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Chapter 4

Chapter 4. Ms. Cuervo. -Two figures that have the same size and shape. -Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangle are congruent. Vocabulary: Congruent. Some Ways to Prove Triangles Congruent.

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Chapter 4

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  1. Chapter 4 Ms. Cuervo

  2. -Two figures that have the same size and shape. -Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangle are congruent. Vocabulary: Congruent

  3. Some Ways to Prove Triangles Congruent Lesson 2

  4. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Postulate 12 (SSS Postulate)

  5. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Postulate 13 (SAS Postulate)

  6. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Postulate 14 (ASA Postulate)

  7. Using Congruent Triangles Lesson 3

  8. Given: AB and CD bisect each other at M. Prove: AD is parallel to BC Example A C M D B

  9. Statements Reasons • AB and CD bisect each other at M. • M is the midpoint of AB and CD • AM ≅MB; DM ≅ MC • 4. <AMD ≅ <BMC • 5. ∆AMD ≅ ∆BMC • 6. <A ≅ <B • 7. AB is parallel to BC Given Def. of bisector of a segment Def. of midpoint Vertical <‘s are ≅ SAS Postulate Corr. Parts of ≅ triangles are ≅ If two lines are cut by a transversal and alt. interior angles are ≅, then the lines are parallel.

  10. If and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection. A Line and A Plane are Perpendicular

  11. Identify two triangles in which the two segments of angles are corresponding parts. • Prove that the triangles are congruent. • State that the two parts are congruent, using the reason: corresponding parts of ≅ triangles are ≅ A Way to Prove Two Segments or Two Angles Congruent

  12. The Isosceles Triangle Theorems Lesson 4

  13. Legs-the two congruent sides Base-the third side of the triangle Vocabulary for Isosceles Triangles Vertex Angle Leg Leg Base Angles BASE

  14. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Given: AB ≅ AC Prove: <B ≅ <C Theorem 4-1The Isosceles Triangle Theorem A B C D

  15. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 4-2 Given: <B ≅ <C Prove: AB ≅ AC A B C D

  16. Corollary 1 An equilateral triangle is also equiangular Corollary 2 An equilateral triangle has three 60 degree angles Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. Corollary 4 An equiangular triangle is also equilateral

  17. Others Methods of Proving Triangles Congruent Lesson 5

  18. Hypotenuse-In right triangles, the side opposite the right angle Legs-The other two sides Vocabulary

  19. If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Theorem 4-3 (AAS Theorem) Given: ∆ABC and ∆DEF; <B≅ <E; <C ≅ <F: AC ≅ DF Prove: ∆ABC ≅ ∆DEF

  20. If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent Theorem 4-4HL Theorem Given: ∆ABC and ∆DEF; <C and <F are right <‘s AB ≅ DE BC ≅ EF Prove: ∆ABC≅∆DEF

  21. All Triangles SSS SAS ASA AAS Right Triangles HL Summary of Ways to Prove Two Triangles Congruent

  22. Using More than One Pair of Congruent Triangles Lesson 6

  23. Given : <1 ≅ <2; <5≅<6 Prove: AC is perpendicular to BD Example B 3 1 5 A C 2 6 4 D

  24. Statements Reasons • <1 ≅ <2; <5 ≅ <6 • AC ≅ AC • ∆ABC ≅∆ADC • AB ≅ AD • AO ≅ AO • ∆ABO≅∆ADO • <3 ≅ <4 • AC is perpendicular to BD • Given • Reflexive Property • ASA Postulate • Corr. Parts of ≅ triangles are ≅ • Reflexive Property • SAS Postulate • Corr. Parts of ≅ triangles are ≅ • If two lines form ≅adj <‘s, then the lines are perpendicular

  25. Medians, Altitudes, and Perpendicular Bisectors Lesson 7

  26. Median: the segment from a vertex to the midpoint of the opposite sides Altitude: the perpendicular segment from the vertex to the line that contains the opposite side. Vocabulary

  27. Perpendicular Bisector: a line (or ray or segment) that is perpendicular to the segment at its midpoint. Distance from a Point to a Line (or Plane): the length of the perpendicular segment from the point to the line (or plane). Vocabulary

  28. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Theorem 4-5 Given: Line l is the perpendicular bisector of BC; A is on line l Prove: AB = AC A B C X

  29. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Theorem 4-6 Given: AB =AC Prove: A is on the perpendicular bisector of BC A 1 2 B C X

  30. If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle Theorem 4-7 Given: Ray BZ bisects <ABC P lies on ray BZ PX ⊥BA PY ⊥BC A X P Z B Y C

  31. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. Theorem 4-8 A Given: PX ⊥ BA; PY ⊥ BC PX=PY Prove: BP bisects <ABC X P B Y C

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