Two numbers do almost all the work here: 90 and 180.
Find x
Two angles form a linear pair. One measures (3x + 10)° and the other (2x + 20)°. What is x?
A linear pair sits on a straight line, so the two angles add to 180°: (3x + 10) + (2x + 20) = 180, giving 5x + 30 = 180, so 5x = 150 and x = 30. Check: the angles are 100° and 80°, which add to 180 ✓ A uses 90 instead of 180 — that is the rule for complementary angles, not a linear pair. C sets the two angles equal, which is the rule for vertical angles.
The two words are easy to swap. Complementary and Corner both start with C — a corner is a right angle, 90°.
Two angles side by side on a straight line always add to 180°.
(3x + 10) + (2x + 20) = 180
a straight line
5x + 30 = 180
collect like terms
x = 30
5x = 150
angles: 100° and 80°
substitute back
The question here asked for x, so 30 is right. But if it asks for the angle, you must substitute: the answer would be 100° or 80°, not 30. Read the last line before choosing.
Two lines crossing make two pairs of equal angles, opposite each other.
Why: each 65° angle and each 115° angle form a linear pair, so both add to 180°. If two angles are supplementary to the same angle, they must be equal.
Ray BD bisects ∠ABC = 84°
find ∠ABD
∠ABD = 84 ÷ 2 = 42°
and ∠DBC is also 42°
∠ABD = (2x + 5)°, ∠DBC = (x + 17)°
BD bisects, so they are equal
2x + 5 = x + 17
set them equal
x = 12, each half = 29°
so ∠ABC = 58°
When the lines are parallel, every one of the eight angles equals ∠1 or is supplementary to it.
| Pair | Where | Rule |
|---|---|---|
| Corresponding: 1 & 5, 2 & 6, 3 & 7, 4 & 8 | same position at each crossing | equal |
| Alternate interior: 3 & 6, 4 & 5 | between the lines, opposite sides | equal |
| Alternate exterior: 1 & 8, 2 & 7 | outside the lines, opposite sides | equal |
| Same-side interior: 3 & 5, 4 & 6 | between the lines, same side | add to 180° |
In the figure, ∠1 = 62°. Find ∠4, ∠5 and ∠6.
∠4 = 62°
vertical to ∠1
∠5 = 62°
corresponding to ∠1 (and alternate interior to ∠4)
∠6 = 180° − 62° = 118°
same-side interior with ∠4
The gold angles in the figure are all 62° and the blue ones are all 118°. Any gold plus any blue makes 180°.
Treating ∠4 and ∠6 as a matching pair gives ∠6 = 62°. They sit on the same side, between the lines, and together they fill a straight angle's worth: 62° + 118° = 180°.
Angles all the way around a point add to 360°.
Four angles meet at a point: x, 2x, 3x and a right angle. Find the largest.
x + 2x + 3x + 90 = 360
the whole is a full turn
6x = 270, x = 45
subtract 90, divide by 6
largest = 3x = 135°
check: 45 + 90 + 135 + 90 = 360 ✓
In words: an angle is 24° more than twice its supplement. Find the angle.
supplement = x, angle = 2x + 24
name the smaller one
x + 2x + 24 = 180
supplementary: they add to 180
3x = 156, x = 52, angle = 128°
check: 2(52) + 24 = 128 and 128 + 52 = 180 ✓
The question wanted the angle, not its supplement; 52° is the right number to the wrong question.
The same equal-or-add-up facts, at their edges.
Vertical angles measure (5x − 12)° and (3x + 20)°. Find each angle.
5x − 12 = 3x + 20
vertical angles are equal, so no 180 here
2x = 32, x = 16
gather the x terms
5(16) − 12 = 68°
check: 3(16) + 20 = 68 ✓
Setting the two expressions to add to 180 would treat them as a linear pair; read the figure first.
An angle equals its own supplement. How big is it? And one that equals its own complement?
x + x = 180, x = 90°
two equal halves of a straight angle
x + x = 90, x = 45°
two equal halves of a right angle
Equal partners always split the whole in half.
What is the complement of 104°?
90 − 104 = −14
an angle cannot be negative
no complement
its supplement, 76°, does exist
Only acute angles have complements; any angle under 180° has a supplement.
When a ray sits inside a larger angle, the two pieces add to it.
∠ABC = 110°, ∠ABD = (4x)°, ∠DBC = (3x − 2)°
D is inside the angle
4x + (3x − 2) = 110
part + part = whole
7x − 2 = 110
collect the x terms
7x = 112, x = 16
add 2, then divide by 7
∠ABD = 64°, ∠DBC = 46°
check: 64 + 46 = 110 ✓
Complementary, supplementary, linear pairs and interior rays are all the same equation — part + part = whole. Only the whole changes: 90, 180, or whatever the diagram says.
Angle items give a figure with expressions in x — a linear pair, vertical angles, or parallel lines with a transversal — and ask for x or for an angle. The fastest route is to decide "equal" or "adds to 90, 180 or 360" before writing anything; the 10-second check is to substitute x back and see that the angles really are equal or really do fill the whole.
Find the angle
Two complementary angles are (2x)° and (3x + 10)°. What is the larger angle?
Complementary means they add to 90°: 2x + 3x + 10 = 90, so 5x = 80 and x = 16. The angles are 2(16) = 32° and 3(16) + 10 = 58°, which add to 90 ✓ A is x itself, not an angle. B is the smaller angle. C comes from using 180 instead of 90.
Hint: three steps — add to 90, solve for x, then substitute into the expression the question actually asked about.
