Unit 4-Review - Triangle Theorems and Rules Name of Theorem or relationship In words/ Symbols Diagrams/ Hints/ Techniques 1. Side angle relationship 2. Triangle inequality Theorem 3. Pythagorean Theorem to find a missing side. Pythagorean Inequality to Classify triangles 4.Isosceles triangle Theorem The longest side is across from the largest angle. The medium length side is across from the medium-sized angle. The shortest side is across from the smallest angle The sum of the lengths of the two smaller sides of a triangle is greater than the length of the largest side. To find a range of possible sides, add two given sides, subtract them. a) c 2 = a 2 + b 2 C is longest side ( hypotenuse-across from the right angle) b) If c 2 a 2 + b 2 it is acute If c 2 a 2 + b 2 it is obtuse If c 2 a 2 + b 2 it is right Remember, c 2 must be on the left of = The base angles of an isosceles triangle are equal in measure. The sides opposite the base angles in an isosceles triangle (called legs) are equal in length. 1. Solve for ALL angles in the triangle. 2. Draw arrows! Add up the two smaller sides and compare to the largest side. If the sum is greater, it s a triangle! 2,3,4 (2+3) > 4? yes! If you see a right angle, it s a right triangle! use Pythagorean Theorem to solve for a missing side. WATCH OUT! If asked does this make a triangle you must use Theorem # 2- NOT PYTHAGOREAN. ERROR ALERT: you must SQUARE (power of 2) a, b and c, DIFFERENT FROM Theorem #2 seen above. Angles opposite are congruent! If you see expressions, make them equal to each other! 5. Exterior angle theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle. IN + IN =OUT
1) In triangle RQS, RQ= 10 inches, SQ =9 inches and RS= 8 inches, arrange the angles form smallest to greatest. 2) The diagram at the right shows a right triangle with representations for two angles. What is the value of x? 3) In triangle DOG, m D =, m O=, and m G =. a) State the longest side of the triangle b) State the shortest side of the triangle- how did you find it? c) Classify the triangle based on its SIDES 4) State whether each of the following could be the sides of a triangle and why. a) {6,6,6} b) {2,4,2} c) {5,9,10} d) {4,4,9} Side lengths in ( a,b,c,d?) are triangles because..
5) Two sides of a triangle have lengths 2 and 7. Write an inequality for all possible integer lengths of the third side. 6) Given the following information find the degrees in each angle of the triangle. What is the longest side of the triangle? 7) Determine whether the following sides form a right, acute or obtuse triangle. Show how you arrived at your answer. a) 5, 11, 12 b) 5, 12, 13 c) 7, 15, 18
8) Solve for x. Give your answer in the simplest radical form. 9) is equilateral, and bisects. a.) Find and b.) Find x. What important concept about special segments in an equilateral triangle helped you solve part b? What other type of triangle has the same property? 10) In a triangle, the ratio of the angles is 1:3:5. Find the measure of all the angles and Classify the triangle according to its angles.
11) 12) Consider the following diagram with 2 right triangles labeled, and the sides included in the diagram. Solve for the whole length of side AC.