Unit 5 Lesson 5 -13 Similarity; Triangle Theorems

Lesson 5 Meaning of Similarity

Similar polygons (shown by ~ ):

polygons whose vertices can be matched in 1:1 correspondence so that corresponding angles are equal and corresponding sides are in proportion.

Two figures are similar if their dimensions are proportional. When the figures are enlarged or shrunk to fit, all dimensions must be changed by the same ratio. 

POSTULATE 15: If the three angles of one triangle are equal to the three corresponding angles of another triangle, then the triangles are similar. (AAA for similar triangles).

Each of the corresponding angles of one triangle equal the corresponding angles of the other triangle.

We have studied a theorem that states that if two angles of a triangle are equal to two angles of another triangle, then the third angles are equal also.One of the pairs of angles in P-15 is not needed. We can write this conclusion as a

theorem. THEOREM 5-1: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. (AA for similar triangles).

Corollary 1: If a line intersects two sides of a triangle and is parallel to the third side, then a triangle similar to the given triangle is formed.Corollary 2: Triangles similar to the same triangle are similar to each other.Given: l || AC, intersect nABC at X and Y

Lesson 6 Meaning of Similarity Theorems

THEOREM 5-2

If two sides of one triangle are proportional to two sides of another

triangle and included angles are equal, then the triangles are similar. (SAS for similar triangles)

 

                                                                                     

STATEMENT REASON1. Draw XY || ST so that RX = AC. 1. Auxiliary line

2. C-1 of AA theorem

3. Definition of similar polygons

4. Given

5. Substitution (Step 1 into 4)

6. Substitution (Step 3 into 5)

7. RY = AB 7. Multiplication property of equality

8. Given

9. SAS

10. CPCTE

11. AA theorem

12. C-2 of AA theorem

 

THEOREM 5-3  

If three sides of one triangle are proportional to three sides of another

triangle,then the triangles are similar. (SSS for similar triangles)

We do not need the ASA or the AAS theorems to prove similar triangles now, because we would already have two angles equal, enough to prove the triangles similar by the AA theorem.

The theorems and postulates used to prove similar triangles are much the same as those used to prove congruency. The difference is found in the definition of similarity. Remember, for a polygon to be similar, the angles must be the same and the sides must be in proportion (to be congruent they must be equal.)

Lesson 8 Theorems- Similar Polygons

If polygon A ~ polygon B and polygon B ~ polygon C, then polygon A ~ polygon C.

Here is formal proof of this theorem:

Given: Polygon ABC . . . ~ Polygon JKL . . .             Polygon JKL . . . ~ Polygon RST . . . 

Prove: Polygon ABC . . . ~ Polygon RST . . . 

STATEMENT REASON1. Corresponding  's of similar polygons are =

2. Transitive property of equality3. Sides of similar polygons are

4. POP

5. Transitive property of equality

6. POP

7. Definition of similar polygons

THEOREM 5 - 5 

If two polygons are similar, then the ratio of their perimeters equals the ratio of any pair of corresponding sides.Given: Polygon ABCDE ~ Polygon QRSTU

 

 

STATEMENT REASON1. Polygon ABCDE ~ Polygon QRSTU

1. Given

2. Definition of similar polygons

3. POP

4. Definition of perimeter

5. Substitution

Lesson 9 Special Segments in Triangle

For example, take segments AB and CD. Let X divide AB into two parts and Y divide CD into two parts. Now segment ABand segment CD are said to be divided proportionally if

 

THEOREM 5- 6

If a line is parallel to one side of a triangle and intersects the other two sides, it divides them proportionally.

Given: 

Prove: 

XB 

Two other equivalent forms of this type of proportion can be formed. We can use Step 3

of the proof   and use POP to get  .

So we now have three ways of writing the conclusion of Theorem 5-6.

Model 1:

Model 2:   

 

Converse of Theorem 5-6: If a line divides two sides of a triangle proportionally, it is parallel to the third side.

The converse of theorem 5-6 uses the same content as the original theorem, however it would be used to prove the lines are parallel given the side measurements.

For this first example, use the figure at the left.

