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4.3 What Makes Triangles Similar? Pg. 12 Conditions for Triangle Similarity

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4.3–What Makes Triangles Similar? Conditions for Triangle Similarity Now that you know what similar shapes have in common, you are ready to turn to a related question: How much information do I need to know that two triangles are similar? As you work through today's lesson, remember that similar polygons have corresponding angles that are equal and corresponding sides that are proportional.

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4.21 – ARE THEY SIMILAR? Erica thinks the triangles below might be similar. However, she knows not to trust the way figures look in a diagram, so she asks for your help.

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a. If two shapes are similar, what must be true about their angles? Corresponding angles must be congruent Corresponding sides must be proportional b. What must be true about their sides?

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120° 20° 40° 11.2cm 5.6cm 4cm 2cm 8.6cm 4.3cm b. Measure the angles and sides of Erica's Triangles and help her decide if the triangles are similar or not. Similar

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4.22 – HOW MUCH IS ENOUGH? Jessica is tired of measuring all the angles and sides to determine if two triangles are similar. "There must be an easier way," she thinks.

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h_help/gizmos/triangleSimilarity.swf help/gizmos/triangleSimilarity.swf

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Side-Side-Side Similarity: A B C D E F If all 3 corresponding sides are proportional, then the triangles are similar

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Angle-Angle Similarity: If 2 corresponding angles are congruent, then the triangles are similar A B C D E F

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4.23 – WHAT'S YOUR ANGLE? Determine whether the triangles are similar. If they are, explain why and write a similarity statement.

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55° 77° AA~

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31° 47°

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AA~

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4.24 – ARE THEY SIMILAR? Based on your conclusions, decide if each pair of triangles below are similar. If they are make a similarity statement. Then determine if you are using Angle-Angle similarity or Side-Side-Side similarity.

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SSS~

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96° 52° AA~

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d. Yes, AA~

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4.25 – FLOWCHARTS Examine the triangles at right. a. Are these triangles similar? Why? Yes, AA~

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b. Julio decided to use a diagram (called a flowchart) to explain his reasoning. Compare your explanation to Julio's flowchart. Did Julio use the same reasoning you used? Yes

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c. What appears to go in the bubbles of a flowchart? What goes outside the bubbles? Inside: Statements Outside: Reasons

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4.26 – WRITING FLOWCHARTS Besides showing your reasoning, a flowchart can be used to organize your work as you determine whether or not triangles are similar. a. Are these triangles similar? Why? Yes, SSS~

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b. What facts must you know to use the triangle similarity conjecture you chose? Julio started to list the facts in a flowchart at right. Complete the third oval.

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c. Once you have the needed facts in place, you can conclude that you have similar triangles. Add to your flowchart by making an oval and filling in your conclusion.

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d. Finally, draw arrows to show the flow of the facts that lead to your conclusion and record the similarity conjecture you used, following Julio's example.

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4.27 – START FROM SCRATCH a. What is the best conjecture to test for these triangles? SSS~ or AA~? AA~, Only know angles

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b. Are these triangles similar? Justify your conclusion using a flow chart.

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ΔJKL ~ AA~ given ΔYZX

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4.27 – HOW MANY OVALS? a. What is the best conjecture to test for these triangles? SSS~ or AA~? SSS~, Only know sides

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b. Are these triangles similar? Justify your conclusion using a flow chart.

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9696 = = 3232 ΔABC ~ SSS~ given ΔDEF 12 8 = 3232

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