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Geometry Name: Classifying Triangles, Angles of Triangles, and Isosceles Triangles - Notes Classifying Triangles Angles Measure of one angle is 90 Measure of one angle is greater than 90 Sides Measure of all angles is less than 90 At least 2 sides are congruent No sides are congruent 3 congruent sides 3 congruent angles 1. Find x and the measure of each side of equilateral triangle JAG if JA= x + 9, AG = 2x, and JG = 3x – 9. A x= JA = AG = JG = G J 2. Find x and the measure of the unknown sides of isosceles triangle EFG if EF = 4x, FG = 2x + 6, and EG = 14. F x= EF = FG = E G Geometry Name: Classifying Triangles, Angles of Triangles, and Isosceles Triangles - Notes Angle Sum Theorem: The sum of the measures of the angles of a triangle is _____. W mW mA mY 180 A Y Find the measures of the following angles. R 3. 4. 84 1 C 110 54º 79 2 30 3 O mR ______ ∆CRO is an __________ triangle m1 ______ m2 ______ m3 ______ Exterior Angle Theorem: the measure of an exterior angle (1) of a triangle is equal to the sum of the measures of the two remote interior angles (2 & 3). 3 m1 m2 m3 2 1 Find the measures of the following angles. 5. 3 6. 3 2 50º 60º 1 120º 4 5 125º 56º m3 ______ 78º m1 ______ m2 ______ m3 ______ m4 ______ m5 ______ Geometry Name: Classifying Triangles, Angles of Triangles, and Isosceles Triangles - Notes The acute angles of a right triangle are ________________________. 4.1 Corollary 1 m1 m2 90 2 There can be at most one right or obtuse angle in a triangle. 4.2 Corollary R 1 40 C 2 110 30 O Corollary 4.3 Corollary 4.4 A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60º 60º 60º 60º Is it a Right Triangle? Put sides into a2 + b2 = c2, with “c” being the hypotenuse or longest side. Simplify and see if they are equal on both sides. If they are equal, then it is a right triangle. Determine if the following side lengths will form a right triangle. Show all work. 7. 5, 12, 13 YES NO 8. 4, 6, 8 YES NO Geometry Name: Classifying Triangles, Angles of Triangles, and Isosceles Triangles - Notes Properties of Isosceles Triangles The angle formed by the congruent sides is the _______________ angle. I ________ ________ D S The two angles formed by the base and one of the congruent sides are called the ___________________ angles. Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Inverse: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If FS ST then F T S S F T If F T then FS ST T F 9. In an Isosceles Triangle, the __________ are congruent and the __________ ______________ are congruent. ̅. ∆DIS is an isosceles triangle with ̅̅̅̅ 𝐷𝐼 ≅ 𝐼𝑆 10. 11. 12. 13. 14. I What is the vertex angle? ____ If DI = 7, then IS = _____ If mD 70 , then mS _____ If mD 80 , then mI _____ If mI 100 , then mD _____ and mS _____ 15. If DI IS , then _______≅ _______ D 16. If D S , then _______≅ _______ 17. If mD 3x 20 and mS 2x 5 then find x and mD. x = _____ and mD = _____ S