Term
Triangle Congruence Shortcuts |
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Definition
SSS (side side side)
SAS(two sides and included angle)
ASA(two angles and included side)
AAS(two angles and non included side)
HL( hypotenuse and leg *only right triangles) |
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Term
What is CPCTC? How do you use it? |
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Definition
Corresponding parts of congruent triangles are congruent.
You use this AFTER your prove triangles congruent and it allows you to say all sides and angles are congruent to one another. |
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Term
Properties of Right Triangles: |
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Definition
*Right Triangles Have one right angle.
*We can use Pythagoren Theorem with these triangles to find the missing length of the third side. a^2+b^2=c^2. C is always the hypotenuse. |
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Term
Tips on Proving Triangles Congruent |
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Definition
*always look for triangles to prove congruent
*drawtriangles seperately to see congruences clearly
*the only way to prove triangles congruent are SSS,SAS,ASA,AAS,HL
*CPCTC can be used after triangles proven congruent
*common things to look for are: vertical angles, perpendicular lines, angle/segment bisectors, midpoints, parallel lines, shared parts (reflexive property) |
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Term
If a triangle is equilateral then it is______ |
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Definition
| equiangular and vice versa |
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Term
Isosceles triangle properties |
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Definition
*Base angles congruent
*legs congruent
*bisector of the vertex angle in an isosceles triangle cuts the triangles into two congruent triangles |
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Term
Difference between a perpendicular bisector, altitude, and median? |
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Definition
*Perpendicular Bisector is perpendicular to a side of a triangle and bisects it , doesnt have to start at vertex.
* Altitude starts at vertex and is perpendicular to opposite side
*Median begins at vertex and bisects opposite side |
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Term
How does a unique angle bisector relate to a perpendicular bisector, median, and altitude? |
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Definition
| * A unique angle bisector cuts an angle into congruent parts but also extends to cut the segment opposite into congruent parts |
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Term
What can we tell from corresponding altitudes of congruent triangles? |
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Definition
| That the altitudes are congruent because if the triangles are congruent then the triangles altitudes are going to be congruent as well because the distance from the vertex to the opposite side is going to be the same |
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Term
Given two sides of a triangle are congruent, what can we conclude? Is the vice versa true as well? |
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Definition
| That the angles opposite these sides are congruent. Yes, if you have two angles that are congruent in one triangle then the opposite sides are congruent. |
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Term
Exterior Angle Inequality |
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Definition
| the measure of an exterior angle of a triangle is greater than either non adjacent angle |
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Term
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Definition
| the sum of the lengths of any two sides of a triangle is greater than the third length |
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Term
If one side of a triangle is longer than another side, then (relate to angles) and is the vice versa true or false? |
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Definition
| the measure of the angle opposite the longer side is greater the the angle opposite the shorter side, the vice versa is true, if one angle is bigger than the other then the side opposite the bigger angle is going to be bigger than the side opposite the smaller angles |
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Term
How are perpendicular lines and distance relatable ? |
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Definition
| because a perpendicular segment from a point to a line or plane is the shortest distance that can be drawn from that point to that plane or line |
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Term
What is the Hinge Theorem? |
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Definition
| it works with the idea of SAS inequality, that says that if two sides of a triangle are congruent to two sides of another triangle, and the included angle is greater than the included angle of the other triangle then the side opposite is greater than the side of the other triangle with the smaller angle |
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Term
Properties of a quadrilateral
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Definition
* four sides
* interior angles sum to 360 ( Sum of Interior Angles Formula : S = (n-2)180 |
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Term
Properties of Parallelogram |
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Definition
* All sides parallel
*Diagonal seperates it into two congruent triangles
*opposite angles congruent
*diagonals bisect each other
*opposite sides congruent
*interior angles sum to 360
*consecutive angles supplementary |
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Term
Altitude of a parallelogram |
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Definition
| segment that begins at a vertex and is perpendicular to the opposite side[image] |
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Term
| In a parallelogram with unequal consecutive angles..... |
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Definition
| the longer diagonal lies opposite the obtuse angle and the smaller angle creates the smaller angle[image] |
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Term
| Relating Parallel Lines to Distance |
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Definition
| they are everywhere equidistant |
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Term
| Converses for Parallelograms |
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Definition
if two sides of a quadrilateral are both congruent and parallel then its a parallelogram
