Proofs With Lines and Angles (Grade 10)
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Proofs With Lines and Angles
1.
Choose the correct missing reason from step 7.
Given the parallel lines [math]k[/math] and [math]m[/math], and transversal [math]t[/math], prove that [math]ang 3 ~= ang 5[/math].

Given the parallel lines [math]k[/math] and [math]m[/math], and transversal [math]t[/math], prove that [math]ang 3 ~= ang 5[/math].

[math] " Statement " [/math] | [math] " Reason "[/math] |
[math]1. m ang 2 + m ang 3 = 180°[/math] | [math]1. "Linear pairs of angels are supplementary"[/math] |
[math]2. m ang 5 + m ang 6 = 180°[/math] | [math]2. "Linear pairs of angles are supplementary"[/math] |
[math]3. m ang 5 + m ang 6 = m ang 2 + m ang 3[/math] | [math]3. "Substitution property of equality"[/math] |
[math]4. k " || " m [/math] | [math]4. "Given"[/math] |
[math]5. ang 6 ~= ang 2[/math] | [math]5. "Corresponding angles are congruent"[/math] |
[math]6. m ang 6 = m ang 2[/math] | [math]6. "Definition of congruent angles"[/math] |
[math]7. m ang 5 + m ang 6 = m ang 6 + m ang 3[/math] | [math]7. \ \ \ \ [/math] |
[math]8. m ang 5 = m ang 3[/math] | [math]8. "Subtraction property of equality"[/math] |
[math]9. ang 5 ~= ang 3[/math] | [math]9. "Definition of congruent angles"[/math] |
- Same side interior angles are supplementary
- Supplementary angles theorem
- Substitution property of equality
- Addition property of equality
2.
Choose the correct missing statement from step 3.
Given the two intersecting lines and the four resulting angles, prove that [math]ang2 ~=ang 4[/math].

Given the two intersecting lines and the four resulting angles, prove that [math]ang2 ~=ang 4[/math].

[math] " Statement " [/math] | [math] " Reason "[/math] |
[math]1. m ang 1 + m ang 2 = 180°[/math] | [math]1. "Linear pairs of angels are supplementary"[/math] |
[math]2. m ang 4 + m ang 1 = 180°[/math] | [math]2. "Linear pairs of angles are supplementary"[/math] |
[math]3. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]3. "Substitution property of equality"[/math] |
[math]4. m ang 2 = m ang 4[/math] | [math]4. "Subtraction property of equality"[/math] |
[math]5. ang 2 ~= ang 4[/math] | [math]5. "Definition of congruent angles"[/math] |
- [math] m ang 2 + 180° = m ang 4[/math]
- [math]m ang 4+ m ang 2 = 180°[/math]
- [math]m ang 1 + m ang 2 - 180° = m ang 4 + m ang 1[/math]
- [math]m ang 1+ m ang 2 = m ang 4 + m ang 1[/math]
3.
Choose the correct missing reason from step 3.
Given the parallel lines [math]k[/math] and [math]m[/math], and transversal [math]t[/math], prove that [math]ang 1 ~= ang 7[/math].

Given the parallel lines [math]k[/math] and [math]m[/math], and transversal [math]t[/math], prove that [math]ang 1 ~= ang 7[/math].

[math] \ \ \ \ \ \ " Statement " \ \ \ \ \ \ [/math] | [math] " Reason "[/math] |
[math]1. k " || " m[/math] | [math]1. "Given"[/math] |
[math]2. ang 1 ~= ang 5[/math] | [math]2. "Corresponding angles are congruent"[/math] |
[math]3. ang5 ~= ang 7 [/math] | [math]3. [/math] |
[math]4. ang 1 ~= ang 7 [/math] | [math]4. "Substitution property of equality"[/math] |
- Vertical angles are congruent
- Given
- Same side interior angles are congruent
- Supplementary angles are congruent
4.
Given line segment [math]bar{AC}[/math] with midpoint [math]B[/math] and perpendicular bisector [math]l[/math] (not pictured), which intersects [math]bar{AC}[/math] at [math]B[/math], prove that any point on line [math]l[/math] is equidistant from points [math]A[/math] and [math]C[/math].


