Complementary angles total 90° and supplementary angles total 180°. Angles on a straight line are supplementary.
Step 1: Let's Learn
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Vertical angles
Two crossing lines make vertical angles, which are always equal.
Parallel lines
A transversal across parallel lines makes corresponding and alternate angles equal, and co-interior angles supplementary.
The triangle sum
The angles of a triangle total 180°. The proof draws a parallel line through one vertex and uses alternate angles.
The exterior angle
An exterior angle equals the sum of the two opposite interior angles, which follows directly from the triangle sum.
Isosceles triangles
Equal sides face equal angles, and the converse holds too. This one is used constantly.
The angle results
Angles on a straight line total 180°, angles round a point 360°, vertical angles are equal, and a transversal across parallel lines produces only two distinct sizes. These are the building blocks.
The triangle sum
The three angles of any triangle total 180°. Tearing the corners off a paper triangle and placing them together demonstrates it, and the proof uses a parallel line through one vertex.
Isosceles triangles
Equal sides force equal base angles, and the converse also holds. Both directions are true here, which is not automatic and is why each needs its own proof.
The exterior angle theorem
An exterior angle equals the sum of the two remote interior angles. It follows from the triangle sum and the straight line, and it shortens many proofs considerably.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Marcus Chase Angles
A triangle has angles of 50° and 60°, with an exterior angle at the third vertex.
- Step 1
The angles total 180°, so the third interior angle is 180 − 110 = 70°.
Step 4: Your Turn
Practice makes it stick.
The Third Angle
Problem 1 of 2
A triangle has angles 50° and 60°. What is the third, in degrees?
degrees
The Supplement
Problem 2 of 2
An angle is 110°. What is its supplement, in degrees?
degrees
Chase the Angles
1 of 8
An angle is 35°. Its complement, in degrees?
2 of 8
An angle is 140°. Its supplement, in degrees?
3 of 8
A triangle has angles 90° and 35°. The third, in degrees?
4 of 8
Interior angles 45° and 65°. The exterior angle at the third vertex, in degrees?
5 of 8
One vertical angle is 72°. What is the other, in degrees?
6 of 8
An isosceles triangle has an apex angle of 40°. Each base angle, in degrees?
7 of 8
Sort each angle pair by whether they are equal or supplementary.
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8 of 8
A triangle has two angles of 60°. The third, in degrees?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A triangle has angles 80° and 55°. What is the third, in degrees?
Question 2 of 2
What does an exterior angle equal?
What You Learned
- Vertical angles are equal; angles on a line are supplementary.
- Parallel lines make corresponding and alternate angles equal.
- A triangle totals 180°, and an exterior angle equals the two opposite interior ones.