Properties of triangles1
- 1. Geometry Angles of Triangles The Triangle Angle Sum Theorem The Exterior Angle Theorem
- 2. Remember! When you look at a figure, you cannot assume segments or angles are congruent based on appearance. They must be marked as congruent. IN ADDITION: Do not assume anything in Geometry is congruent – unless they are marked. This is true for parallel & perpendicular lines.
- 3. Triangle Sum Theorem The sum of the angle measures of a triangle is 180 degrees. A B C
- 4. Example – Find the measure of the missing angle. 46° 91° 43°
- 5. Example: Find m<1 Find m<2 Find m<3 28 82 682 1 3 m<1 = 70 m<2 = 70 m<3 = 42
- 6. One of the acute angles in a right triangle measures 25°. What is the measure of the other acute angle? Solve the Following Problem: 25° 65°
- 7. Example 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? y = 12, Store = 30°
- 8. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.
- 10. Example: Applying the Third Angles Theorem Find m∠P and m∠T.
- 11. The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? Example: 26.3°
- 12. 1 2 3 4 5 6 Exterior Angles Interior Angles
- 13. Sum of Interior Angles = Sum of Interior & Exterior Angles = 180° 1 2 3 4 5 6 180° 180° 180° 540° Sum of Exterior Angles = 360°540°- 180°= Sums of Exterior Angles 180•3 = 540
- 14. 180° 180° 180° 180° Sum of Interior Angles = Sum of Interior & Exterior Angles = 360° 720° Sum of Exterior Angles = 360°720°- 360°= Sums of Exterior Angles 180•4 = 720
- 15. The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Interior Exterior
- 16. An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. Interior Exterior ∠4 is an exterior angle. ∠3 is an interior angle.
- 17. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Interior Exterior ∠3 is an interior angle. ∠4 is an exterior angle. The remote interior angles of ∠4 are ∠1 and ∠2.
- 18. 18 A B C Using Angle Measures of Triangles Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.
- 19. 70° 50° 120°
- 20. 20 Ex. 3 Finding an Angle Measure. 65° x° Exterior Angle theorem: m∠1 = m ∠A +m ∠1 (2x+10)° x° + 65° = (2x + 10)° 65 = x +10 55 = x
- 21. Find m∠ACD. Example (2z + 1) + 90 = 6z – 9 2z + 91 = 6z – 9 91 = 4z – 9 100 = 4z z = 25 m∠ACD = 6(25) – 9 141 °
- 22. Lesson Review 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° 2 3 33 ° 1 3
- 23. Find m∠B. Example: Applying the Exterior Angle Theorem 2x + 3 + 15 = 5x – 60 2x + 18 = 5x – 60 18 = 3x – 60 78 = 3x x = 26 m∠B = 2(26) + 3 m∠B = 55°
- 24. Solve for x. 42° x° 120° x = 78°
- 25. Find the m∠MNP y = 9 mm∠MNP = 2(9) + 2 = 20
- 26. m∠1 = 1 2 3 110° (5x - 5)° (4x + 15)° (8x - 10)° pentagon 5x - 5 + 4x + 15 + 8x - 10 + 110 + 90 = 540 17x + 200= 540 -200 -200 17x = 340 x = 20 17 17 5(20) - 5 = 95° Find m∠1.
- 27. Triangle Sum Theorem The sum of the angle measures of a triangle is 180 degrees. A B C
- 28. Example – Find the measure of the missing angle. 46° 91° 43°
- 29. One of the acute angles in a right triangle measures 25°. What is the measure of the other acute angle? Solve the Following Problem: 25° 65°
- 30. Angle Relationships Lesson 2-1 angles relationships 31
- 31. interior INTERIOR –The space INSIDE the 2 lines EXTERIOR -The space OUTSIDE the 2 lines exterior exterior
- 32. Another practice problem Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements. 40° 120°
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