Triangles. Learning Objectives. Pre-Activity

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How an she use geometry to ensure that the boards are aligned properly? Geena an use the Pythagorean Theorem (see information below).

The mathematial field of trigonometry is the study of right triangle relationships. Eah dek board must make a square orner with the brae.

Geena an use the Pythagorean Theorem to ensure that the orner is a right angle by measuring the two sides of a triangle formed by the brae and board.

If the line drawn onneting the two points measures exatly 5, then the angle opposite that line is a right angle.

1 Setion 3.2 Pre-tivity Preparation Triangles Geena needs to make sure that the dek she is building is perfetly square to the brae holding the dek in plae. How an she use geometry to ensure that the boards are aligned properly? Geena an use the Pythagorean Theorem (see information below). Every right triangle has speial properties related to the lengths of its sides, its angles, and even the relationship of its angles to its sides. The mathematial field of trigonometry is the study of right triangle relationships. Eah dek board must make a square orner with the brae. square orner is a right angle. Geena an use the Pythagorean Theorem to ensure that the orner is a right angle by measuring the two sides of a triangle formed by the brae and board. She will measure 4 along one side and make a point, and 3 along the other side and make a point. If the line drawn onneting the two points measures exatly 5, then the angle opposite that line is a right angle Learning Objetives Learn how to apply the Pythagorean Theorem to find the length of a side of a right triangle, given its other two sides Use the rules for ongruent triangles to determine if two triangles are ongruent Find the perimeter and area of a triangle given its base and height Terminology Previously Used aute angle angle obtuse angle perpendiular square root vertex New Terms to Learn area base ongruent triangles orresponding sides equilateral height hypotenuse isoseles legs perimeter Pythagorean Theorem right triangle salene side 173

174 hapter 3 Geometry uilding Mathematial Language Triangles Three distint non-parallel lines, lying in the same plane, will interset at three distint points and form a triangle.

) The interior angles of the triangle are a, b, and.

2 174 hapter 3 Geometry uilding Mathematial Language Triangles Three distint non-parallel lines, lying in the same plane, will interset at three distint points and form a triangle. Triangles are named based upon these points of intersetion. The triangle to the right is alled triangle or. (While triangle names are usually alphabetized rather than this is not always the ase.) The interior angles of the triangle are a, b, and. The sum of the interior angles of a triangle is always 180 : a + b + = 180 b a The base or bottom of triangle is the segment,, or depending on the orientation of the triangle. (Orientation means the way the triangle is turned). The height (also alled the altitude) is the perpendiular distane from the top point to the base. Find the height by measuring the length of a line drawn from the top vertex that intersets the base at a right angle. Eah pair of interseting lines forms four angles two pairs of vertial angles. Eah interior angle is adjaent to two supplementary exterior angles and aross from one vertial exterior angle. Types of Triangles Name Piture Speial Features Observations Salene Isoseles No sides are equal and no angles are equal Two equal sides and two equal angles Salene triangles are the generi triangle and have no speial features. Usually presented with the base and equal angles at the bottom. The equal sides are opposite the equal angles. Equilateral ll angles are equal and all sides are equal n equilateral triangle has angles that measure 60. ute ll three angles are aute Eah angle measures less than 90. n equilateral triangle is aute. Obtuse One angle is obtuse One obtuse angle; therefore, two aute angles.

