Proofs Involving Triangles (Grade 10)

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Proofs Involving Triangles

1. 
Given the triangle below, prove that the sum of the interior angles is 180°.

Equilateral Triangle ABC v1

[math] " Statement " [/math][math] " Reason "[/math]
[math]1. "Draw a line though point B parallel to "bar{AC} "[/math][math]1. [/math]
[math]2. "Label new line " bar{DE}, "where D lies to the left,"[/math] [math]\ \ \ " and E to the right, of B"[/math][math]2. "Infinite points on a line"[/math]
[math]3. m ang DBA + m ang ABC + m ang CBE = 180deg[/math][math]3. [/math]
[math]4. ang DBA ~= ang BAC[/math][math]4. "Alternate interior angles"[/math]
[math]5. [/math][math]5. "Alternate interior angles"[/math]
[math]6. m ang DBA = m ang BAC[/math][math]6. "Definition of congruent angles"[/math]
[math]7. m ang EBC = m ang BCA[/math][math]7. "Definition of congruent angles"[/math]
[math]8. m ang BAC + m ang ABC + m ang BCA = 180 deg[/math][math]8. [/math]
A. 
What is the missing reason in step 1?
  1. Two lines intersect in a point
  2. Parallel Postulate equivalent
  3. There exist an infinite number of lines in a plane
  4. Skew lines do not intersect
B. 
What is the missing reason in step 3?
  1. Definition supplementary angles
  2. Definition perpendicular lines
  3. Definition vertical angles
  4. Angles on a straight line add up to 180°
C. 
What is the missing statement in step 5?
  1. [math] ang EBC ~= ang BCA[/math]
  2. [math] ang ABC ~= ang BCA[/math]
  3. [math] ang EBA ~= ang ACB[/math]
  4. [math]ang DBC ~= ang CAB[/math]
D. 
What is the missing reason in step 8?
  1. Sum of angles in a triangle is 180°
  2. Summation Property
  3. Definition of supplementary angles
  4. Substitution Property of Equality
2. 
Given triangle ABC below, where [math]bar{AB} ~= bar{BC}[/math], prove that [math]ang BAC ~= ang BCA[/math].

Isosceles Triangle Height v1

[math] " Statement " [/math][math] " Reason "[/math]
[math]1. "Draw a line though point B perpendicular to "bar{AC} \ \ \ "[/math][math]1. [/math]
[math]2. "Label point of intersection D"[/math][math]2. "Two lines intersect at a point"[/math]
[math]3. angBDA, angBDC " are right angles"[/math][math]3. [/math]
[math]4.triangle ABD, triangle CBD " are right triangles"[/math][math]4. "Definition right triangles"[/math]
[math]5. bar{AB} ~= bar{BC}[/math][math]5. [/math]
[math]6. [/math][math]6. "Reflexive Property"[/math]
[math]7. [/math][math]7. "HL"[/math]
[math]8. ang BAC ~= ang BCA[/math][math]8. [/math]
A. 
What is the missing reason in step 1?
  1. Through any point and a line, there exists one perpendicular line
  2. Assume from diagram
  3. Given
  4. Two points define a line
B. 
What is the missing reason in step 3?
  1. Assume from diagram
  2. Given
  3. Definition of perpendicular lines
  4. Definition supplementary angles
C. 
What is the missing reason in step 5?
  1. Definition of isosceles triangle
  2. Given
  3. Assume from diagram
  4. Sides intersecting an altitude of a triangle are congruent
D. 
What is the missing statement in step 6?
  1. [math]bar{BD}~=bar{BD}[/math]
  2. [math]bar{AC}~=bar{AC}[/math]
  3. [math]bar{BC}~=bar{BC}[/math]
  4. [math]bar{AB}~=bar{AB}[/math]
E. 
What is the missing statement in step 7?
  1. [math]triangle ABD ~= triangle BDC [/math]
  2. [math]triangle ABD ~= triangle DBC [/math]
  3. [math] triangle ABD ~= triangle CDB [/math]
  4. [math] triangle ABD ~= triangle CBD [/math]
F. 
What is the missing reason in step 8?
  1. Corresponding angles of congruent triangles are congruent
  2. Corresponding angles of similar triangles are congruent
  3. AAA
  4. Definition of congruent angles

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