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Undefined terms, segment and angle addition, midpoints; angle pairs; transversals and proving lines parallel.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn the building blocks of geometry: point, line and plane, which are taken as undefined, and the segments, rays and angles defined from them. You write your first geometric equations from segment addition, angle addition, midpoints, complementary angles and linear pairs. Then you study what happens when a transversal crosses parallel lines, which angle pairs are equal and which are supplementary, and how the converses let you prove that two lines are parallel.
You can plot and read a point on the coordinate plane, you can measure and add angles, and you can solve a one-step or two-step equation such as $5x + 30 = 180$. Geometry supplies the sentence; the algebra you already own turns it into a number.
Undefined term: a word geometry takes as given rather than defines — point, line and plane. Everything else is built from them.
Segment: the piece of a line between two points, written $AB$. Ray: starts at a point and goes on forever. Angle: two rays from a shared vertex.
Between: $B$ is between $A$ and $C$ when all three are on one line and $B$ is inside the segment $AC$.
Midpoint: the point that cuts a segment into two equal pieces.
Complementary: two angles adding to $90^\circ$. Supplementary: two adding to $180^\circ$.
Linear pair: two angles that share a ray and whose other rays make a straight line — always supplementary.
Point, line and plane are undefined terms: everything else is built from them. A segment is the part of a line between two points; a ray starts at a point and continues forever; an angle is two rays from a common vertex. Segment addition says that if $B$ is between $A$ and $C$ then $AB + BC = AC$, so $AB = 7$ and $AC = 19$ give $BC = 12$. Angle addition works the same way for a ray inside an angle: $\angle AOB = 35^\circ$ inside $\angle AOC = 80^\circ$ leaves $\angle BOC = 45^\circ$. The midpoint averages the coordinates: $(2, 5)$ and $(8, 1)$ give $(5, 3)$. Complementary angles add to $90^\circ$, supplementary to $180^\circ$, and a linear pair is supplementary, so $3x + (2x + 30) = 180$ gives $x = 30$.
Another way: picture
A line with $A$, $B$ and $C$ marked in that order, $AB = 7$ written under the first piece and $AC = 19$ spanning the whole: the missing piece is what is left, $12$.
Another way: story
Undefined terms are the words a dictionary cannot define without using them. You already know what a point is; what the axioms add is how points behave.
Adding when the question gives you the whole. If $AC$ is the whole and $AB$ is a part, then $BC = AC - AB$. Read which length is the whole before you pick an operation; drawing the three points in order settles it in two seconds.
Confusing complementary with supplementary. Ninety and one hundred and eighty are easy to swap. Complementary is the smaller pair, and a corner is $90^\circ$; supplementary is the straight line, $180^\circ$.
Reading a linear pair as two equal angles. A linear pair adds to $180^\circ$; it is only two right angles when the two expressions happen to come out equal. Set up the equation rather than assuming the halves.
A linear pair is supplementary, so $3x + (2x + 30) = 180$.
Turn the relationship into an equation before touching the numbers.
Collect: $5x + 30 = 180$, so $5x = 150$.
$x = 30$, and the two angles are $90^\circ$ and $90^\circ$.
Check: they add to $180$.
Average the $x$ values: $(2 + 8) \div 2 = 5$.
Average the $y$ values: $(5 + 1) \div 2 = 3$.
$M = (5, 3)$.
It should sit halfway along, and it does.
Segment addition: $9 + BC = 23$.
$BC = 23 - 9 = 14$.
Point $B$ lies between $A$ and $C$. $AB = 14$ and $AC = 21$. What is $BC$?
answer
Ray $OB$ lies inside angle $AOC$. Angle $AOB = 27^\circ$ and angle $AOC = 56^\circ$. What is angle $BOC$?
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$M$ is the midpoint of $AB$ with $A = (-5, -7)$ and $B = (5, 1)$. Fill in $M$.
(x, y)
Two angles form a linear pair. One measures $5x$ degrees and the other $4x + 45$ degrees. Find $x$.
answer
You have just used supplementary angles and linear pairs, and you can solve an equation with the unknown on both sides, such as $4x + 10 = 6x - 30$. Every question here is one of those two things applied to a picture with eight angles in it.
