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Unit 5

Unit 5. Proving Triangles Congruent. Midterm Reflection. What was your goal? What was your actual grade? Why did you meet/not meet your goal? What were your strengths? What areas do you need to work on? What are you going to do to succeed on the next test?. Section 1.

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Unit 5

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  1. Unit 5 Proving Triangles Congruent

  2. Midterm Reflection • What was your goal? • What was your actual grade? • Why did you meet/not meet your goal? • What were your strengths? • What areas do you need to work on? • What are you going to do to succeed on the next test?

  3. Section 1 Angles of a Triangle

  4. Warm-Up • Write down everything you remember about triangles!

  5. Types of Triangles • By Side: • Equilateral—all sides congruent • Isosceles—two sides congruent • Scalene—no sides congruent • By Angle: • Obtuse—one angle greater than 90° • Right—one angle equal to 90° • Acute—all angles less than 90°

  6. Angles of a Triangle • Interior Angles add up to 180° • Exterior Angles add up to 360° • Examples: • In triangle DEF; ∠D = 45°, ∠E = 30°. Find ∠F. • In triangle ABC; ∠C = 3x – 5, ∠B = x + 40 and ∠A = 2x + 25. Find the measures of all three angles. 2x + 5 x + 40 3x – 5

  7. $1 to the first person that gets this:

  8. Section 1 CPCTC and SSS

  9. Mad Minute: Marking Congruent and Parallel Parts • Congruent line segments are marked with a small dash • Congruent angles are marked with an arc • Parallel lines are marked with arrows • To separate different pairs of congruent line segments or angles, we use different numbers of dashes or arcs

  10. CPCTC: Corresponding Parts of Congruent Triangles are Congruent • Corresponding = matching • Order of the letters matters! • Example: ΔABC = ΔDEF. Which angles are congruent? Which sides are congruent?

  11. Side-Side-Side (SSS) • SSS Postulate: If all three corresponding sides of two triangles are congruent, then the triangles are congruent • Example: ΔFEG = ΔKJL because of SSS.

  12. SSS or Not?

  13. Section 2 SAS, ASA, AAS, HL

  14. Mad Minute • Name all the corresponding angles if ΔIJH ≅ ΔKJL.

  15. Yesterday’s Exit Slip • ∠SRU ≅ ∠STU • ∠RSU ≅ ∠TSU • ∠RUS ≅ ∠TUS • Yes, can be proven through SSS • ∠BRD ≅ ∠DYB • ∠RBD ≅ ∠YDB • ∠RDB ≅ ∠YBD • BR ≅ DY • BY ≅ DR • BD ≅ BD

  16. Triangle Congruence Postulates • Side-Side-Side (SSS) • Side-Angle-Side (SAS) • Sandwich! • Angle-Side-Angle (ASA) • Sandwich! • Angle-Angle-Side (AAS) • No sandwich! • Hypotenuse-Leg (HL) • Right triangles only! THERE IS NO ASS (OR SSA) IN THIS CLASS!

  17. Which postulate proves each pair of triangles congruent? WHY?

  18. Section 3 Identity Properties in Triangle Proofs

  19. Identity Properties • Reflexive Property: AB ≅ AB (congruent to itself) • Transitive Property: AB ≅ BC, BC ≅ CD, so AB ≅ CD • Additive Property: Adding the same amount to two congruent parts results in two equal sums • Multiplicative Property: Multiplying two congruent parts by the same number results in two equal products

  20. How to Complete a Formal Proof • Mark diagram with “Given” and write as Step 1. • Figure out how many parts of the triangles you know are congruent, and how many you need to prove congruent. • Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns. • Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL. • Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.

  21. Example Check: Did we use all info in the Given? Does our last step match the Prove? • Given: GJ ≅ JI HJ ┴ GI • Prove: ΔGJH ≅ ΔIJH side 2. ┴ lines form right angles, all right angles are ≅ angle 2. ∠GJH ≅ ∠IJH side 3. HJ ≅ HJ 3. Reflexive property 4. ΔGJH ≅ ΔIJH 4. SAS

  22. Section 4 Line/Angle Theorems in Triangle Proofs

  23. Lines & Points • Midpoint • Halfway point on a line segment • Bisect • Split a line segment or angle into two equal parts V is the midpoint of TW HJ bisects GI

  24. Angles • Vertical Angles • ALWAYS congruent; (“X”) • Alternate Interior Angles • ONLY congruent when we know lines are parallel (“Z”) • ABCD is a parallelogram

  25. How to Complete a Formal Proof • Mark diagram with “Given” and write as Step 1. • Figure out how many parts of the triangles you know are congruent, and how many you need to prove congruent. • Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns. • Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL. • Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.

  26. Example Check: Did we use all info in the Given? Does our last step match the Prove? • Given: HK bisects IL ∠IHJ ≅ ∠JKL. • Prove: ΔIHJ ≅ ΔLKJ angle side 2. IJ ≅ JL 2. Definition of “bisect” angle 3. ∠IJH ≅ ∠LJK 3. Vertical angles congruent 4. ΔIHJ ≅ ΔLKJ 4. AAS

  27. Section 5 Using Quadrilateral Theorems in Triangle Proofs

  28. Quadrilateral Review • Parallelogram • Rhombus • Rectangle • Square • ALSO WATCH OUT FOR: • Alternate Interior Angles • Vertical Angles

  29. Properties of a Parallelogram • Opposite sides are parallel and congruent • Opposite angles are congruent • Diagonals bisect each other • Bisect = to split in half

  30. Rhombus • Has all the properties of a parallelogram, plus: • FOUR congruent sides • Diagonals are perpendicular and bisect

  31. Rectangle • Has all properties of a parallelogram, plus: • Four right angles • Congruent diagonals that bisect

  32. SQUARE • Four congruent sides and four right angles • Diagonals are congruent and perpendicular; also bisect

  33. Example Check: Did we use all info in the Given? Does our last step match the Prove? • Given: FLSH is a parallelogram; LG ┴ FS, AH ┴ FS • Prove: ΔLGS ≅ ΔHAF side 2. LG ≅ FH 2. Opp. sides of p.gram are ≅ 3. ┴ lines form right ∠’s, all right ∠’s ≅ angle 3. ∠LGS ≅ ∠HAF 4. ∠LSG ≅ ∠HFA 4. Alt. int. ∠’s ≅ when lines || angle 5. AAS 5. ΔLGS ≅ ΔHAF

  34. Section 6 Using Circle Theorems in Triangle Proofs

  35. Chords Chords intercepting congruent arcs are congruent

  36. Tangents Tangent is perpendicular to the radius at the point where it touches the circle

  37. Arcs • Arcs between parallel lines are congruent

  38. Inscribed Angles • Inscribed angle is half the intercepted arc. • Two inscribed angles that intercept the same arc are congruent

  39. Example Check: Did we use all info in the Given? Does our last step match the Prove? • Given: arc BR = 70°, arc YD = 70°; BD is the diameter of circle O • Prove: ΔRBD ≅ ΔYDB 2. BD ≅ BD side 2. Reflexive 3. ∠YBD = 35°, ∠RDB = 35° 3. Inscribed angles = ½ arc angle 4. ∠YBD ≅ ∠RDB 4. ≅ arcs have same measure 5. Inscribed angles intercepting same arc are ≅ angle 5. ∠BYD ≅ ∠BRD 6. AAS 5. ΔRBD ≅ ΔYDB

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