
10. grand?DBE = 0 Subtraction property of equality

4. The supplement of a right bending is a right angle.

?A and ?B are supplementary angles, and ?A is a right angle.

Given: ?A and ?B are supplementary angles, and ?A is a right angle.

Evidence: ?B is a right angle.

Statements Reasons
one. ?A and ?B are supplementary angles, and ?A is a correct angle Given
2. m?A + m?B = 180 Definition of supplementary angles
three. k?A = 90 Definition of right bending
4. 90 + m?B = 180 Substitution (steps ii and 3)
5. grand?B = 90 Algebra
6. ?B is a right angle Definition of right angle

Proving Relationships Between Lines

  1. m?6 = 105 , grand?8 = 75
  2. Theorem x.3: If ii parallel lines are cut by a transversal, then the alternating exterior angles are congruent.

l ? ? m cut past a transversal t.

Given: fifty ? ? m cutting by a transversal t.

Testify: ?1 ~= ?3.

Statements Reasons
i. l ? ? m cutting by a transversal t Given
2. ?1 and ?2 are vertical angles Definition of vertical angles
three. ?ii and ?three are corresponding angles Definition of corresponding angles
4. ?2 ~= ?3 Postulate 10.1
five. ?1 ~= ?2 Theorem 8.1
6. ?1 ~= ?3 Transitive holding of three.

3. Theorem 10.5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles.

l ? ? k cut by a transversal t.

Given: fifty ? ? grand cut by a transversal t.

Prove: ?one and ?iii are supplementary.

Statement Reasons
1. l ? ? 1000 cutting by a transversal t Given
2. ?1 and ?2 are supplementary angles, and m?ane + m?two = 180 Definition of supplementary angles
3. ?2 and ?three are corresponding angles Definition of corresponding angles
4. ?2 ~= ?iii Postulate ten.1
5. thousand?2 ~= m?3 Definition of ~=
half-dozen. m?1 + m?3 = 180 Substitution (steps two and five)
7. ?1 and ?3 are supplementary Definition of supplementary

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4. Theorem 10.9: If two lines are cutting by a transversal so that the alternate outside angles are congruent, then these lines are parallel.

Lines l and m are cut by a transversal t.

Lines fifty and m are cutting by a transversal t.

Given: Lines fifty and yard are cut by a transversal t, with ?i ~= ?3.

Prove: l ? ? m.

Argument Reasons
1. Lines l and 1000 are cut by a transversal t, with ?1 ~= ?3 Given
2. ?1 and ?2 are vertical angles Definition of vertical angles
3. ?1 ~= ?two Theorem viii.1
4. ?2 ~= ?iii Transitive belongings of ~=.
5. ?2 and ?3 are corresponding angles Definition of corresponding angles
half-dozen. l ? ? k Theorem 10.7

five. Theorem 10.11: If two lines are cutting by a transversal and so that the exterior angles on the same side of the transversal are supplementary, and so these lines are parallel.

Lines l and m are cut by a t transversal t.

Lines fifty and m are cut by a t transversal t.

Given: Lines fifty and chiliad are cut past a transversal t, ?ane and ?3 are supplementary angles.

Evidence: l ? ? thousand.

Statement Reasons
1. Lines l and m are cut past a transversal t, and ?1 are ?three supplementary angles Given
2. ?2 and ?1 are supplementary angles Definition of supplementary angles
3. ?3 ~= ?ii Case two
4. ?iii and ?2 are corresponding angles Definition of corresponding angles
five. l ? ? grand Theorem 10.seven

Two'southward Company. Iii's a Triangle

  1. An isosceles obtuse triangle
  2. The acute angles of a right triangle are complementary.

?ABC is a right triangle.

Given: ?ABC is a right triangle, and ?B is a correct angle.

Prove: ?A and ?C are complementary angles.

Argument Reasons
1. ?ABC is a correct triangle, and ?B is a correct angle Given
ii. grand?B = 90 Definition of right angle
3. grand?A + k?B + thousand?C = 180 Theorem 11.1
four. yard?A + ninety + 1000?C = 180 Substitution (steps 2 and 3)
5. one thousand?A + m?C = ninety Algebra
half dozen. ?A and ?C are complementary angles Definition of complementary angles

3. Theorem 11.3: The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.

?ABC with outside angle ?BCD.

