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°.


[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?
- Two lines intersect in a point
- Parallel Postulate equivalent
- There exist an infinite number of lines in a plane
- Skew lines do not intersect
B.
What is the missing reason in step 3?
- Definition supplementary angles
- Definition perpendicular lines
- Definition vertical angles
- Angles on a straight line add up to 180°
C.
What is the missing statement in step 5?
- [math] ang EBC ~= ang BCA[/math]
- [math] ang ABC ~= ang BCA[/math]
- [math] ang EBA ~= ang ACB[/math]
- [math]ang DBC ~= ang CAB[/math]
D.
What is the missing reason in step 8?
- Sum of angles in a triangle is 180°
- Summation Property
- Definition of supplementary angles
- Substitution Property of Equality
2.
Given triangle ABC below, where [math]bar{AB} ~= bar{BC}[/math], prove that [math]ang BAC ~= ang BCA[/math].


[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?
- Through any point and a line, there exists one perpendicular line
- Assume from diagram
- Given
- Two points define a line
B.
What is the missing reason in step 3?
- Assume from diagram
- Given
- Definition of perpendicular lines
- Definition supplementary angles
C.
What is the missing reason in step 5?
- Definition of isosceles triangle
- Given
- Assume from diagram
- Sides intersecting an altitude of a triangle are congruent
D.
What is the missing statement in step 6?
- [math]bar{BD}~=bar{BD}[/math]
- [math]bar{AC}~=bar{AC}[/math]
- [math]bar{BC}~=bar{BC}[/math]
- [math]bar{AB}~=bar{AB}[/math]
E.
What is the missing statement in step 7?
- [math]triangle ABD ~= triangle BDC [/math]
- [math]triangle ABD ~= triangle DBC [/math]
- [math] triangle ABD ~= triangle CDB [/math]
- [math] triangle ABD ~= triangle CBD [/math]
F.
What is the missing reason in step 8?
- Corresponding angles of congruent triangles are congruent
- Corresponding angles of similar triangles are congruent
- AAA
- Definition of congruent angles
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