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G E O M E T R Y Chapter 4: Triangles

G E O M E T R Y Chapter 4: Triangles. Definition of Congruent Triangles. A. P. Δ A B C Δ P Q R. B. Q. R. C. If then the corresponding sides and corresponding angles are congruent. Δ A B C Δ P Q R ,. Types of Triangles classified by the sides.

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G E O M E T R Y Chapter 4: Triangles

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  1. GEOMETRY Chapter 4: Triangles

  2. Definition of Congruent Triangles A P ΔABCΔ PQR B Q R C If then the corresponding sides and corresponding angles are congruent ΔABCΔ PQR ,

  3. Types of Triangles classified by the sides • Equilateral triangle has 3 congruent sides • Isosceles trianglehas at least 2 congruent sides • Scalene triangle has no sides congruent Types of Triangles classified by the angles • Acute triangle has 3 acute angles. If these angles are congruent, then the triangle is also equiangular. • Right trianglehas exactly one right angle • Obtuse triangle has exactly one obtuse angle AcuteEquiangularRightObtuse

  4. Sides of Right triangles and Isosceles triangles are given special names C Adjacent sides of < A Opposite side of < A A B Base Legs Hypotenuse Isosceles Triangle, when only 2 sides = , these sides called “legs” and the remaining side is called “base” Legs Right Triangle, the side opposite the right angle is called “hypotenuse”, the adjacent sides called “legs”

  5. Theorem: Properties of Congruent Triangles • Every triangle is congruent to itself [ reflexive property ] • If then [ symmetric property ] • If and then ΔABCΔ PQR ΔPQRΔ ABC ΔABCΔ PQR ΔABCΔ TUV ΔPQRΔ TUV [ transitive property ]

  6. Interior and Exterior Angles of a Triangle Exterior angle Exterior angle Original angle Extend the angles 1 Interior angle 2 3 2 3 1 < 1 + < 2 + < 3 = line If you rotate each angle of a triangle and line them up, they add up to a straight line. Since 1 = 1, 2 = 2, 3 = 3, then angles of a triangle add up to a line or 180 ° 2 3 1 When an auxiliary line is drawn parallel to the opposite side of the triangle, then alternate interior angles are equal. 2 3

  7. Theorem: Triangle Sum The sum of the measures of the interior angles of a triangle is 180 ° A 2 3 1 3 2 B C

  8. Example 1: Using the Triangle Sum Theorem In this triangle, find m < 1, m < 2, m < 3 by using the Triangle Sum Theorem m < 3 = 180°─ ( 51 ° + 42 ° ) = 87 ° With m < 3, use Linear Pair postulate to get m < 2 = 180°─ 87 ° = 93 ° Again, Using the Triangle Sum Theoremm < 1 = 180°─ ( 28 ° + 93 ° ) = 59 ° 51 28 1 2 3 42 Third Angles Theorem: If 2 angles of 1st triangle are congruent to 2 angles of a 2nd triangle, then the 3rd angles are congruent Theorem: Acute angles of a right triangle are complementary

  9. Exterior Angles Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non-adjacent) interior angles Exterior angle of < C = sum of the 2 remote interior angles < A + < B B A C

  10. Exterior Angles Theorem: As you can see by moving duplicate copies of the triangle, it clearly shows how the exterior angle of angle C = the 2 remote angles of the triangle, < A + < B. C A B B B A C A

  11. Proving that Triangles are Congruent (same shape and size) Side – Side – Side ( SSS ) Congruence Postulate: If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the two triangles are congruent. Side – Angle – Side ( SAS ) Congruence Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a second triangle, then the two triangles are congruent.

  12. Proving that Triangles are Congruent (same shape and size) Angle – Side – Angle ( ASA ) Congruence Postulate: If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the two triangles are congruent. Angle – Angle – Side ( AAS ) Congruence Postulate: If 2 angles and a non-included side of one triangle are congruent to 2 angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

  13. Examples where AAA and SSA don’t work thus Triangles are NOT Congruent (same shape and size) B D As you can see, all 3 angles of both triangles are congruent to corresponding angles of other triangle, yet the triangles are NOT congruent. A C F B As you can see, triangles ABC and ABD are NOT congruent even though they have 2 congruent sides, BC BD and AB AB and a non-included angle, < A < A A D C

  14. CPCTC: Corresponding Parts of Congruent Triangles are Congruent By definition, 2 triangles are congruent if and only if their corresponding parts are congruent. M R A S T

  15. Base Angles Theorem (Isosceles Triangle) If 2 sides of a triangle are congruent, then the angles opposite them are congruent C N H Y

  16. Theorem: If 2 angles of a triangle are congruent, then the sides opposite them are congruent. Corollary to Base Angles theorem: If a triangle is equilateral, then it is also equiangular. Corollary to theorem above: If a triangle is equiangular, then it is also equilateral. Definition of a Corollary: A corollary is a theorem that follows easily from a theorem that has been proven.

  17. Hypotenuse – Leg (HL) Congruence Theorem If the hypotenuse + a leg of a right Δ are congruent to the hypotenuse + leg of a 2nd right Δ, the 2 triangles are congruent D A F E C B G

  18. m  π 

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