[math] " Statement " [/math] | [math] " Reason "[/math] |
[math]1. "Let " P " be an arbitrary point on line " [/math] [math] \ \ \ \ \ l " that does not lie on "bar{AC}[/math] | [math]1. "A line consists of an infinite number of points"[/math] |
[math]2. "Construct "bar{AP}" and "bar{CP}[/math] | [math]2. "Two points define a line segment"[/math] |
[math]3. [/math] | [math]3. "Reflexive property"[/math] |
[math]4. B " is the midpoint of " bar{AC}[/math] | [math]4. "Given" [/math] |
[math]5. bar{AB} ~= bar{BC} [/math] | [math]5. "Definition of midpoint"[/math] |
[math]6. l " is the perpendicular bisector of " bar{AC} [/math] | [math]6. "Given"[/math] |
[math]7. ang ABP " and " ang PBC " are right angles" [/math] | [math]7. "Definition of perpendicular"[/math] |
[math]8. ang ABP ~= ang PBC [/math] | [math]8. "All right angles are congruent"[/math] |
[math]9. Delta ABP ~= Delta CBP [/math] | [math]9. ""[/math] |
[math]10. bar{AP} ~= bar{PC} [/math] | [math]10. "Corresponding sides of congruent triangles " [/math] [math] \ \ \ \ " are congruent"[/math] |
[math]11. P " is equidistant from " A " and " C [/math] | [math]11. "Definition of equidistant"[/math] |
A.
What is the missing statement in step 3?
- [math]l ~= l [/math]
- [math]bar{BP} ~= bar{BP[/math]
- [math]bar{AB} ~= bar{BC} [/math]
- [math]bar{AC} ~= bar{AC}[/math]
B.
What is the missing reason in step 9?
- SSS
- ASA
- HL
- SAS
C.
What still needs to be done in order to complete the proof?
- Consider other points on [math]l[/math], showing the same thing holds true for these other points.
- Redraw the diagram, showing that the results are still true if [math]l[/math] is horizontal and [math]bar{AC}[/math] is vertical.
- State that, if the point on [math]l[/math] lies on [math]bar{AC}[/math], that it is equidistant from points [math]A[/math] and [math]B[/math], due to [math]l[/math] bisecting [math]bar{AC}[/math] (from given information).
- Nothing, it is complete.
5.
In the diagram given, lines [math]k[/math] and [math]m[/math] are parallel, and line [math]t[/math] is a transversal. Let point [math]A[/math] be the intersection of lines [math]k[/math] and [math]t[/math], and point [math]B[/math] be the intersection of lines [math]m[/math] and [math]t[/math] (points not labeled in the diagram). Also, let [math]h[/math] be the line parallel to lines [math]k[/math] and [math]m[/math] which is equidistant from these two lines (not pictured).

A.
If one were to perform a translation by vector [math]vec{AB}[/math], what would this vector look like, in terms of separate horizontal and vertical translations? (Assume that [math]m ang 4 < m ang 3[/math] for this purpose.)
- Translate down by the distance between lines [math]k[/math] and [math]m[/math] and translate left by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate up by the distance between lines [math]k[/math] and [math]m[/math] and translate left by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate up by the distance between lines [math]k[/math] and [math]m[/math] and translate right by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate down by the distance between lines [math]k[/math] and [math]m[/math] and translate right by the horizontal distance between [math]A[/math] and [math]B[/math].
B.
If the upper half plane (everything above line [math]h[/math], not pictured) is translated by the vector [math]vec{AB}[/math], which of the following is true? There may be more than one correct answer.
- [math]A[/math] maps to [math]B[/math].
- [math]k[/math] maps to [math]m[/math].
- [math]m[/math] maps to a new line, but parallel to itself.
- [math]t[/math] maps to [math]t[/math].
C.
Which of the following reasons ensures that lines [math]k[/math] and [math]m[/math] are coincident after the translation by [math]vec{AB}[/math], given that line [math]k[/math] now passes through point [math]B?[/math]
- Rigid transformations don't change size or shape.
- No rotations or reflections were used in the transformation, thus the slope of the line cannot change.
- Point [math]A[/math] maps to point [math]B[/math], and point [math]A[/math] was on line [math]k[/math], thus these two points define a line, which is [math]k[/math].
- The Parallel Postulate and the fact that the slope of line [math]k[/math] remains unchanged by translations.
D.
What is the reason we can conclude, including the information from parts A - C, that [math]m ang 1 = m ang 5,[/math] (and therefore, that [math]ang1 ~= ang 5[/math] by the definition of congruent angles).
- Substitution Property of Equality.
- Congruent Supplements Theorem.
- Rigid transformations preserve angle measure.
- Given by the diagram.
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