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4-2. Angle Relationships in Triangles. Holt Geometry. Warm Up. Lesson Presentation. Lesson Quiz. Warm Up 1. Find the measure of exterior  DBA of BCD , if m DBC = 30°, m C = 70°, and mD = 80°. 2. What is the complement of an angle with measure 17°?

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4-2

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  1. 4-2 Angle Relationships in Triangles Holt Geometry Warm Up Lesson Presentation Lesson Quiz

  2. Warm Up 1. Find the measure of exterior DBA of BCD, if mDBC = 30°, mC= 70°, and mD = 80°. 2. What is the complement of an angle with measure 17°? 3. How many lines can be drawn through N parallel to MP? Why? 150° 73° 1; Parallel Post.

  3. Solve 1-3 in your head, Solve #4 in your notes. Classify each triangle by its angles and sides. 1. MNQ 2.NQP 3. MNP 4. Find the lengths of each side of the triangle. acute; equilateral obtuse; scalene acute; scalene 3x + 2 = 4x – 7 (isosceles ) 9 = x, sides are 23, 29, & 29

  4. Objectives Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles.

  5. Vocabulary auxiliary line corollary interior exterior interior angle exterior angle remote interior angle

  6. An exterior angle is formed by one side of the triangle and extension of an adjacent side. The remote interior angles are the non adjacent interior angles. The remote interior angles of 4 are 1 and 2. Exterior Angle 1 + 2= 4 Remote Interior Angles 3 is an interior angle.

  7. An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem

  8. Sum. Thm Example 1A: Application Draw the picture for the next 2 examples. Find mXYZ mXYZ + mYZX + mZXY = 180° Substitute. mXYZ + 40+ 62= 180 mXYZ + 102= 180 Simplify. mXYZ = 78° Subtract

  9. ? Example 1B: Application Find mWXY. Lin. Pair Thm. and  Add. Post. mYXZ + mWXY = 180° 62 + mWXY = 180 Substitute mWXY = 118° Subtract

  10. 118° Sum. Thm Example 1B: Application Continued Find mYWZ. mYWX + mWXY + mXYW = 180° Substitute. mYWX + 118+ 12= 180 mYWX + 130= 180 Simplify. Subtract. mYWX = 50°

  11. Sum. Thm Check It Out! Example 1 Just look… Use the diagram to find mMJK. mMJK + mJKM + mKMJ = 180° Substitute 104 for mJKM and 44 for mKMJ. mMJK + 104+ 44= 180 mMJK + 148= 180 Simplify. Subtract 148 from both sides. mMJK = 32°

  12. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

  13. Acute s of rt. are comp. Example 2: Finding Angle Measures in Right Triangles 2x Let the acute angles be A and B, with mA = 2x°. mA + mB = 90° 2x+ mB = 90 Substitute 2x for mA. mB = (90 – 2x)° Subtract 2x from both sides.

  14. Check It Out! Example 2a The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? mA + mB = 90° 63.7 + mB = 90 mB = 26.3°

  15. Check It Out! Example 2b x Let the acute angles be A and B, with mA = x°. mA + mB = 90° x+ mB = 90 mB = (90 – x)°

  16. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. Exterior Interior 3 is an interior angle.

  17. Example 3: Applying the Exterior Angle Theorem Find mB. mA + mB = mBCD Ext.  Thm. Substitute 15 + 2x + 3= 5x – 60 2x + 18= 5x – 60 Simplify. Subtract 2x and add 60 to both sides. 78 = 3x 26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55° Notes End

  18. Check It Out! Example 3 Find mACD. mACD = mA + mB Ext.  Thm. Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z – 9 = 2z + 1+ 90 6z – 9= 2z + 91 Simplify. Subtract 2z and add 9 to both sides. 4z = 100 z = 25 Divide by 4. mACD = 6z – 9 = 6(25) – 9 = 141°

  19. Example 4: Applying the Third Angles Theorem Find mK and mJ. K  J Third s Thm. mK = mJ Def. of s. 4y2= 6y2 – 40 Substitute 4y2 for mK and 6y2 – 40 for mJ. –2y2 = –40 Subtract 6y2 from both sides. y2 = 20 Divide both sides by -2. So mK = 4y2 = 4(20) = 80°. Since mJ = mK,mJ =80°.

  20. Check It Out! Example 4 Find mP and mT. P  T Third s Thm. mP = mT Def. of s. 2x2= 4x2 – 32 Substitute 2x2 for mP and 4x2 – 32 for mT. –2x2 = –32 Subtract 4x2 from both sides. x2 = 16 Divide both sides by -2. So mP = 2x2 = 2(16) = 32°. Since mP = mT,mT =32°.

  21. 1 3 33 ° 2 3 Lesson Quiz: Part I 1. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle? 2. Find mABD. 3. Find mN and mP. 124° 75°; 75°

  22. Lesson Quiz: Part II 4. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store? 30°

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