Geometry: Lines, Angles and Triangles
Parallel and perpendicular lines, the Pythagorean theorem, and the right, isosceles and equilateral triangle properties E3n names by name.
9 topics · 37 questions
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Angles, Complements, Supplements and Vertical Angles
- How it worksHow do vertical angles come out equal every single time two straight lines cross?
- Worked exampleFind the missing angle when three angles sitting on a straight line measure 47, x and 68 degrees.
- Worked exampleFind the complement and the supplement of a 34 degree angle.
- Common mistakeWhy doesn't an angle get bigger when you draw its two rays twice as long?
Your turneasy1 of 3
Three angles sit side by side on a straight line and measure 47, x and 72 degrees. What is x?Parallel Lines Cut by a Transversal
- How it worksHow do corresponding and alternate interior angles line up when a transversal cuts two parallels?
- Worked exampleFind all eight angles when a transversal meets two parallel lines at 63 degrees.
- Common mistakeWhy doesn't the rule about equal alternate interior angles survive when the two lines are not parallel?
- Going furtherWhat happens to those angle pairs when the transversal crosses both lines at a right angle?
Your turneasy1 of 3
A transversal meets two parallel lines, making an angle of 114 degrees with the first. What is the co-interior angle beside it on the second line?Perpendicular Lines and Right Angles
- How it worksHow does a perpendicular bisector cut a segment in half and square with it at the same moment?
- Worked exampleFind the distance from the point (3, 7) straight down to the horizontal line y = 2.
- Going furtherWhat does being perpendicular look like on a coordinate grid once you compare the two slopes?
Your turneasy1 of 3
What is the distance from the point (3, 22) straight down to the horizontal line y = 2?Triangle Angle Sum and Exterior Angles
- How it worksHow does tearing the three corners off a paper triangle and lining them up form a straight line?
- Worked exampleFind the third angle of a triangle whose other two measure 52 and 79 degrees.
- Worked exampleFind the exterior angle of a triangle whose two remote interior angles are 35 and 60 degrees.
- Common mistakeWhy isn't a triangle with sides of 2, 3 and 9 possible when those numbers look perfectly ordinary?
Your turneasy1 of 3
Two angles of a triangle measure 52 and 89 degrees. What is the third?Isosceles and Equilateral Triangles
- How it worksHow does an equilateral triangle end up with three 60 degree angles out of pure symmetry?
- Common mistakeWhy isn't every triangle with two equal angles automatically equilateral?
- Worked exampleFind the base angles of an isosceles triangle whose apex angle measures 36 degrees.
- Worked exampleFind the other two angles of an isosceles triangle that has one angle of 100 degrees.
- Going furtherWhat happens to an isosceles triangle when you drop a height from the apex to the base?
Your turneasy1 of 3
An isosceles triangle has an apex angle of 104 degrees. What does each base angle measure?The Pythagorean Theorem
- How it worksHow do the two small squares on the legs pour perfectly into the big square on the hypotenuse?
- Worked exampleFind the hypotenuse of a right triangle whose legs measure 9 and 12.
- Worked exampleFind the missing leg of a right triangle with hypotenuse 26 and one leg of 10.
- Common mistakeWhy isn't the relation between a squared, b squared and c squared usable without a right angle?
- Going furtherWhat happens to the theorem when you build semicircles on the three sides instead of squares?
Your turneasy1 of 3
A right triangle has legs of 12 and b. Write an expression, in terms of b, for the hypotenuse.Pythagorean Triples and Scaling Them Up
- How it worksHow does scaling a 3-4-5 triangle up by 7 keep it a right triangle with whole number sides?
- Worked exampleFind the hypotenuse for legs of 15 and 20 by spotting the triple hiding inside them.
- Common mistakeWhy doesn't adding 1 to each side of a 3-4-5 triangle leave it a right triangle?
- Worked exampleFind the third side when a right triangle already has sides of 24 and 26.
Your turneasy1 of 3
A right triangle has legs of 7 times k and 14 times k. Write an expression, in terms of k, for its hypotenuse.45-45-90 and 30-60-90 Triangles
- How it worksHow does cutting an equilateral triangle down the middle create the 30-60-90 side ratios?
- Worked exampleFind the diagonal of a square whose side measures 7.
- Worked exampleFind both legs of a 30-60-90 triangle whose hypotenuse is 10.
- Going furtherWhat changes about the 45-45-90 ratio when you are handed the hypotenuse rather than a leg?
Your turneasy1 of 3
A rectangle has sides of s and 5 times s. Write an expression, in terms of s, for the length of its diagonal.Similar Triangles and Scale Factor
- How it worksHow does a scale factor of 3 triple every length while multiplying the area by 9?
- Common mistakeWhy doesn't doubling every side of a triangle simply double the area inside it?
- Worked exampleFind x when a triangle with sides 6 and 10 is similar to one with matching sides 9 and x.
- Worked exampleWalk me through using a 12 foot shadow cast by a 6 foot person to find a nearby tree's height.
Your turneasy1 of 3
A triangle with sides 7 and 12 is similar to one whose matching sides are s and x. Write an expression for x, in terms of s.AP®, SAT® and PSAT/NMSQT® are trademarks registered by the College Board (PSAT/NMSQT® jointly with the National Merit Scholarship Corporation). ACT® is a registered trademark of ACT Education Corp. GRE® is a registered trademark of ETS. LSAT® is a registered trademark of the Law School Admission Council. MCAT® is a registered trademark of the Association of American Medical Colleges. ISEE® and SSAT® are registered trademarks of their owners. None of these organizations is affiliated with, or endorses, Jentoo.