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Proofs Involving Circles (Grade 10)

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Proofs Involving Circles

1. 
Given the circle pictured below with diameter [math]bar{AD}[/math] (not drawn), with [math]bar{AB}[/math] tangent to the circle, and line [math]bar{AC}[/math] also not drawn. Prove that [math]triangle ABD \ ~ triangle CAD [/math].

Circle with Secant CD and Tangent AB

[math] " Statement " [/math][math] " Reason "[/math]
[math]1. "Draw lines " bar(AD) \ & \ bar{AC}[/math][math]1. "Two points create a line"[/math]
[math]2. bar(AD) \ \ "is a diameter"[/math][math]2. "Given"[/math]
[math]3. [/math][math]3. "Inscribed angle subtended by a diameter is a right angle"[/math]
[math]4. bar(AB) \ \ "is tangent to the circle"[/math][math]4. "Given"[/math]
[math]5. bar{AB} _|_ bar{AD}[/math][math]5. [/math]
[math]6. ang BAD \ \ "is a right angle"[/math][math]6. "Definition of perpendicular lines"[/math]
[math]7. [/math][math]7. "All right angles are congruent"[/math]
[math]8. ang ADC ~= ang ADB[/math][math]8. "Reflexive property"[/math]
[math]9. triangle ABD \ ~ triangle CAD[/math][math]9. [/math]
A. 
What is the missing statement in step 3?
  1. [math]ang ACD \ \ "is a right angle"[/math]
  2. [math]ang BAD \ \ "is a right angle"[/math]
  3. [math]ang ABD \ \ "is a right angle"[/math]
  4. [math]ang DAC \ \ "is a right angle"[/math]
B. 
What is the missing reason in step 5?
  1. Triangle ABD looks like a right triangle
  2. Tangent lines intersect all lines at a right angle
  3. A tangent line is perpendicular to the diameter
  4. The Pythagorean Theorem
C. 
What is the missing statement in step 7?
  1. [math] ang ADC ~= ang BAD [/math]
  2. [math]ang ACD ~= ang BAD [/math]
  3. [math] ang ADC ~= ang ABD [/math]
  4. [math] ang BAD ~= ang BAD [/math]
D. 
What is the missing reason in step 9?
  1. ASA
  2. AAS
  3. SSS
  4. AA
2. 
Given the circle below, where [math]m \stackrel{\frown}{EB} = m \stackrel{\frown}{DC}[/math] and with lines [math]bar{EC}, bar{BD}, & \ bar{ED}[/math] not drawn, prove that [math]triangle EBD ~= triangle DCE[/math].

Circle with Two Secants ABCDE

[math] " Statement " [/math][math] " Reason "[/math]
[math]1. "Draw lines " bar{EC}, bar{DB}, & \ bar{ED}[/math][math]1. "Two points define a line"[/math]
[math]2. [/math][math]2. "Given"[/math]
[math]3. m ang EDB = 1/2 m \stackrel{\frown}{EB}[/math][math]3. [/math]
[math]4. m ang CED = 1/2 m \stackrel{\frown}{CD}[/math][math]4. "Measure of inscribed angle is 1/2 the measure of intercepted arc"[/math]
[math]5. 1/2 m \stackrel{\frown}{EB} = 1/2 m \stackrel{\frown}{DC}[/math][math]5.
[/math]
[math]6. [/math][math]6. "Substitution Property of Equality"[/math]
[math]7. ang EDB ~= ang CED[/math][math]7. "Definition of congruent angles"[/math]
[math]8. ang EBD ~= ang ECD[/math][math]8. [/math]
[math]9. bar{ED} ~= bar{ED}[/math][math]9. "Reflexive Property"[/math]
[math]10. triangle EBD ~= triangle DCE[/math][math]10. [/math]
A. 
What is the missing statement in step 2?
  1. Draw center O
  2. [math]m \stackrel{\frown}{EB} = m \stackrel{\frown}{DC}[/math]
  3. [math]triangle EBD ~= triangle DCE[/math]
  4. [math]bar{EB} ~= bar{DC} [/math]
B. 
What is the missing reason in step 3?
  1. Measure of central angle is 1/2 the measure of intercepted arc
  2. Intersecting Secant Theorem
  3. Sum of the angles in a triangle is half of the sum of the non-overlapping central angles in a circle
  4. Measure of inscribed angle is 1/2 the measure of intercepted arc
C. 
What is the missing reason in step 5?
  1. Multiplicative Identity Property
  2. Multiplicative Inverse Property
  3. Multiplicative Property of Equality
  4. Closure Property of Multiplication
D. 
What is the missing statement in step 6?
  1. [math]m ang EDB = m ang CED [/math]
  2. [math]m ang EDB = 1/2 m ang CED[/math]
  3. [math]m ang CED = m \stackrel{\frown}{EB}[/math]
  4. [math]m stackrel{\frown}{EB} = 1/2 m \stackrel{\frown}{CD}[/math]
E. 
What is the missing reason in step 8?
  1. Implied by diagram
  2. Intersecting chords create congruent angles
  3. All inscribed angles of the same circle are congruent
  4. Inscribed angles intercepting the same arc are congruent
F. 
What is the missing reason in step 10?
  1. SSA
  2. AAS
  3. SSS
  4. AAA

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