Proofs With Lines and Angles (Grade 10)

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Proofs With Lines and Angles

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].

Parallel Lines - Skew Transversal

[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

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].

Vertical Angle #2

[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]

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].

Parallel Lines - Skew Transversal

[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

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].

Segment ABC

[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]

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]

What is the missing reason in step 9?

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.

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).
Parallel Lines - Skew Transversal

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].

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].

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.

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).