1. 1 Triangle Angle Sum Theorem The sum of the measures of the angles of a triangle is 180°. m ∠A + m ∠B + m ∠C = 180 A B C Ex: If m ∠A = 30 and m∠B = 70; what is m∠C ? m ∠ A + m ∠ B + m ∠ C = 180 30 + 70 + m∠C = 180 100 + m∠C = 180 m∠C = 180 – 100 = 80

2. Exterior Angle Theorem 1 2 34 P QR In the triangle below, recall that  1,  2, and  3 are _______ angles of ΔPQR. interior Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________, (they add up to 180) with one of the angles of the triangle. linear pair ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR,  1, and  2 are the remote interior angles with respect to  4. In ΔPQR,  4 is an exterior angle because  3 +  4 = 180. The measure of an exterior angle of a triangle is equal to sum of its ___________________ remote interior angles

3. Exterior Angle Theorem 1 2 345 In the figure, which angle is the exterior angle? 55 which angles are the remote the interior angles?  2 and  3 If  2 = 20 and  3 = 65, find  5 65 20 If  5 = 90 and  3 = 60, find  2 85 90 60 30

4. Triangle Inequality Theorem Triangle Inequality Theorem The sum of the measures of any two sides of a triangle is _______ than the measure of the third side. greater a b c a + b > c a + c > b b + c > a

5. Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10 However, 10 + 5 > 16

6. Medians, Altitudes, Angle Bisectors Perpendicular Bisectors

7. A B C Given  ABC, identify the opposite side 1. of A. 2. of B. 3. of C. BC AC AB Just to make sure we are clear about what an opposite side is…..

8. A new term… Point of concurrency Where 3 or more lines intersect

9. B A C M N L Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the opposite side.

10. The point where all 3 medians intersect Centroid Is the point of concurrency

11. The centroid is 2/3 the distance from the vertex to the side. from the vertex to the side. 2x x 10 5 32 X16

12. angle bisector of a triangle a segment that bisects an angle of the triangle and goes to the opposite side.

13. The Incenter is where all 3 Angle bisectors intersect Incenter Is the point of concurency

14. Any point on an angle bisector is equidistance from both sides of the angle

15. Any triangle has three altitudes. Definition of an Altitude of a Triangle A altitude of a triangle is a segment that has one endpoint at a vertex and the other creates a right angle at the opposite side. The altitude is perpendicular to the opposite side while going through the vertex Acute Triangle

16. Orthocenter is where all the altitudes intersect. Orthocenter

17. A Perpendicular bisector of a side does not have to start at a vertex. It will form a 90° angles and bisect the side. a 90° angles and bisect the side. Circumcenter Is the point of concurrency

18. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. A B C D AB is the perpendicular bisector of CD

19. The Midsegment of a Triangle is a segment that connects the midpoints of two sides of the triangle. D B C E A D and E are midpoints DE is the midsegment The midsegment of a triangle is parallel to the third side and is half as long as that side.

20. Example 1 In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU. PR = ________ TU = ________16 ft 5 ft

21. Give the best name for AB ABABABABAB || | | || Median Altitude None AngleBisector PerpendicularBisector