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Unit 4

Unit 4. Congruent & Similar Triangles. Lesson 4.1. Day 1: Congruent Triangles. Lesson 4.1 Objectives. Identify corresponding parts of congruent figures. Characterize congruent figures based on a congruence statement.

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Unit 4

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  1. Unit 4 Congruent & Similar Triangles

  2. Lesson 4.1 Day 1: Congruent Triangles

  3. Lesson 4.1 Objectives • Identify corresponding parts of congruent figures. • Characterize congruent figures based on a congruence statement. • Identify congruent triangles by using congruence theorems and postulates. (G2.3.1)

  4. Congruent Triangles • When two triangles are congruent, then • Corresponding angles are congruent. • Corresponding sides are congruent. • Corresponding, remember, means that objects are in the same location. • So you must verify that when the triangles are drawn in the same way, what pieces match up?

  5. Naming Congruent Parts That way we know: We also know: D A and and F C E B • Be sure to pay attention to the proper notation when naming parts. • For instance: •  ABC  DEF • By the way, this is called a congruence statement. • The order of the first triangle is usually done in alphabetical order. • The order of the second triangle must match up the correspondingangles.

  6. Example 4.1 In the figure above, TJM  PHS. Complete the following statements. • Segment JM  ___________ • segment HS • P  ________ • T • mM  ________ • 48o • mP = ________ • 73o • MT = ________ • 5 cm • HPS  __________ • JTM • Yes, the order is important!

  7. Postulate 19:Side-Side-Side Congruence Postulate • If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. • Abbreviated • SSS • That means, if (SIDE) (SIDE) (SIDE) and then

  8. Postulate 20:Side-Angle-Side Congruence Postulate • If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. • Abbreviated • SAS • That means, if (SIDE) (ANGLE) (SIDE) and then

  9. Example 4.2 It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. When two lines intersect, they form vertical angles, and vertical angles are always congruent. So that makes the third congruence we need to find the congruent triangles. Is there enough information given to prove the triangles are congruent?If so, state the postulate or theorem that would prove them so. No, SSA does not guarantee congruent s. Yes, SSS Yes, SAS No, SSA does not guarantee congruent s.

  10. Lesson 4.1a Homework • Lesson 4.1: Day 1 – Congruent Triangles • p1-2 • Due Tomorrow

  11. Lesson 4.1 Day 2: More Congruent Triangles

  12. Postulate 21:Angle-Side-Angle Congruence • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. • Abbreviated • ASA • That means, if (ANGLE) (SIDE) (ANGLE) and then

  13. Theorem 4.5:Angle-Angle-Side Congruence • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of the second triangle, then the two triangles are congruent. • Abbreviated • AAS • That means, if (ANGLE) (ANGLE) (SIDE) and then

  14. Example 4.3 It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. When two lines intersect, they form vertical angles, and vertical angles are always congruent. So that makes the third congruence we need to find the congruent triangles. Is there enough information given to prove the triangles are congruent?If so, state the postulate or theorem that would prove them so. Yes, AAS Yes, AAS No, there needs to be at least one pair of congruent sides. Yes, ASA

  15. Theorem 4.8:Hypotenuse-Leg Congruence Theorem • If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. • Abbreviate using • HL • It still means, if (HYPOTENUSE) (LEG) (RIGHT ANGLE) with then

  16. Example 4.4 It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. Determine if enough information is given to conclude the triangles are congruent using HL Congruence? No, becauseHL Congruenceonly works in right triangles. Yes Yes Yes

  17. Which One Do I Use?…How Can I Tell? If there are moresides marked than angles, then use one of the following: SSS If ALLthreesides are marked as congruent pairs. SAS If ONLYtwosides are marked with the angle IN BETWEEN. HL Only works in RIGHT TRIANGLES Notice the angle is NOT IN BETWEEN the twosides. It looks like SSA. If there are moreangles marked than sides, then use one of the following: ASA If ANYtwoangles are marked with the sideIN BETWEEN. AAS If ONLY twoangles are marked along with a side that is NOT IN BETWEEN. Notice, the side must come NEXT to the same angle that is marked the same way in both triangles. The best way to determine which congruence postulate/theorem is to identify the number of each part that is marked.

  18. Lesson 4.1b Homework • Lesson 4.1: Day 2 – More Congruent Triangles • p3-4 • Due Tomorrow

  19. Lesson 4.2 Proving Triangles are Congruent

  20. Lesson 4.2 Objectives • Create a proof for congruent triangles. (G2.3.1) • Identify corresponding parts of congruent triangles. (G2.3.2)

  21. Remembering Proofs • Do you remember how to write a two-column proof? • What is the first step? • Rewrite the problem because… • That was what was GIVEN to you. • What should you write in the left-hand column? • The STATEMENTS you are making as you try to solve the problem. • What must you show in the right-hand column? • The REASONS for why you made that statement.