if both pairs of opposite sides of a quadrilateral are congruent then its a parallelogram
if the diagonals of a quadrilateral bisect each other then its a parallelogram |
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Term
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Definition
*interior angles sum to 360
*two pairs of congruent adjacent sides
* one pair of opposite angles congruent
[image] |
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Term
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Definition
| segment that joins the midpoints of two sides of a triangle , it is parallel to the third side and half the length |
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Term
| 3 special types of parallelograms |
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Definition
*Rectangles
*Rhombus
*Square |
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Term
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Definition
Parallelogram
§ All sides parallel
§ Opposite angles congruent
§ Opposite sides are congruent
§ Diagonals bisect each other
§ Interior angles sum to 360
§ Consecutive angles are supplementary
§ Diagonals separate into two congruent triangles
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All right angles
Diagonals are congruent
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Term
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Definition
- Parallelogram
§ All sides parallel
§ Opposite angles congruent
§ Opposite sides are congruent
§ Diagonals bisect each other
§ Interior angles sum to 360
§ Consecutive angles are supplementary
§ Diagonals separate into two congruent triangles
- Two congruent adjacent sides
- All sides congruent
- Diagonals perpendicular
- Diagonals create 4 congruent right triangles
- Can apply Pythagorean theorem to the triangles** |
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Term
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Definition
- Parallelogram
§ All sides parallel
§ Opposite angles congruent
§ Opposite sides are congruent
§ Diagonals bisect each other
§ Interior angles sum to 360
§ Consecutive angles are supplementary
§ Diagonals separate into two congruent triangles
- Rectangle + Rhombus
§ Two congruent adjacent sides
§ All sides congruent
§ Diagonals perpendicular
§ Diagonals create 4 congruent right triangles
§ Can apply Pythagorean theorem to the triangles**
§ All right angles
§ Diagonals are congruent |
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Term
Properties of a Trapezoid: |
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Definition
- Quadrilateral
§ Interior angles sum to 360
- One pair of parallel sides
- Consecutive angles are supplementary |
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Term
Medians Of Trapezoids
[image] |
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Definition
Is the line segment that joins the midpoints of the legs. It is parallel to the bases and equal to half the sum of them.
Hence the formula given >>
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Term
If three or more parallel lines intercept a transversal and create congruent segments, ...
[image]
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Definition
| then that it true for any other transversals you find in the same situation. |
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Term
| Isosceles Trapezoid Properties |
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Definition
- Congruent legs
- Quadrilateral
§ Interior angles sum to 360
- One pair of parallel sides
- Consecutive angles are supplementary
- Diagonals are congruent
- Base angles congruent
[image] |
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Term
| How do you write a ratio? What is something you must remember when dealing with ratios? |
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Definition
a:b or a/b
you must always remember to convert to the same units |
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Term
| What happens when you cannot convert the ratios units into the same units ? |
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Definition
| It becomes a rate like two gallons per mile |
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Term
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Definition
| a statment that equates two ratios |
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Term
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Definition
product of the means = product of the extremes (proportions)
[image][image] |
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Term
| Label the means and the extremes in a proportion |
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Definition
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Term
| Whats the quadratic equation |
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Definition
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Term
| When dealing with proportions remember to keep them......explain |
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Definition
A/B = C/ D
A and C are type 1 so they are both on top. B and D are type two so they are both on bottom. OR A and B are both type 1 so they are both in the first fraction and C and D are type 2 so they are both in the second fraction |
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Term
| What is the geometric mean? |
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Definition
, represented by x, of two numbers, a and b, is found by using the following proportion: a/x = x/b
so your replacing the means with an x and the a and b with the two numbers you are trying to find the geometric mean of |
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Term
| What are extended propoertions? |
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Definition
| Proportions that are really long, you can use any two equal to each other to solve |
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Term
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Definition
it is like a ratio but a lot of ratios mashed together and the key is that you can set them like this:
ax+bx+cx+dx =
x representing the common factor |
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Term
| What does it mean when two things are similar? |
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Definition
| They are the exact same shape just blown up or shrunken down |
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Term
| Similar Polygon Properties |
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Definition
*Corresponding angles are congruent
*corresponding sides are proportional |
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Term
Jacob is 4 ft tall and casts a shadow of 1.5ft. At the same time, a tree that is x tall casts a shadow of 20 ft. How tall is the tree?
