4-2 Angles in a Triangle Mr. Dorn Chapter 4.

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Presentation transcript:

4-2 Angles in a Triangle Mr. Dorn Chapter 4

4-2 Angles in a Triangle Angle Sum Theorem: The sum of the angles in a triangle is 180. x + y + z = 180 y x z

4-2 Angles in a Triangle Third Angle Theorem: If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent. A Y Given: C B Conclusion: Z X

4-2 Angles in a Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m = x + y y x z m

4-3 Congruent Triangles

Corresponding Parts of Congruent Triangles are Congruent. CPCTC: Corresponding Parts of Congruent Triangles are Congruent. A X Z B C Y

Example 1 Complete the sentence. Q P N L M R

Example 2 Given: CA = 14, AT = 18, TC = 21, and DG = 2x + 7 Find x. O 14 = 2x 7 = x 14 18 G C T D 21 2x + 7

Example 3 Given: AC = 7, BC = 10, DF = 2x + 4, and DE = 4x. Find x and AB. AB = 4x 7 = 2x + 4 AB = 4(1.5) 3 = 2x AB = 6 B 1.5 = x E 10 4x F A C D 7 2x + 4

4-4 Proving Triangles Congruent

4-4 Proving Congruent Triangles SSS Postulate: (Side-Side-Side) If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Q M P R

4-4 Proving Congruent Triangles SAS Postulate: (Side-Angle-Side) If two sides of one triangle and the included angle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. B C X Z Y

4-4 Proving Congruent Triangles ASA Postulate: (Angle-Side-Angle) 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 triangles are congruent. B C X Z Y

4-4 Proving Congruent Triangles AAS Postulate: (Angle-Angle-Side) If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the triangles are congruent. B C X Z Y

4-4 Proving Congruent Triangles Four Postulates that will work for any type of triangle: SSS, SAS, ASA, AAS *Just remember: Any Combination works as longs as it does not spell a bad word forward or back.

Example 1 Find x. x = 30 Use the Angle Sum Theorem! 3x 3x + x +2x = 180 2x 6x = 180 x x = 30

Example 2 Find x. x = 22 Use the Exterior Angle Theorem! (103 – x) +2x = 6x - 7 (103-x) 2x 103 + x = 6x - 7 103 = 5x - 7 110 = 5x (6x-7) x = 22

Example 3 Find x. x = 65 Use the Angle Sum Theorem! y + 53 + 80 = 180 50 n + 50 + 62 = 180 53 n + 112 = 180 x n = 68 Vertical angles are congruent, so… 62 47 y n 80 68 x + 47 + 68 = 180 x + 115 = 180

Example 4 Are the two triangles congruent? If so, what postulate identifies the two triangles as congruent? Yes, SAS

Example 5 Are the two triangles congruent? If so, what Postulate identifies the two triangles as congruent? Not Congruent!

Example 6 Given: Prove: ; S is the midpoint of Statements Reasons 1. QRS  TRS Q T S Statements Reasons 1. 1. Given ; S is the midpoint of 2. 2. Midpoint Theorem  3. Reflexive Property 3.  4. QRS  TRS 4. SSS

Example 7 Given: bisects BAC and BDC. Prove: BAD  CAD Proof: Since bisects BAC and BDC, BAD  CAD and BDA  CDA.  by the Reflexive Property. By ASA, BAD  CAD.