  22. The Understood Properties of Congruence IF AND THEN

  23. Example 4.5

  24. Example 4.6 SHOWING what the term does to your picture. Any time you are GIVEN a vocabulary word… You should DEFINE that term in your proof by…

  25. Proof Building Tips • Here are a few helpful hints to building a proof: • If there is a mark already on the picture, then there should be a step in the proof to explain the mark. • Those marks should be from the GIVEN, but if not…the reason in the proof would still be GIVEN. • Any time you add a mark to the picture, you need a step for that it in your proof. • This would be a good time for the REFLEXIVE PROPERTY. • Or things like • Vertical Angles • Midpoint • Parallel Line Theorems • Transitive Property (MAYBE) • Always be sure the last STATEMENT in the proof is an exact match to what you are trying to PROVE in the problem.

  26. Example 4.7 Once you have3 congruencies,you should have enoughto prove the trianglesare congruent.

  27. Lesson 4.2 Homework • Lesson 4.2 – Proving Congruent Triangles • p5-6 • Due Tomorrow

  28. Lesson 4.3 Similar Triangles

  29. Lesson 4.3 Objectives • Show triangles are similar using the correct postulate/theorem. (G2.3.3) • Solve similar triangles. (G2.3.4) • Utilize the scale factor and proportions to solve similar triangles. (G2.3.5)

  30. Ratio • If aand bare two quantities measured in the same units, then the ratio ofa to bisa/b. • It can also be written as a:b. • A ratio is a fraction, so the denominator cannot be zero. • Ratiosshould alwaysbe written in simplified form. • 5/10  1/2

  31. Similarity of Polygons • Two polygons are similar when the following two conditions exist • Corresponding angles are congruent. • Corresponding sides are proportional. • Means that all corresponding sides fit the same ratio. • Basically we are saying we have two polygons that are the same shapebutdifferent size. • We use similarity statements to name similarpolygons. • GHIJ ~ KLMN • You must match the order of the second polygon with that of the first to show corresponding angles and sides!

  32. Scale Factor • Remember, polygons will be similar when their corresponding sides all fit the same ratio. • That commonratio is called a scale factor. • We use the variable k to represent the scale factor. • So k = 3/2 or k = 2/3 , depending on which way you look at it. • Remember, it’s a ratio!

  33. Postulate 25:Angle-Angle Similarity Postulate • If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

  34. Theorem 8.2:Side-Side-Side Similarity • If all the corresponding sides of two triangles are proportional, then the triangles are similar. • Your job is to verify that all corresponding sides fit the sameexact ratio! Shortest sides 2nd shortest sides Longest sides

  35. Theorem 8.3:Side-Angle-Side Similarity • If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the corresponding sidesincluding these angles are proportional, then the triangles are similar. • Your task is to verify that two sides fit the same exact ratio and the angles between those two sides are congruent! SIDE ANGLE SIDE

  36. Example 4.8 • Are the triangles similar? • If so, list the congruent angles. • Also, what is the scale factor? YES, by SSS

  37. Example 4.9 Yes, by SSS Similarity. State if the triangles are similar. If so, write a similarity statement AND the postulate or theorem that makes them similar. VA Yes, by AA. VA Yes, by SAS Similarity.

  38. Example 4.10 C B D A A E Corresponding Angles Postulate Reflexive Property State if the triangles are similar. If so, write a similarity statement AND the postulate or theorem that makes them similar. Scale factors must beequal! Yes, by AA Similarity Not similar

  39. Lesson 4.3 Homework • Lesson 4.3 – Similar Triangles • p7-8 • Due Tomorrow

  40. Lesson 5.4 Using Similar Triangles

  41. Lesson 4.4 Objectives • Identify corresponding parts of congruent figures. • Solve similar triangles. (G2.3.4) • Utilize the scale factor and proportions to solve similar triangles. (G2.3.5)

  42. Proportion = a c b d • An equation that has two ratios equal to each other is called a proportion. • A proportioncan be broken down into two parts: • Extremes • Identifies the partnership of the numerator of the first ratio and the denominator of the second ratio. • Means • Identifies the partnership of the denominator of the first ratio and numerator of the second ratio.

  43. Solving Proportions a c = b d MULTIPLY MULTIPLY ad bc = • To solve a proportion, you must use the cross product property. • More commonly referred to as cross-multiplying. • So multiply the extremes together and set them equal to the means.

  44. Example 4.11 Solve the following proportions using the Cross Product Property

  45. Solving Similar Triangles • You must use a proportionto solve similar triangles. • To set-up the proportionit is best to match corresponding sides in each ratio. • BE CAREFUL TO SET UP EACH RATIO THE SAME WAY • So make the top of each ratio represent the small triangle (for instance) and the bottom of each ratio represent the larger triangle.

  46. Example 4.12 Either way, it should result in the same ratio for similar triangles. Find the scale factor. Or partner the smallest sides together IN THE SAME ORDER. Largest sides must work together.

  47. Example 4.13 Find KJ. Notice, each ratiowas made withcorresponding parts in theSAME ORDER!

  48. Example 4.14 The other ratiocomes fromusing thescale factor. Solve for x using the given scale factor.

  49. Example 4.15 Find the height, h, of the flagpole. Notice, h is a part ofthe large triangle. So becareful when selecting thelength of another side whenbuilding yourproportion.

  50. Lesson 4.4 Homework • Lesson 4.4 – Using Similar Triangles • p9-10 • Due Tomorrow

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