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Name ______________________________ Geometry Date ___________ Period ___________ Triangle Vocabulary: Equiangular Equilateral Isosceles Scalene Vertex Angle Base Angle – #1) Triangle ABC is an isosceles triangle. Angle A is the vertex angle, AB=4x-14 and AC=x+10. Find the length of the legs. #2) The perimeter of an Isosceles triangle is 70. Two of its sides are equal and the third side is five less than the equal sides. Find the length of the three sides. #3) A scalene triangle has three unequal sides. The first side of a triangle is three times the second side. The third side is twelve more than the second side. If the perimeter is 57 inches, find the dimensions of the scalene triangle. 1 Name ______________________________ Geometry Date ___________ Period ___________ Parts of a Triangle: Interior angles Exterior angles Adjacent angles Supplementary angles Remote interior angles Theorems and Corollaries: Angle Sum Theorem: The angles of a triangle add up to a _________. #1) #2) #3) #4) 2 Name ______________________________ Date ___________ Geometry Period ___________ Example #5: Find the measure of each numbered angle in the figure if AB || CD . A B 5 C 2 60° 135 ° 1 3 4 D Example #6: Find the measure of each angle. 3 4 1 5 2 6 68° Exterior Angle Theorem: the measure of an _____________ is = to the sum of the two _______________. #7) #8) 3 Name ______________________________ Geometry Date ___________ Period ___________ #8) Theorems: Isosceles Triangle Theorem- If two sides of a triangle are ___________, then the _______________ opposite those sides are _____________. Theorem: If two angles of a ___________ are congruent, then the __________ opposite those angles are ____________. Directions: Find x and y. Example #1) Example #2) M 14y + 29 y 7x - 10 O 65 x 3x + 40 17y - 78 x P 4 Name ______________________________ Geometry Example #3) 6x - 6 Date ___________ Period ___________ 32° 24 y 30 Example #4) If Angle X is the vertex angle of an isosceles triangle and angle X=52, find angle Y and angle Z. X Y Z Example #5) In isosceles triangle DEF, Angle D is the vertex angle. If Angle E=2x+40 and angle F=3x+22, find the measure of each angle of the triangle. Example #6) In isosceles triangle RST, Angle R is the vertex angle. If Angle S=7x-17 and Angle T=3x+35, find the measure of each angle of the triangle. 5 Name ______________________________ Geometry #7) Date ___________ Period ___________ Congruent Triangles: Two triangles are _________________ if and only if their ____________________ parts are congruent. Directions: Write a congruence statement. Example #1) E B F A D C Example #2) D A C E F B Example #3: Use the congruence statement to complete the correspondences if RST ABC. C ____ R ____ AC ____ ST ____ RS ____ 6 Name ______________________________ Geometry Date ___________ Period ___________ Congruency Theorems: #1) __________: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. #2) ________: If two sides and the included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent. #3) ___________: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. #4) ____________: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent. #5) ____________: If the hypotenuse and a leg of a right triangle are congruent to the corresponding hypotenuse and leg of a second triangle, the two triangles are congruent. (Must be a right triangle to use this theorem!!!!) * ____________: This is not a valid theorem because __________________________. Directions: Determine whether the triangles are congruent, and if so by which theorem. ) Example #1) Example #2) Example #3) 7 Name ______________________________ Geometry Date ___________ Period ___________ Directions: For each diagram, write a congruence statement. Then list the theorem that is used to show the triangles are congruent. Example #1) Example #2) D Z N K F G E X Y F O L Example #3) Example #4) Z P X O Y S #5) #6) 8 Name ______________________________ Geometry Directions: Write a two column proof. Date ___________ Period ___________ Example #7) Given: Z C ABC and XYZ are right triangles. A and X are right angles. BC YZ B Y Prove: A ABC XYZ B X Y Example #8) C Given: BD is a perpendicular bisector of Triangle ABC. Prove: D ABD CBD A B Directions: Mark all congruent parts. Indicate the postulate that can be used to prove their congruence. Example #4) T R L RL ST ST bi sec ts RSL S 9 Name ______________________________ Geometry Example #5) Date ___________ Period ___________ D A AD || GR AD GR G R Example #6) AB BC B AD CD A D C Directions: Write a two column proof. S Example #7) Given: 1 and 2 are right 5 angles , ST TP. Prove: RTS RTP T 1 3 2 4 R 6 P W Example #8) Given: WZ YZ VZ ZX Prove: VZW XZY V Z X Y 10 Name ______________________________ Geometry Example #9) Given: BE bi sec ts AD, A D Date ___________ Period ___________ A C E Prove: 1 AB DC 2 B D Example #10) Given: Q and S are right angles. QR SR P Prove: P T Q R S T 11