3 Setion 3.2 Triangles 175 Name Piture Speial Features Observations Right Isoseles Right One angle is a right angle. The side opposite the right angle is the hypotenuse. The other two sides an be alled legs. The legs of the right angle are equal and the two aute angles are equal and measure 45. Many onepts in mathematis and many real world appliations use right triangles. Dividing a square on one of its diagonals reates two isoseles right triangles. Dividing a square on both its diagonals yields four isoeles right triangles. Perimeter of a Triangle Perimeter is the distane around an objet in a plane. Perimeter is a measure of length and, therefore, measured in feet (ft), inhes (in), meters (m), entimeters (m), et. In this ontext, the Greek peri- means around and -meter means measure, so the word perimeter means measure around. The perimeter of a triangle is the sum of the measures of the sides: Perimeter P = a + b + The perimeter of triangle P = a + b + = 12 a = 4.9 P = P = 24.7 b = 7.8 rea of a Triangle rea measures the amount of surfae of an objet in a plane. Measure area in terms of square feet (ft 2 ), square meters (m 2 ), square miles (mi 2 ), et. The formula for finding the area of a triangle is one-half times the base times the height: rea = ½bh rea = ½ base height The area of right triangle = ½bh = (½)(12m)(4m) = (6 4)m 2 = 24m 2 Validate: 24m 2 4m = 6m; 6m 12m = ½ height = 4m base (b) = 12m

176 hapter 3 Geometry ongruent Triangles Triangles that are the same size and shape are ongruent triangles. ongruent triangles have equal orresponding angles and equal orresponding side lengths.

4 176 hapter 3 Geometry ongruent Triangles Triangles that are the same size and shape are ongruent triangles. ongruent triangles have equal orresponding angles and equal orresponding side lengths. Use the symbol, to indiate ongrueny. b = 7 a = 5 = 3.5 F e = 7 Triangle, DEF d = 5 D f = 3.5 E, D and a, d, E and b, e, F and, f Tehniques Determining ongrueny between Triangles To determine if two triangles are ongruent, math their orresponding parts and then use one of the following rules. Types of Triangle ongrueny Name Desription Visual Observations S-S-S side side side Two triangles are ongruent if orresponding sides are equal. The tik marks indiate whih sides are onguent. If you an determine that orresponding side measures are equal, then the triangles are ongruent. S--S side-angle-side Two sides and the enlosed angle of one triangle are equal to the orresponding two sides and enlosed angle of the other triangle. The tik marks indiate onguent sides and angles. The equal angle must be between the two equal sides. -S- angle-side-angle Two angles and the enlosed side of one triangle are equal to the orresponding two angles and enlosed side of the other triangle. The tik marks indiate onguent angles and sides. The equal side must be between the two equal angles.

While various soures ontain proofs of the relationship (some of whih ertainly predate Pythagoras work), the theorem was named after him, as it was his work that beame most widely known.

5 Setion 3.2 Triangles Pythagorean Theorem 177 The Pythagorean Theorem is the most widely used and important relationship in geometry. Knowledge of this theory has existed for millennia, with the first reorded statement found on a abylonian tablet ( ). While various soures ontain proofs of the relationship (some of whih ertainly predate Pythagoras work), the theorem was named after him, as it was his work that beame most widely known. In any right triangle, where is the side opposite the right angle: a 2 + b 2 = 2 (Note: side a is aross from angle Pythagorean Theorem and side b is aross from angle.) Example: Given right triangle, and a = 3, b = 4, and = 5: a 2 + b 2 = = = = 25 a 5 4 b Try it! Write down the Pythagorean Theorem for the triangle below. Label the right angle as angle and = 10, b = 8, and a = 6. a 2 a 2 a 2 + b 2 = 2 rea of Square a + rea of Square b = rea of Square You an easily learn more about the Pythagorean Theorem online. Here are some fun sites to get you started: b b a 2 + b 2 = 2

6 178 hapter 3 Geometry Methodology Using the Pythagorean Theorem: a 2 + b 2 = 2 Example 1: Find if a = 7 and b = 12. Round to the nearest tenth, if neessary. Example 2: Find if a = 11 and b = 15. Round to the nearest tenth, if neessary. Label eah triangle with as its right angle. Try It! Steps in the Methodology Example 1 Example 2 Step 1 Identify a, b and on the given triangle. Step 2 Substitute the given side lengths into the formula. Step 3 Solve for the square of the unknown side using order of operations. Step 4 Use your alulator to find the positive square root. Step 5 Validate ny of the three sides an be the unknown value. Make a sketh if neessary. If is not the unknown side, use the equivalent formula to find side a or b: 2 b 2 = a 2 or 2 a 2 = b 2 Solve for a 2 or b 2 if you are using the alternative formulas. Use your alulator if neessary. Round off to the desired number of plaes. If you round off, use not =. Use the original information to hek for equality. a = 7 a = 7 b =12 b = 12 = unknown a 2 + b 2 = = = = = = ? = ? = =?