Argument Reasons
1. ?ABC with exterior angle ?BCD Given
2. ?DCA is a straight angle, and m?DCA = 180 Definition of direct angle
iii. m?BCA + m?BCD = m?DCA Angle Improver Postulate
iv. k?BCA + m?BCD = 180 Exchange (steps two and 3)
5. m?BAC + m?ABC + g?BCA = 180 Theorem xi.i
half dozen. chiliad?BAC + 1000?ABC + m?BCA = grand?BCA + grand?BCD Substitution (steps four and 5)
7. chiliad?BAC + m?ABC = m?BCD Subtraction property of equality

iv. 12 units2

5. xxx units2

half-dozen. No, a triangle with these side lengths would violate the triangle inequality.

Coinciding Triangles

one. Reflexive belongings: ?ABC ~= ?ABC.

Symmetric property: If ?ABC ~= ?DEF, then ?DEF ~= ?ABC.

Transitive property: If ?ABC ~= ?DEF and ?DEF ~= ?RST, then ?ABC ~= ?RST.

two. Proof: If AC ~= CD and ?ACB ~= ?DCB as shown in Effigy 12.5, then ?ACB ~= ?DCB.

Statement Reasons
one. Air conditioning ~= CD and ?ACB ~= ?DCB Given
ii. BC ~= BC Reflexive holding of ~=
three. ?ACB ~= ?DCB SAS Postulate

3. If CB ? Advertising and ?ACB ~= ?DCB, as shown in Effigy 12.8, then ?ACB ~= ?DCB.

Statement Reasons
1. CB ? AD and ?ACB ~= ?DCB Given
2. ?ABC and ?DBC are right angles Definition of ?
3. 1000?ABC = ninety and m?DBC = ninety Definition of right angles
iv. m?ABC = 1000?DBC Exchange (stride 3)
5. ?ABC ~= ?DBC Definition of ~=
half dozen. BC ~= BC Reflexive property of ~=
vii. ?ACB ~= ?DCB ASA Postulate

4. If CB ? AD and ?CAB ~= ?CDB, as shown in Figure 12.10, and so ?ACB~= ?DCB.

Argument Reasons
one. CB ? AD and ?CAB ~= ?CDB Given
2. ?ABC and ?DBC are right angles Definition of ?
iii. m?ABC = 90 and m?DBC = 90 Definition of right angles
four. m?ABC = chiliad?DBC Commutation (footstep 3)
5. ?ABC ~= ?DBC Definition of ~=
vi. BC ~= BC Reflexive property of ~=
7. ?ACB ~= ?DCB AAS Theorem

5. If CB ? AD and Air conditioning ~= CD, as shown in Figure 12.12, and so ?ACB ~= ?DCB.

Argument Reasons
1. CB ? Advertizing and AC ~= CD Given
2. ?ABC and ?DBC are right triangles Definition of right triangle
3. BC ~= BC Reflexive belongings of ~=
4. ?ACB ~= ?DCB HL Theorem for right triangles

6. If ?P ~= ?R and M is the midpoint of PR, equally shown in Figure 12.17, then ?N ~= ?Q.

Argument Reasons
1. ?P ~= ?R and M is the midpoint of PR Given
2. PM ~= MR Definition of midpoint
three. ?NMP and ?RMQ are vertical angles Definition of vertical angles
4. ?NMP ~= ?RMQ Theorem 8.1
five. ?PMN ~= RMQ ASA Postulate
half dozen. ?N ~= ?Q CPOCTAC

Smiliar Triangles

  1. ten = 11
  2. x = 12
  3. 40 and 140
  4. If ?A ~= ?D as shown in Figure 13.6, then BC/AB = CE/DE.
Statement Reasons
1. ?A ~= ?D Given
ii. ?BCA and ?DCE are vertical angles Definition of vertical angles
3. ?BCA ~= ?DCE Theorem viii.1
4. ?ACB ~ ?DCE AA Similarity Theorem
5. BC/AB = CE/DE CSSTAP

5. 150 feet.

Opening Doors with Similar Triangles

  1. If a line is parallel to i side of a triangle and passes through the midpoint of a second side, and then information technology volition pass through the midpoint of the third side.

DE ? ? AC and D is the midpoint of AB.

Given: DE ? ? AC and D is the midpoint of AB.

Prove: E is the midpoint of BC.

