Two pages of visual notes for students to keep: the key words with pictures, how to tell which rule to use, four solved shots from the Math Billiards game, and where these angles show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Angle Relationships
Grade 8 · CCSS 7.G.B.5, 8.G.A.5 · Pairs with the Math Billiards game
Name
The big idea: when lines cross, the angles they make are connected. Some pairs are equal, and some add up to 90° or 180°. Know one angle and you can find the others. In Math Billiards every shot is one of these problems: find x, and the ball goes where you aimed.
Key vocabulary (the colors show the angles each word is about)
Complementary angles
Two angles that add to 90°. Look for the little box: 35° + 55° = 90°.
Supplementary angles
Two angles that add to 180°. Side by side on a straight line they form a linear pair: 50° + 130° = 180°.
Vertical angles
The angles across from each other where two lines cross. They are always equal.
Transversal
A line that crosses two or more other lines (the blue line). The little arrows mean the two black lines are parallel.
Alternate interior angles
Between the parallel lines, on opposite sides of the transversal. They are equal.
Same-side interior angles
Between the parallel lines, on the same side of the transversal. They add to 180°.
Which rule do I use? (look at the picture first)
A little box?The parts add to 90°.
A straight line?The parts add to 180°.
Two lines make an X?Angles across are equal.
Parallel arrows and a Z?Alternate interior angles are equal.
An F?Corresponding angles (same spot at each crossing) are equal.
Spot it: the dashed guide makes a box with the cushion, so x and 40° are complementary.
Write it: x + 40 = 90
Solve it: x = 90 − 40 = 50°
✓ Check: 50 + 40 = 90
2. A guide line crosses the shot
x is across from 62°: vertical angles, so x = 62°.
y sits next to 62° on a straight line: supplementary, so y + 62 = 180.
y = 180 − 62 = 118°
✓ Check: 62 + 118 = 180
3. A guide parallel to the cushion
∠2 = 70°. Find ∠6 and ∠5.
∠2 and ∠6 are in the same spot at each crossing (an F): corresponding, so ∠6 = 70°.
∠5 and ∠6 make a straight line: ∠5 = 180 − 70 = 110°.
✓ Check: ∠3 = 70° (across from ∠2), and same-side interior ∠3 + ∠5 = 180
4. Challenge: write an equation
The box means the parts add to 90: x + (3x + 10) = 90
Combine like terms: 4x + 10 = 90
Subtract 10: 4x = 80. Divide by 4: x = 20
The angles: x = 20° and 3(20) + 10 = 70°
✓ Check: 20 + 70 = 90
Watch out
A little box means 90°, not 180°. Subtracting from the wrong number is the most common mistake.
Check with your eyes: equal angles look the same size. If one is clearly bigger, the pair adds to 180° instead.
Only use the Z, F and C rules when the lines are parallel (look for the arrows).
Where you see it in real life
Pool: a bank shot leaves the cushion at the angle it came in.Scissors: blades and handles make equal vertical angles.Streets: an avenue is a transversal; the marked corners are equal.Ladder: its angles with the ground and wall add to 90°.Ramp: its angles with the flat ground add to 180°.