Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:An angle tells how much one ray must rotate to meet another ray.
Common stuck point:The procedure for angles is the easy part; the trap is judging angle size by ray length. Asking "Am I measuring turn between rays rather than length?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I measuring turn between rays rather than length?
Worked Examples
Example 1
easy
Two angles are supplementary. One measures 115°. Find the other.
Answer
x=65°
First step
1
Supplementary angles add up to 180°.
Full solution
2
Let the unknown angle be x: 115+x=180.
3
Solve: x=180−115=65°.
Supplementary angles form a straight line (180°). This relationship appears frequently when working with parallel lines and transversals.
Example 2
medium
Two parallel lines are cut by a transversal. One of the alternate interior angles measures 72°. Find all eight angles formed.
Example 3
easy
At 6:00 the hour and minute hands point opposite directions. What angle do they form?
Example 4
medium
Two parallel lines are cut by a transversal. A pair of co-interior (same-side interior) angles are (2x+10)∘ and (3x−30)∘. Find x.
Example 5
medium
Two parallel lines are cut by a transversal. A pair of corresponding angles are (5x−12)∘ and (3x+28)∘. Find x and the angle measure.Corresponding angles between parallel lines — find x and the angle measure
Example 6
medium
Two parallel lines are cut by a transversal. Alternate exterior angles measure (4x+5)∘ and (6x−25)∘. Find each angle.Alternate exterior angles between parallel lines — find each angle
Example 7
medium
Two adjacent angles together form a right angle, and the larger is twice the smaller. Find both angles.
Example 8
hard
Three rays from a point divide the plane into three angles in the ratio 2:3:5. Find each angle.
Example 9
hard
Two lines meet at 40∘. A third line through the same point bisects one of the obtuse angles. What angles does the third line make with the original two lines?
Example 10
hard
The complement of an angle is 41 of its supplement. Find the angle.
Example 11
challenge
How many times between 12:00 noon and 12:00 midnight do the hour and minute hands of a clock form a right angle?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Two angles are complementary. One measures 37°. Find the other.
Example 2
medium
An exterior angle of a triangle measures 124°. One remote interior angle measures 47°. Find the other remote interior angle.
Example 3
easy
What is the measure of a right angle?
Example 4
easy
An angle measures 45∘. Is it acute, right, or obtuse?
Example 5
easy
How many degrees are in a straight angle?
Example 6
easy
Two angles together form a right angle. One is 30∘. What is the other?
Example 7
easy
An angle is 120∘. Acute, right, or obtuse?
Example 8
easy
A clock's hands point at 3:00. What angle do they make?
Example 9
easy
Does a longer pair of rays make a bigger angle than a shorter pair, if both open the same amount?
Example 10
easy
Two angles lie on a straight line. One is 110∘. What is the other?Linear pair on a straight line — find the missing angle
Example 11
medium
Two angles are complementary and one is twice the other. Find both.
Example 12
medium
Two lines cross. One of the four angles is 65∘. Find the angle directly opposite it.Two intersecting lines — find the vertical angle
Example 13
medium
Three angles around a single point measure 120∘, x, and 150∘. They go all the way around. Find x.Three angles around a point — find x
Example 14
medium
At what time after 12:00 do the clock hands first form a straight angle?
Example 15
medium
An angle is 40∘ more than its complement. Find the angle.
Example 16
medium
A pizza is cut into 8 equal slices. What is the angle at the tip of one slice?One slice of a pizza cut into 8 equal pieces — find the tip angle
Example 17
medium
Two angles are supplementary and in the ratio 2:3. Find the larger angle.
Example 18
medium
A ray turns clockwise from due North to due East. Through how many degrees does it turn?
Example 19
challenge
At 3:30, what is the angle between the hour and minute hands?
Example 20
challenge
Two lines cross. The angles are in the ratio 1:1:1:1 in one figure and 2:7:2:7 in another. Which figure is possible, and what are its angles?
Example 21
challenge
The interior angles of a quadrilateral are in the ratio 1:2:3:4. Find the largest angle.
Example 22
challenge
How many times between 12:00 noon and 12:00 midnight do the clock hands point in exactly the same direction (overlap)?
Example 23
easy
An angle measures 89∘. Is it acute, right, or obtuse?
Example 24
easy
Two angles are complementary and one is 22∘. Find the other.
Example 25
easy
A full turn corresponds to how many right angles?
Example 26
easy
Two angles form a linear pair. One is 145∘. Find the other.
Example 27
easy
Two lines intersect. One of the four angles is 63∘. What are the other three?
Example 28
medium
The measure of an angle is 15∘ more than twice its complement. Find the angle.
Example 29
medium
An angle is its own supplement. Find the angle.
Example 30
medium
A clock shows 4:00. What is the smaller angle between the hour and minute hands?
Example 31
medium
The supplement of an angle is 40∘ less than three times its complement. Find the angle.
Example 32
hard
At what time between 3:00 and 4:00 are the clock's hands first perpendicular?
Example 33
hard
Two parallel lines ℓ1 and ℓ2 are cut by a transversal. The bisectors of a pair of co-interior angles meet at P. Find ∠ at P between the two bisectors.