4.6

How Can I Use Equivalent Ratios?Pg. 23

Triangle Similarity and Congruence

4.6 – How Can I Use Equivalent Ratios?__Triangle Similarity and Congruence

By looking at side ratios and at angles, you are now able to determine whether two figures are similar. But how can you tell if two shapes are the same shape and the same size? In this lesson you will examine properties that guarantee that shapes are exact replicas of one another.

4.41 – MORE THAN SIMILARExamine the triangles. a. Are these triangles similar? How do you know? Use a flowchart to organize your explanation.

33

= 1 44

= 1 = 155

SSS~

given given given

~ABC FDE

b. Cameron says, "These triangles aren't just similar – they're congruent!" Is Cameron correct? What special value in your flowchart indicates that the triangles are congruent?

Similar with side ratio of 1.

c. Write a conjecture (in "If...,then..." form) that explains how you know when two shapes are similar.

"If two shapes are ____________ with a

side ratio of ________, then the two shapes

are _____________."

similar

1

congruent

d. Cameron wanted to write a statement to convey that these two triangles are congruent. He started with "∆CAB...", but then got stuck because he did not know the symbol for congruence. Now that you know the symbol for congruence, complete Cameron's statement for him.

CAB EFD

BCDE

= 1

given

BC= DE

given

4.42 – ANOTHER WAY OF PROOF Stephanie is tired of drawing flowcharts because the bubbles can be messy. She decides to organize her proof in columns instead. Compare your proof in the previous problems with the one below. How do they alike? How are they different?

Statements Reasons

1. BC = DE 1. given

2. AB = FD 2. given

3. AC = FE 3. given

4. ∆ABC ≅ ∆FDE 4. SSS≅

4.43 – A QUICKER WAY Examine the triangles at right.a. Are these triangles similar? Explain your reasoning.

yes

SAS~

b. Are the triangles congruent? Explain your reasoning.

yes

Similar with a side ratio of 1

c. Derek wants to find general shortcuts that can help determine if triangles are congruent. To help, he draws the diagram at right to show the relationship between the triangles in part (a). If two triangles have the relationship shown in the diagram, do they have to be congruent? How do you know?

yes

SAS~ with ratio of 1

d. Complete the conjecture below based on this relationship. What is a good abbreviation for this shortcut?

"If two triangles have two pairs of equal

____________ and the angles between them

are __________, then the triangles are

______________.

sides

equal

congruent

4.44 – ARE THEY CONGRUENT OR JUST SIMILAR?Determine if the triangles are similar, congruent, or neither. Justify your answers.

AA~ scale factor of 1

AA~

SAS~ with side ratio of 2/3

4 2

6 3

6 2

9 3

SSS~ or AA~

SSS~ or AA~with side ratio of 1

4.45 – ARE THERE OTHERS?Derek wonders, "What other types of information can determine that two triangles are congruent?" Your Task: Examine the pairs of triangles below to decide what other types of information force triangles to be congruent. Notice that since no measurements are given in the diagrams, you are considering the general cases of each type of pairing. For each pair of triangles below that you can prove are congruent, come up with a shortcut name to use for now on.

SSS

Pythagorean theorem to find missing side

HL

AA~

Not

AAS

SAAor

There is no SSA or ASS in geometry!

http://hotmath.com/util/hm_flash_movie.html?movie=/hotmath_help/gizmos/similarTriangle.swf&title=Proving%20Two%20Triangles%20are%20Similar

ASSSSA

ASA

Ways to Prove Triangles are Congruent

SSS SAS HL ASA AAS

4.46 – STATE THE CONJECTUREUse your triangle congruence conjectures to state why the following triangles are congruent.

SAS

SSS

HL

AAS

SAS

ASA

SAS

SSS

No

ASA

AAS

SSS

Not congruent

ASS

Not congruent

AAS

47

_____, _____, _____U V W

_____, _____, _____UV VW UW

X Y Z

XY YZ XZ