Setion 3.2 Triangles 179 Models Model 1 Find the perimeter and area of the given right triangle. a = h = 14.7 in = 24.5 in b = 19.6 in Perimeter P = a + b + Validate: 58.8 in 24.5 in = 34.

8 in ) 2 = 19. 6 in 1 2 Model 2 Find the perimeter and area of the given triangle.

7 Setion 3.2 Triangles 179 Models Model 1 Find the perimeter and area of the given right triangle. a = h = 14.7 in = 24.5 in b = 19.6 in Perimeter P = a + b + Validate: 58.8 in 24.5 in = 34.3 in P = in 19.6 in = 14.7 in P = 58.8 inhes rea = ½bh = ½(19.6)(14.7) = (9.8)(14.7) = square inhes 2 Validate: ( in ) = 9. 8 in in ( 9. 8 in) = ( 9. 8 in ) 2 = in 1 2 Model 2 Find the perimeter and area of the given triangle. a = 50 in h = 40 in = 41 in 30 in 9 in Perimeter determine side b (base): Validate: 130 in 41 in = 89 in 30 in + 9 in = 39 in 89 in 50 in = 39 in P = a + b + P = P = 130 inhes rea = ½bh = ½(39)(40) = (19.5)(40) = 780 square inhes 2 Validate: ( 780 in ) = in 40 in ( in) = ( in ) 2 = 39 in 1 2

8 180 hapter 3 Geometry Model 3 Given a right triangle, find side b if side a = 3.9 and side (hypotenuse) = 6.5. pproximate the answer to the nearest tenth, if neessary. Step 1 Identify a = 3.9 b = unknown = 6.5 Step 2 Substitute 2 a 2 = b 2 (lternative formula to find side b.) = b 2 Step 3 Solve = b = b 2 Step 4 Square root 5.2 = b Step 5 Validate ? = ? = = a = 3.9 b = 6.5 Model 4 Given a right triangle, find side a if side b = 11 and side (hypotenuse) = pproximate the answer to the nearest tenth, if neessary. Step 1 Identify a = unknown b = 11 = 18.2 Step 2 Substitute 2 b 2 = a = a 2 Step 3 Solve = a = a 2 Step 4 Square root 14.5 a Step 5 Validate ? = ? = a b = 11 = 18.2

9 Setion 3.2 Triangles 181 Model 5a E re the triangles in the pair at right ongruent? Use what you know about triangles, inluding types of ongrueny and how to solve for unknown sides in right triangles, in order to determine if the triangles at right are ongruent with eah other. Provide an explanation for your answer(s). a 45 b = 3m 45 d = 3m 1 2 F e f 45 D Reason The missing angle in triangle 1 ( ) is 45 : = = 45 Triangle 1 is therefore an isoseles triangle and side a = 3m. Triangle 2 is also an isoseles triangle and side e = 3m. With the enlosed angle of 90, the S--S rule of ongruity is satisfied. See the diagram below. E Type of ongrueny S--S Side 1: a, d ngle:, F Side 2: b, e 1, SIDE 1 SIDE 1 a = 3m d = 3m = 90 b = 3m F = 90 e = 3m NGLE SIDE 2 NGLE SIDE 2 D We an also apply the Pythagorean Theorem to find the length of the hypotenuse, to show S-S-S ongrueny. a = 3m, b = 3m 2 = = m d = 3m, e = 3m f 2 = = 18 f 4.24 m 45 = 4.24m 45 f = 4.24m SIDE 3 SIDE 3 a = 3m d = 3m SIDE 1 SIDE b = 3m F e = 3m SIDE 2 SIDE 2 One we have used the Pythagorean Theorem to find sides and f, we have all angle and side measurements for eah triangle. This allows us to show -S- ongrueny, in three different ways: E D S-S-S Side 1: a, d Side 2: b, e Side 3:, f 1, 2 -S- 1, 2 -S- ngle 1:, E Side: a, d ngle 2:, F -S- ngle 1:, F Side: b, e ngle 2:, D -S- ngle 1:, D Side:, f ngle 2:, E Knowing the missing hypotenuse for eah triangle also enables us to show S--S ongrueny in two additional ways: S--S Side 1: a, d ngle:, E Side 2:, f S--S Side 1:, f ngle:, D Side 2: b, e