Statement Reasons
one. DE ? ? Air-conditioning and D is the midpoint of AB. Given
two. DE ? ? Air conditioning and is cut by transversal ?AB Definition of transversal
3. ?BDE and ?BAC are corresponding angles Definition of corresponding angles
4. ?BDE ~= ?BAC Postulate x.1
5. ?B ~= ?B Reflexive belongings of ~=
6. ?ABC ~ ?DBE AA Similarity Theorem
7. DB/AB = Be/BC CSSTAP
eight. DB = AB/ii Theorem 9.1
9. DB/AB = i/ii Algebra
10. 1/2 = Exist/BC Substitution (steps vii and 9)
11. BC = 2BE Algebra
12. BE + EC = BC Segment Addition Postulate
13. BE + EC = 2BE Substitution (steps 11 and 12)
14. EC = Be Algebra
15. E is the midpoint of BC Definition of midpoint

2. Air conditioning = 4?iii , AB = 8? , RS = 16, RT = viii?3

3. Air conditioning = 4?ii , BC = 4?2

Putting Quadrilaterals in the Forefront

  1. Advertizement = 63, BC = 27, RS = 45
  2. AX, CZ, and DY

Trapezoid ABCD with its XB CY four altitudes shown.

Trapezoid ABCD with its XB CY four altitudes shown.

three. Theorem 15.5: In a kite, i pair of opposite angles is congruent.

Kite ABCD.

Kite ABCD.

Given: Kite ABCD.

Prove: ?B ~= ?D.

Statement Reasons
1. ABCD is a kite Given
2. AB ~= AD and BC ~= DC Definition of a kite
3. AC ~= Air conditioning Reflexive property of ~=
4. ?ABC ~= ?ADC SSS Postulate
5. ?B ~= ?D CPOCTAC

4. Theorem 15.6: The diagonals of a kite are perpendicular, and the diagonal opposite the congruent angles bisects the other diagonal.

Kite ABCD.

Kite ABCD.

Given: Kite ABCD.

Prove: BD ? AC and BM ~= MD.

Argument Reasons
1. ABCD is a kite Given
two. AB ~= Advert and BC ~= DC Definition of a kite
3. Air-conditioning ~= Ac Reflexive property of ~=
4. ?ABC ~= ?ADC SSS Postulate
v. ?BAC ~= ?DAC CPOCTAC
6. AM ~= AM Reflexive property of ~=
7. ?ABM ~= ?ADM SAS Postulate
eight. BM ~= Medico CPOCTAC
9. ?BMA ~= ?DMA CPOCTAC
ten. m?BMA = m?DMA Definition of ~=
11. ?MBD is a straight angle, and m?BMD = 180 Definition of straight angle
12. k?BMA + m?DMA = g?BMD Angle Addition Postulate
thirteen. thou?BMA + m?DMA = 180 Substitution (steps 9 and 10)
14. 2m?BMA = 180 Commutation (steps 9 and 12)
xv. m?BMA = xc Algebra
16. ?BMA is a right bending Definition of right bending
17. BD ? Air conditioning Definition of ?

5. Theorem xv.9: Contrary angles of a parallelogram are coinciding.

Parallelogram ABCD.

Given: Parallelogram ABCD.

Bear witness: ?ABC ~= ?ADC.

Argument Reasons
1. Parallelogram ABCD has diagonal AC. Given
two. ?ABC ~= ?CDA Theorem xv.7
3. ?ABC ~= ?ADC CPOCTAC

6. 144 unitsii

7. 180 unitstwo

eight. Kite ABCD has area 48 unitstwo.

Parallelogram ABCD has surface area 150 units2.

Rectangle ABCD has area 104 units2.

Rhombus ABCD has area 35/2 unitsii.

Anatomy of a Circle

  1. Circumference: 20? anxiety, length of ?RST = 155/xviii? feet
  2. 9? feet2
  3. 15? feetii
  4. 28

The Unit of measurement Circumvolve and Trigonometry

  1. 3/?34 = 3?34/34
  2. 1/?3 = ?3/iii
  3. tangent ratio = ?40/three, sine ratio = ?40/vii
  4. tangent ratio = 5/?56 = v?56/56, cosine ratio = ?56/9

Excerpted from The Complete Idiot'southward Guide to Geometry 2004 past Denise Szecsei, Ph.D.. All rights reserved including the correct of reproduction in whole or in part in any form. Used by system with Blastoff Books, a member of Penguin Group (Us) Inc.

To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can as well purchase this book at Amazon.com and Barnes & Noble.

  • Geometry: Using and Proving Angle Supplements

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Source: https://www.infoplease.com/math-science/mathematics/geometry/geometry-answer-key

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