Given: Prove: SR || BA

Statements Reasons

1. 1. Given

2. 2. Reciprocals

3. 3. Add one to both sides by Addition Property of Equality

4. 4. Add fractions with like denominators

5. 5. Segment Addition Property

6.  6. Reflexive Property

7.  7.SAS~

8.  8. Similar figures have congruent angles.

9.  9. Corresponding Angles Postulate

THEOREM 5-7

If a ray bisects an angle of a triangle, it divides the opposite side into segment lengths proportional to the lengths of the other two sides of the triangle. (Triangle with angle-bisector theorem.)

Given: 

Prove: 

THEOREM 5-8

If two triangles are similar, the length of corresponding altitudes have the same ratio as the length of any pair of corresponding sides.

Given: 

Prove: 

Lesson 10 Similar Right Triangles

altitude of a triangle

An altitude of a triangle is a perpendicular line from a vertex to the opposite side.

geometric meanFor any positive real numbers a, b, and x, if   then x is

called the geometric mean between a and b.  Notice

that 

Projection of a point on a line

The point where a perpendicular through the point to the line intersects the line.

Projection of a segment on a line

The portion of a line with endpoints that are the projections of the endpoints of the segment.

When we have a proportion and both mean positions are occupied with the same number, that number is the geometric mean between the other two numbers. The geometric mean can be found by taking the square root of the product of the two numbers.

Model 1: Find the geometric mean between 4 and 9.

Solution:

                

 

Model 2: Five is the geometric mean between 8 and what other number?

Solution:

                

The projection of a point on a line is the point where a perpendicular through the given point to the given line intersects the given line. X is the projection of P on l. If the point is on the line, then the point is its own projection.

     The projection of a segment on a line is the segment whose endpoints are the

projections of the endpoints of the given segment.   is the projection of   on line l. Notice the length of the projection when the segment is in different positions. The projection can be equal to or less than the given segment, but it cannot be longer than the given segment.

THEOREM 5-9

If the altitude to the hypotenuse of a right triangle is drawn, the two triangles are similar to each other and similar to the given triangle.

Given:                CD altitude to hypotenuse

                                                            

The next three corollaries to Theorem 5-9 will be very useful in finding the length of segments in right triangles.

Corollary 1

The length of a leg of a right triangle is the geometric mean between the length of the hypotenuse and the length of the projection of that leg on the hypotenuse.

Model 1:

                        

Corollary 2

The length of the altitude to the hypotenuse is the geometric mean between the length of the segments of the hypotenuse.

Model 2:

                    

Corollary 3

In a right triangle, the product of the hypotenuse and the altitude to the hypotenuse is equal to the product of the lengths of the legs.

Model  3:

                            

                    AB · CD = AC · BC

        The figure shows all three of these corollaries. The small letters represent the length of the segments.

 

C -1

C -2

C - 3  

Lesson 12 Theorems about 30-60-90 Right Triangles

THEOREM 5-12 In a 30°-60°-90° right triangle, the measure of the hypotenuse is twice the measure of the short leg, and the measure of the longer leg is the measure of the short leg times  .

Given:

Right   ABC AC = s 

 B = 30° A = 60°C = 90°

Prove: AB = 2s 

BC = s 

STATEMENT REASON

1. Draw BD so that   CBD = 30° 1. Auxiliary lines

Extend   to intersect 

2. In   ADB  D = 60° 2. Sum  's of triangle = 180°

3.  ADB is equilateral 3. Sides opposite equal  s are =

4. CD = s 4. Altitude of equilateral triangle

5. AD = 2s 5. Betweenness and addition

6. AB = AD = 2s 6. Definition of equilateral triangle

7. (AB) 2 = (AC) 2 + (BC) 2 7. Pythagorean Theorem

8. (2s) 2 = (s) 2 + (BC) 2 8. Substitution

9. 4s 2 = s 2 + (BC)2  9. Algebra

3s 2 = (BC)2

s =BC

Lesson 13 Theorems about 45-45-90 Right Triangles

THEOREM 5-13

If each acute angle of a right triangle has a measure of 45°,

then the measure of the hypotenuse is   times as long as the measure of a leg. (45°-45°-90° Theorem)

Given:   ABC A = 45° B = 45°

AC = s

Prove: AB = s