Transversal: a line that crosses two or more other lines.
Corresponding angles: the pair in matching positions at the two crossings — equal when the lines are parallel.
Alternate interior angles: on opposite sides of the transversal, between the two lines — equal. Alternate exterior angles: opposite sides, outside the two lines — also equal.
Same-side interior (co-interior) angles: same side of the transversal, between the lines — these add to $180^\circ$.
Converse: the theorem read backwards. Parallel lines give equal corresponding angles has the converse equal corresponding angles prove the lines parallel.
A transversal crossing two parallel lines makes eight angles in two families of four: within a family every angle is equal, and one from each family adds to $180^\circ$. Corresponding angles, in matching positions at each crossing, are equal, so $68^\circ$ pairs with $68^\circ$. Alternate interior angles, on opposite sides of the transversal between the lines, are equal, and so are alternate exterior angles. Same-side interior angles are supplementary: $112^\circ$ pairs with $68^\circ$. A transversal perpendicular to one of two parallel lines is perpendicular to the other. Each of these has a converse that runs the other way: if the alternate interior angles are equal, the two lines must be parallel — which is how parallelism gets proved rather than assumed.
Another way: picture
Two parallel lines cut by a slanted transversal, the eight angles shaded in two colours. Every angle of one colour is equal to every other of that colour, and one of each colour adds to $180^\circ$.
Another way: story
Slide the whole crossing at the top line down along the transversal until it reaches the bottom line. It lands exactly on the other crossing — which is why corresponding angles match.
Using the rules without the parallel marks. None of this holds for two lines that merely look parallel. Find the arrowheads on the diagram, or a statement in the question, before you use any of it.
Assuming every pair is equal. Half the pairs are equal and half are supplementary. A quick check: if one angle looks sharp and the other blunt, they add to $180^\circ$; if both look about the same, they are equal.
Mixing up the theorem and its converse. These lines are parallel, so the angles are equal is the theorem; these angles are equal, so the lines are parallel is the converse. You are given one and asked for the other, and saying which you used is half of a geometry proof.
Alternate exterior angles at parallel lines are equal, so $4x + 10 = 6x - 30$.
Name the pair first; the name chooses the equation.
Take $4x$ from both sides: $10 = 2x - 30$, so $2x = 40$.
$x = 20$, and each angle is $4(20) + 10 = 90^\circ$.
Check with the other expression: $6(20) - 30 = 90$.
The corresponding angle at the other parallel line is also $112^\circ$.
Corresponding angles are equal.
The same-side interior angle is $180 - 112 = 68^\circ$.
They complete a straight line together.
Same-side interior angles are supplementary.
$180 - 73 = 107^\circ$.
A transversal crosses two parallel lines. One interior angle measures $124^\circ$. What does the same-side interior angle measure?
Answer:
Two parallel lines are cut by a transversal. A pair of alternate exterior angles measure $6x + 30$ and $7x + 7$ degrees. Find $x$.
Answer:
Two parallel lines are cut by a transversal, and one angle measures $61^\circ$. Which angle is also $61^\circ$ rather than $119^\circ$?
A transversal crosses two parallel lines. One interior angle measures $51^\circ$. What does the same-side interior angle measure?
Answer:
A cable runs from one wall of a corridor to the opposite wall, and the two walls are parallel. It leaves the first wall at $43^\circ$ to that wall, bends once in the middle of the corridor, and meets the second wall at $21^\circ$ to it. What is the angle at the bend, measured inside the cable?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Point $B$ lies between $A$ and $C$. $AB = 4$ and $AC = 18$. What is $BC$?
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Two parallel lines are cut by a transversal. A pair of alternate exterior angles measure $6x + 14$ and $9x - 49$ degrees. Find $x$.
Answer:
You can reason with segments, angles and parallel lines. Explain why same-side interior angles must be supplementary if corresponding angles are equal, and say which fact you would use to prove two lines parallel.
8. Your turn: $B$ is between $A$ and $C$ with $AB = 9$ and $AC = 23$, step 2
19. Your turn: a same-side interior angle of $73^\circ$; what is the other?, step 2