Similarity and Proportions Retake Guide

geometric mean

...of two numbers a and b is the value of x such that a over x is equal to x over b

You can compare side lengths of similar triangles to each other (side of one triangle to side of another triangle) but you can also compare...

...the ratio of two sides in a single triangle to the ratio of two corresponding sides in the other triangle.

What is the geometric mean of 3/4 and 4/8

4!

ITs important to remember that in the previous case, the two triangles formed are or are not similar?

Are NOT because they share a corresponding side length that has the same exact length (1 to 1 ratio, and the other side lengths may not have a 1 to 1 ratio).

The geometric mean will always be ______ from itself in a ratio

DIAGONAL

If you are not sure which side of a triangle is longer and therefore can't figure out how to label the angles a and b so you can orient them correctly, what is a strategy you can use?

Draw a line from the tip of the top angle straight downwards. It will intersect the base at a 90 degree angle, but one segment will be longer than the other. The longer segment will be connected to the longer side and the shorter segment will be connected to the shorter side. This way, you can identify the corresponding sides of two triangles (long to long, short to short, etc).

What is the converse of the triangle proportionality theorem?

If a line divides two sides of a triangle so that they are proportional, that line is parallel to the base of the triangle.

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths. *So this means that if you know the base is parallel to the line intersecting the legs, then the segments the legs are made of are proportional (you could say two segments on one side, their ratio is equal to the two corresponding ratios on the other side, or you could say the ratio between the opposite segments are equal)

SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. **This makes sense because if you have two sides set at an angle, there is only one side length that could connect them and so the angles are predetermined.

If then statement for this theorem... (ray bisecting angle)

If angle such and such is congruent to angle this and that, then the two segments on the base have an equal ratio to the other two sides OR the two two segments on one side of the bisector have an equal ratio to the two segments on the other bisector.

If then statement for converse of the triangle proportionality theorem...

If side one over side two is proportional to side 3 over side four, then the intersect line II to base

SSS Similarity Theorem

If the corresponding sides of two triangles are in proportion, then the triangles are similar. **The SCALE FACTOR of the corresponding sides is the same, then you know it is not necessarily the same size but definitely the same shape.

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. *Because consider this- third angles theorem means those final angles are pre-determined. POSTULATE!

Why is the Triangle Proportionality Theorem important?

It can help you find missing side lengths. The name relates to the definition because it is talking about triangle pieces being proportional.

So the three lines intersecting two transversals have to be _______

PARALLEL in order for the ratios of the segments to be proportional

IF a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are _________ to the lengths of the other two sides.

PROPORTIONAL

If 3 parallel lines intersect two transversals, then they divide the transversals _____________ly.

PROPORTIONAly

If you draw a line (aka height/altitude) from the right angle to the hypotenuse of a triangle, you create three _______ right triangles?

SIMILAR

How would you solve for missing side lengths when comparing two similar triangles?

Set up a ratio, and then cross multiply, and use algebra to single out the variable. You may want to check with the calculator to see if the end ratios comparing the corresponding sides of the triangle are really equal.

scale factor

The ratio of any two corresponding lengths in two similar geometric figures.

Say you find a side length two similar triangles both share...how do you know the other numbers in the ratio?

The shared side length is not in the same location for each of these triangles. So, you find the location of the shared side length on one triangle, find its corresponding side, and set a ratio, then you find the (different) location of the shared side length in the other triangle, find its corresponding side length and do another ratio. Just make sure you go in the right direction (triangle 1 value over triangle 2 value for each ratio or the other way around).

When applying this whole geometric mean thing to triangles, the geometric mean is...

The side length two similar triangles both share (when you use the whole altitude making three right triangles), whether it be a number or even two letters (like CB) representing vertices.

If then statement for this theorem...

WV ll RT ll YK, then ration is equal to ratio of corresponding segments

Why is the geometric mean important to similarity and proportions?

When you find the geometric mean (lets say it is a variable) you can set up an equation that says something like ysquared=27 and then you can separate square root of 27 into other square roots and your answer might have a radical.

What can you do to compare similar triangles made using the process above?

You can label the angles (a,b) so that you can easily orient/draw them out (a big and small kinda thing) so that they are both oriented in the same way. This way you can find the corresponding sides and solve for any missing side lengths.

You have a triangle with side lengths, and two other triangles with different side lengths. How could you figure out which of the two are similar to the first.

You could find the largest, middle and smallest side in each of the two and then set those as corresponding sides to the original. Set up ratios comparing the corresponding sides between the first and second, and then the first and third triangle. Whichever ratios simplify to be all equal (consistently) represent that those two triangles are congruent.

Similarity Statement Example

triangle ABC ~ triangle DEF- remember, the vertices have to correspond!