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Chapter 4 Section 4.1 – Congruent Figures

Chapter 4 Section 4.1 – Congruent Figures. Objectives: To recognize congruent figures and their corresponding parts. Congruent figures -> have the same size and shape. When two figures are congruent, you can move one so that it fits exactly on the other one.

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Chapter 4 Section 4.1 – Congruent Figures

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  1. Chapter 4 Section 4.1 – Congruent Figures Objectives: To recognize congruent figures and their corresponding parts

  2. Congruent figures -> have the same size and shape. When two figures are congruent, you can move one so that it fits exactly on the other one. • Congruent polygons -> have congruent corresponding parts (matching sides and angles). Matching vertices are corresponding vertices. *When you name congruent polygons, always list corresponding vertices in the same order.*

  3. D R • Ex: List the congruent corresponding parts F C J T Sides: TJ congruent to RC JD congruent to CF DT congruent to FR Angles: T congruent to R J congruent to C D congruent to F

  4. The fins of the Space Shuttle suggest congruent pentagons. Find m<B. A C R W 132 E B 88 S P T D

  5. Theorem 4-1 • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. A D B C F E <C congruent to <F

  6. Example of proving triangles congruent: • Given: PQ congruent PS QR congruent SR <Q congruent <S <QPR congruent <SPR Prove: Triangle PQR congruent Triangle PSR P Q S R

  7. Homework #19 • Due Thursday (Oct 17) • Page 200 – 201 • #1 – 27 odd

  8. Section 4.2 – Triangle Congruence by SSS and SAS • Objectives: To prove two triangles congruent using the SSS and SAS Postulates

  9. In the last section, we learned that if two triangles have three pairs of congruent corresponding angles and three pairs of congruent corresponding sides, then the triangles are congruent. • However, you do notneed to know that all six corresponding parts are congruent in order to conclude that two triangles are congruent.

  10. Postulate 4.1 -> Side-Side-Side (SSS) Postulate • If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. A D B E F C Triangle ABC is congruent to triangle DEF

  11. Example using SSS • Given: HF congruent HJ FG congruent JK H is the midpoint of GK Prove: Triangle FGH congruent triangle JKH F J G K H

  12. Postulate 4.2 -> Side-Angle-Side (SAS) Postulate • If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. D A F B E C Triangle ABC congruent to triangle DEF

  13. Example using SAS • Given: AB congruent BE BC congruent BD • Prove: Triangle ABC congruent triangle DBE D B A E C

  14. Homework #20 • Due Monday (Oct 22) • Page 208 – 209 • # 2 – 24 Even

  15. Section 4.3 – Triangle Congruence by ASA and AAS • Objectives: To prove two triangles congruent using the ASA postulate and the AAS theorem

  16. In the last section, we learned that two triangles are congruent if two pairs of sides are congruent and the included angles are congruent (SAS). • Today, we will find out that two triangles are also congruent if two pairs of angles are congruent and the included sides are congruent (ASA).

  17. Postulate 4.3 – Angle-Side-Angle (ASA) Postulate • If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. A D B C E F Triangle ABC congruent to triangle DEF

  18. Example using ASA • Given: NM congruent NP <M congruent <P • Prove: Triangle NML congruent triangle NPO P L N M O

  19. Theorem 4.2 – Angle-Angle-Side (AAS) Theorem • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. A D B F E C Triangle ABC congruent to triangle DEF

  20. Examples using AAS • Given: <S congruent <Q RP bisects <SRQ • Prove: Triangle SRP congruent to triangle QRP P Q S R

  21. Given: XQ parallel to TR XR bisects QT • Prove: triangle XMQ congruent triangle RMT Q X M R T

  22. Homework #21 • Due Tuesday(Oct 23) • Page 215 – 217 • # 1 – 23 odd

  23. Section 4.4 – Using Congruent Triangles: CPCTC • Objective: To use triangle congruence and CPCTC to prove that parts of two triangles are congruent.

  24. With SSS, SAS, ASA, and AAS, we learned how to use three parts of triangles to show that the triangles are congruent. Once you have triangles congruent, you can make conclusions about their other parts because, by definition, Corresponding Parts of Congruent Trianglesare Congruent. This is abbreviated as CPCTC.

