SSS, SAS, AAS, ASA. Right Triangles & The Pythagorean Theorem

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1 Lesson 1 Lesson 1, page 1 of 7 Glencoe Geometr Chapter 5. & 8.1 Right Triangles & The Pthagorean Theorem B the end of this lesson, ou should be able to 1. Determine if two right triangles are congruent.. Use the Pthagorean Theorem to solve problems. A few episodes ago, we looked at determining whether ANY two triangles were congruent. To review, there were several combinations of sides and angles we needed to draw a conclusion. The were (in no particular order): SSS, SAS, AAS, ASA Toda, we are going to take a special look at the most famous of all triangles: Right Triangles When determining if two right triangles are congruent, we are ALWAYS given at least one angle, the right (or 90 degree) angle. Though all of the above methods work on right triangles, there was a special case that worked ONLY for right triangles, Mr. Korpi,

2 Lesson 1, page of 7 HL: If the hpotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. Remember that this was the SSA or Ambiguous case for non-right triangles. Because this onl works for right triangles, we give it its own special name: HL Eample: Determine the reason that! NKL! NJL. Then, find " m J if m KLN = 0. Because the two right triangles share a common side (a leg), segment NL, and we know that the hpotenuses are congruent, the two triangles are congruent b HL. Remember that means that all corresponding sides and angles are congruent. Notice also that! JKL is isosceles, so the base angles, J and K are congruent. Also notice that segment NL is then a median, perpendicular bisector, altitude, and angle bisector All 4!!! So m KLJ = = 40 leaving = 140 for the two congruent base angles K and J. Therefore, the are both equal to 140/ = 70 degrees. Mr. Korpi,

3 Lesson 1, page 3 of 7 Eample: Find the value of so that the two right triangles! ABC and! XYZ are congruent b the HL postulate. Assume angle B is the right angle. AC = 8, AB= 7+ 4, ZX = 9+ 1, YX = 5( + ) AC and XZ are the hpotenuses and have equal measures. 8 = So = 7 /9 3 Or AB = XY. So, 7+ 4= 5( + ) 7+ 4= = 6 =3 Now onto the best part about Right triangles: THE PYTHAGOREAN THEOREM Recall from Lesson : The sum of the square of the two legs of a right triangle equals the square of the hpotenuse. a + b = c c b a Mr. Korpi,

4 4 Eample: Find the value of. 11 A. 137 C. 44 B. 105 D = 11 = = 105 Lesson 1, page 4 of 7 There are certain combinations of three whole numbers that will alwas satisf the Pthagorean Theorem. These sets of numbers are called Pthagorean Triples. If the measures of a right triangle are whole numbers, the measures form a Pthagorean triple. Do the measures in the above eample, 4, 105,11, form a Pthagorean triple? No, 105 is not a whole number. Eample: 3, 4,5 and 7, 4,5 (It is customar to list the numbers in increasing order, with the measure of the hpotenuse last. Mr. Korpi,

5 Eample: Which set of numbers is a Pthagorean triple? A. 10, 15, 18 Lesson 1, page 5 of 7 B. 10, 0, 30 C. 9, 40, 41 A. 10 B = = = = = = 900 C = = 1681 = 41 so this is the P-triple Checking to be safe (and for etra practice) D. 8, 10, 1 D = = = 1 Eample: If! NKL! NJL, NL = 8ft. and KL = 10ft., find JN. Since segment NL is a median, KN = JN, So we find KN. B the Pthagorean theorem, ( KN ) ( KN ) 8 + = 10 ( KN ) = = 36 = 36 KN = 6 ft. DON'T FORGET YOUR UNITS! So, JN = 6 too This gives us another P-triple: 6,8,10 Mr. Korpi,

6 Eample: Find the value of. 4 3 Lesson 1, page 6 of We must find first. For the outer triangle: ( 4 3) + 4 = ( 6+ ) = = = 0 ( )( ) + 14 = 0 = 14 or = But since our side lengths cannot be negative, = Or finding first: ( ) + 6 = = 48 = = 1 = 3 Now finding from the small triangle: ( ) 3 + = 4 + = 1 16 = 16 1 = 4 = Mr. Korpi,

7 Sa What??!! Lesson 1, page 7 of 7 Hpotenuse comes from the common Greek root hpo(for under, as in hpodermic -under the skin) and the less common tein or ten, for stretch. This last is the source of our modern word tension. The hpotenuse was the line segment "stretched under" the right angle. and Frank and Ernest b Bob Thaves Eample: Computer Link The actual screen of one of the newest models of flat screen computer monitors measures 19.5 inches b 1 inches. Find the measure of the diagonal of the screen. Round to the nearest tenth of an inch. You know that the sides of the screen are the legs of a right triangle and that the diagonal is the hpotenuse of the right triangle. Let a = 19.5 and b = 1. Use the Pthagorean Theorem to find c, the hpotenuse. c = a + b c = (19.5) + (1) c = c = 54.5 c = 54.5 c.9 The length of the diagonal of the screen is about.9 inches 1 in in. c in. *all graphics and man eamples are from Mr. Korpi,