Unit 11: Angle Relationships and Triangles
Every angle two lines can make, how a triangle's three angles are locked together, and the theorem hiding in a tilted square.
7 topics · 24 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Complementary and Supplementary Angles
- How it worksHow do you find a missing angle when you know it's complementary or supplementary to one you already have?
- Worked exampleFind the measure of an angle that is complementary to a 35 degree angle, and one that is supplementary to a 110 degree angle.
- Common mistakeWhy isn't every pair of angles that adds up to 180 degrees automatically sitting right next to each other?
Your turn1 of 2
A carpenter cuts one angle of a picture frame corner at 62 degrees. What is the complementary angle needed to make a right angle?Vertical Angles
- How it worksHow does the straight-line rule for supplementary angles force the pair of opposite angles to match?
- Worked exampleFind the measure of the angle vertical to a 72 degree angle.
- Going furtherWhat happens to all four angles at a crossing when one of them is changed to 90 degrees?
Your turn1 of 2
Two street lines cross at an intersection, forming an angle of 125 degrees. What is the measure of the angle vertical (directly across) from it?Angles Formed by Parallel Lines and a Transversal
- How it worksHow do corresponding, alternate interior, and alternate exterior angles each pair up when a transversal crosses parallel lines?
- Worked exampleFind the measures of all eight angles formed when a transversal crosses two parallel lines at a 65 degree angle.
- Going furtherWhat happens to the angle relationships if the two lines the transversal crosses aren't actually parallel?
- Common mistakeWhy don't co-interior angles on the same side of the transversal come out equal instead of adding to 180 degrees?
Your turn1 of 2
A ladder leans across two parallel railings, making one angle 125 degrees. What is the measure of its corresponding angle on the other railing?Triangle Angle Sum
- How it worksHow does tearing off a triangle's three corners and lining them up along a straight edge prove they sum to 180 degrees?
- Worked exampleFind the missing angle in a triangle with angles of 50 degrees and 65 degrees.
- Common mistakeWhy doesn't a bigger triangle have angles that add up to more than 180 degrees?
Your turn1 of 2
A triangular road sign has two angles measuring 59 degrees and 55 degrees. What is the measure of the third angle?The Pythagorean Theorem
- How it worksHow does rearranging four copies of the same right triangle inside a square prove that a squared plus b squared equals c squared?
- Worked exampleFind the length of the hypotenuse of a right triangle with legs of 6 and 8.
- Worked exampleFind the missing leg of a right triangle with a hypotenuse of 13 and one leg of 5.
- Common mistakeWhy doesn't the Pythagorean theorem work on a triangle that isn't a right triangle?
- Going furtherWhat happens to the Pythagorean theorem's equation if you're working in three dimensions instead of two?
Your turn1 of 2
A ladder leans against a wall, forming a right triangle with legs of 15 feet and 14 feet. What is the length of the ladder, the hypotenuse?The Converse of the Pythagorean Theorem
- How it worksHow does testing whether a squared plus b squared equals c squared tell you if a triangle has a right angle, even before you measure any angles?
- Worked exampleFind whether a triangle with sides 9, 12, and 15 is a right triangle.
- Common mistakeWhy doesn't a triangle whose sides add up correctly in some other way automatically count as a right triangle?
Your turn1 of 2
A triangle has sides 7 times 3, 7 times 4, and 7 times 5. Is it a right triangle?Distance on the Coordinate Plane
- How it worksHow does drawing a right triangle between two points let the Pythagorean theorem measure the distance between them?
- Worked exampleFind the distance between the points negative 1 comma 2 and 3 comma 5.
- Ties togetherHow does the distance formula on the coordinate plane come directly from the Pythagorean theorem you already proved?
Your turn1 of 2
On a treasure map, a chest sits at (0, 3) and a flag sits at (14, -4). What is the distance between them?