  25. Ex: • In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. • Given: SL congruent to SR <1 congruent to <2 • Prove: <3 congruent to <4 C 3 4 L R 5 6 1 2 S

  26. Ex: • Given: <Q congruent <R <QPS congruent <RSP • Prove: SQ congruent to PR P Q R S

  27. According to legend, one of Napoleon’s officers used congruent triangles to estimate the width of a river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that was in line with his eye and the tip of his visor. • Given: <DEG and <DEF are right angles <EDG congruent <EDF • Prove: EF congruent EG D F G E

  28. Homework # 22 • Due Wed (Oct 24) • Page 222 – 224 • # 1 – 19 odd • Quiz Thursday/Friday • Section 4.1 – 4.4

  29. Section 4.5 – Isosceles and Equilateral Triangles • Objectives: To use and apply properties of isosceles triangles

  30. Isosceles triangles are common in the real world. They can be found in everything from bridges to buildings. • Legs -> the congruent sides of an isosceles triangle • Base -> third side (not one of the legs) • Vertex angle -> formed by the two congruent sides (legs) • Base angles -> other two angles in an isosceles triangle

  31. Vertex angle Legs Base Base angles

  32. C • Theorem 4.3 -> Isosceles Triangle Theorem • If two sides of a triangle are congruent, then the angles opposite those sides are congruent. <A congruent to <B A B

  33. C • Theorem 4.4 -> Converse of Isosceles Triangle Theorem • If two angles of a triangle are congruent, then the sides opposite the angles are congruent. AC congruent to BC A B

  34. Theorem 4.5 -> Isosceles Bisector Theorem • The bisector of the vertex angles of an isosceles triangle is the perpendicular bisector of the base. C CD perpendicular to AB CD bisects AB A B D

  35. Ex: Explain why each statement is true. • <WVS congruent to <S • TR congruent to TS T U W R S V

  36. M • Ex: Using Algebra • Find the value of y y 63 N L O

  37. Corollary -> a statement that follows immediately from a theorem. • Corollary to Theorem 4.3 • If a triangle is equilateral, then the triangle is equiangular • Corollary to Theorem 4.4 • If a triangle is equiangular, then the triangle is equilateral

  38. Homework #23 • Due Tuesday (Oct 30) • Page 230 – 232 • #1 – 7 all • #10 – 13 all • #19 – 22 all

  39. Section 4.6 – Congruence in Right Triangles • Objectives: To prove triangles congruent using the HL Theorem

  40. In a right triangle, the side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs.

  41. Theorem 4.6 -> Hypotenuse-Leg (HL) Theorem • If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. • Three conditions must be met in order to use the HL Theorem: • There are two right triangles • The triangles have congruent hypotenuses • There is one pair of congruent legs

  42. A • Ex: • Given: AB congruent to AC • Prove: triangle ADB congruent to triangle ADC B C D

  43. Ex: • Given: CD congruent to EA AD is perpendicular bisector of CE • Prove: triangle CBD congruent to triangle EBA C A D B E

  44. Ex: • Given: WZ congruent to YX <W and <Y are right angles • Prove: triangle XWZ congruent to triangle ZYX X W Z Y

  45. Homework # 24 • Due Wednesday/Thursday • Page 237 – 239 • # 1 – 4 all • #6 – 14 even • #20 – 21 all

  46. Section 4.7 – Using Corresponding Parts of Congruent Triangles • Objectives: To identify congruent overlapping triangles To prove two triangles congruent by first proving two other triangles congruent

  47. Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a c0mmon side or angle. You can simplify your work with overlapping triangles by separating and redrawing triangles.

  48. Example: Identifying Common Parts • Separate and redraw triangle DFG and EHG. Identify the common angle. G H F D E G G <G is the common angle H F D E

  49. Ex: Using Common Parts • Given: <ZXW congruent <YWX <ZWX congruent <YXW • Prove: ZW congruent to YX Z Y W X

  50. Given: triangle ACD congruent to triangle BDC • Prove: CE congruent to DE A B E C D

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