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Definition
| Set up proportion like 4/x = 1.5/20 then solve |
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Term
| Shortcuts to Proving Triangles Similar |
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Definition
- AA~ : Angle Angle Similarity (if one pair of angles congruent to another pair on another triangle)
- SAS~ Side Angle Side Similarity ( one angle congruent to another angle on other triangle and the sides including the angle have to be proportional NOT CONGRUENT)
· So how do you know if its proportional? You can set up a proportion and if the fractions are equal then the sides are proportional. Take 9/15 and 12/20 for example. That’s one triangle with two side measures of 9 and 15 and the other triangle with side measures 12/20. From corresponding sides. If you simply reduce the fractions you get 3/5=3/5 and if you cross multiply you get 180=180. Now you know they are proportional. But if the lengths are congruent, that’s not similarity.
- SSS~ Side Side Side Similarity ( when you have three sides of one triangle proportional to the other three corresponding sides of the other triangle) |
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Term
| After we have proven triangles similar what can we use to get other congruent statements? |
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Definition
CSSTP >> Corresponding Sides of Similar Triangles are Proportional
CASTC >> Corresponding Angles of Similar Triangles are Congruent |
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Term
The lengths of corresponding altitudes of similar triangles have the same ratio as the lengths of any pair of corresponding sides. What does this mean>? |
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Definition
| So its basically saying you can add in the altitude as another side into the proportion. |
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Term
| If a line segment divides two sides of a triangle proportionally....... |
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Definition
| then this line is parallel to the third side |
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Term
| Describe the altitude of a right triangle |
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Definition
| is drawn to the hypotenuse and seperates the triangle into two right triangles that are similar to each other and to the original right triangle |
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Term
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Definition
[image]
(for right triangles only) the lengh of the altitude to the hypotenuse is the geomretric mean of the lengths of the segments of the hypotenuse.
So using figure above ^^ AD/CD = CD/DB we can make this proportion |
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Term
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Definition
The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg formed by the altitude to the hypotenuse
So it would be these two proportions that would give us our answers:
AD/AC=AC/AB
And
DB/CB=CB/AB
*Remember that grandma and elevetor theorems can only be applied to right triangles where the altitude is drawn from the right angle to the hypotenuse.
[image]
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Term
| Converse of Pythagorean Theorem and how you can tell if its a right, obtuse, or acute triangle |
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Definition
says that if a, b, and c are lengths of a triangle with c being the longest and a2 + b2 = c then the triangle is right with hypotenuse c.
1. If a2 + b2 = c then it’s a right triangle
2. If a2 + b2 < c then its an obtuse triangle
3. If a2 + b2 > c then its an acute triangle
**Make sure to check if its even a triangle first! Using the Triangle Inequality |
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Term
| What are Pythagorean Triples? And name the 5 most popular |
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Definition
a set of positive whole numbers where a2 + b2 = c2
- 3-4-5
- 5-12-13
- 7-24-25
- 8-15-17
- 5-20-25 |
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Term
What are the two special right triangles?
Why are they significant ? |
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Definition
45-45-90
30-60-90
Pythagorean Theorem needs two sides to find the third but with these two special right triangles all you need is one. |
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Term
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Definition
Right triangle with these angle measures means you can set up these variables
** We can also apply this thorem to a square because a square’s diagonal creates two isocosleces triangles meaning 45-45-90 tirangles.
[image] |
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Term
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Definition
Set up like dis
[image]
** The altitude of an equilateral triangle creates two 30-60-90 traingles so we can use this theorem for them as well.
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