More Transversals Theorems
First, we considered switching the conditions and conclusions of the original Corresponding Angles Postulate- this would say: “If the corresponding angles are congruent, then the two lines are parallel.” We considered this situation, and decided that it also made sense. We called this the Converse of the Corresponding Angles Postulate.
From there, we reworded and proved the Converse of the Alternate Interior Angles Theorem, and the Converse of the Alternate Interior Angles Theorem. We also talked about the Converse of the Same Side Interior Angles Theorem, but did not prove it or write it down.
The big idea for using reasons (Postulates, Theorems, Definitions, and Properties) in proofs is to make sure that your reason is actually connected to the box it’s under, and the box before it connecting with an arrow!
Ex.) | l//m |–>|<1 is cong to <2 | The reason underneath the second box needs to be the original version of some theorem, because the order of information is going from parallel lines to say something about a congruent angle pair.
Ex.)| <1 is cong to <2|–>| l//m | The reason underneath the second box here needs to be the converse of some theorem, because the order of information is going from congruent angle pairs to say something about parallel lines.
Ex.) You will never see a line going into a box using the vertical angles theorem as a reason. Vertical angles just come from the diagram, and don’t need a box before them in a proof.
Ex.) Almost all proofs contain a step where two boxes are combined into one box. Usually the reason for allowing this is Transitive Prop. of Eq. or Cong., as well as the Substitution Prop. of Equality. Today we also saw use of the Congruent Supplements Theorem.
Generally, make sure you have met the conditions of any reason you are using, and make sure you are also making correct use of the conclusion of your reason.
Triangle Sum Theorem- the angles of a triangle add up to 180 degrees.
This can be used to create equations and solve for angle measures or variables in a problem.
Classifying- two ways: Angles and Sides
Angle Classifications: Acute- all angles are less than 90; Obtuse- One angle is greater than 90; Right- One angle is equal to 90; Equiangular- all angles are congruent (they all equal 60 degrees, these triangles will also always be equilateral)
Side Classifications: Scalene- All sides are different lengths, Isosceles- two sides are congruent; Equilateral- All sides are congruent.
Inequalities on Triangles
Smallest sides are across from smallest angles.
Medium sides are across from medium angles.
Largest sides are across from largest angles.
We can use these facts to list sides and/or angles from least to greatest from info given.
What lengths can make a triangle
For three given lengths to create a valid triangle, the two shorter lengths must have a sum greater than the 3rd side.
Consider why: Imagine the lengths 1, 2 and 3. If you imagine laying the two shorter lengths down end to end (1, followed by 2) they would make a straight segment of length 3 (2+1=3). The third side of a triangle with sides 1 and 2 couldn’t possibly be 3, if 1 and 2 can only make 3 when they are completely lined up! Triangles need some sort of angle between any two sides. As soon as length one and length two create an angle between them, the distance between their ends must be less than 3 (the length it was when they were in a straight line).
Finding the range of possible values for a third side given two sides:
Consider sides of length 13 and 7. Lined up end to end, they would make 20, so the third side must be less than 20. Now imagine folding the 7 side in so that the angle between it and the 13 side gets smaller, and smaller, and smaller, and smaller… Eventually the 7 is just laying on top of 13. The remainder is 6. Now 6 and 7 can’t make a triangle with 13, because they are all just lined up on top of each other. To form an angle and still meet, the third side needs to be bigger than 6.
So the possible range of values for the third side of a triangle with side lengths 13 and 7 is greater than 6 and less than 20, or 6< x < 20.
If two triangles have two pairs of congruent sides, the triangle with the larger angle contained by those two sides will also have the larger 3rd side.
If a triangle is a right triangle with sides a, b, and c, where c is the hypotenuse and a and b are the legs, then a^2 + b^2 = c^2.
We can use this fact for more than just solving for missing sides of right triangles!
Combined with the hinge theorem, we can classify any triangle just from knowing its 3 side lengths.
Steps: 1.) Use the 3 side lengths as a, b, and c in the pythagorean theorem.
2.) If it works, great! It’s a right triangle. If not, go to step three.
3.) Create a diagram of a right triangle where the legs are the shorter lengths of your 3 you’re working with. Create another diagram of your actual 3 lengths (make it look pretty much like a right triangle, we don’t know yet if it’s going to be obtuse or acute, so don’t assume anything yet!). Questions: a.) What would the hypotenuse of your right triangle be? b.) What 3rd side do you actually have? c.) Is your 3rd side bigger or smaller than the right triangle one? Use the Hinge Theorem to interpret what type of triangle you must have.
If your 3rd side is shorter than the right triangle one, then your angle across from that must be less than 90 degrees (Hinge Theorem), so the triangle would be acute.