10 182 hapter 3 Geometry Model 5b re the triangles in the pair at right ongruent? Use what you know about triangles, inluding types of ongrueny and how to solve for unknown sides in right triangles. Provide an explanation for your answer x Reason Type of ongrueny These are not right triangles, so we annot use the Pythagorean Theorem to find missing side lengths. However, we already have equivalent sides of 1.8 and equivalent angles of 52. If we an solve for the missing angle ( x) in triangle 2 and it is equal to 80, then we an show -S- ongrueny. x = = 80. The 80 angles, the 52 angles, and their enlosed sides of 1.8 satisfy the -S- rule of ongrueny. See the diagram below. NGLE 1 -S- ngle 1: 80 = 80 Side: 1.8 = 1.8 ngle 2: 52 = 52 1, 2 SIDE NGLE NGLE SIDE NGLE 2 Model 5 Determine whether or not the triangles in the pair at right ongruent. Use what you know about triangles, inluding types of ongrueny and how to solve for unknown sides in right triangles. Provide an explanation for your answer Reason Type of ongrueny Keep in mind that you annot determine ongrueny by looks. You must use S-S-S, S--S, or -S- to show ongrueny. The largest orresponding angles equal 99, so they are not right triangles and the Pythagorean Theorem annot be used. While the triangle on the right has the given angle enlosed with the two given sides, the triangle on the left does not; therefore, the relationship does not fit any of the three rules for ongrueny. NOT ONGRUENT

sides are equal measures? a b z X a = z = x b = y Y y x Z Orientation matters. Rearrange the triangles so that the largest angles are mathed.

11 Setion 3.2 Triangles 183 ddressing ommon Errors Issue Inorret Proess Resolution orret Proess Validation Misidentifying orresponding sides or angles If triangle is ongruent to triangle XYZ, ( 9 / 9XYZ) whih sides are equal measures? a b z X a = z = x b = y Y y x Z Orientation matters. Rearrange the triangles so that the largest angles are mathed. The seond triangle should be rotated or flipped until angle Z is in the same position as angle. a b x Z a = x b = y = z Y y z X Use of the Pythagorean theorem on non-right triangles If side a = 12 and side b = 6, what is the measure of side? a b = a + b, so = and 2 = = 180 The Pythagorean Theorem is only appliable to RIGHT triangles. areful observation reveals that the triangle is not a right triangle; no indiation of a right angle is present. The Pythagorean Theorem is not useful here. Side annot be determined with the information given. = pplying the Pythagorean Theorem inorretly In right triangle : b a if a = 5 and b = 7 find. The Pythagorean Theorem states that the SQURES of the two legs added together equals the SQURE of the hypotenuse. a 2 + b 2 = = = 2 74 = 2 = ? = ? = a + b = = 12 =

12 184 hapter 3 Geometry Issue Inorret Proess Resolution orret Proess Validation Using the wrong side as the hypotenuse in the Pythagorean formula Triangle RST is a right triangle. Find side s if r = 3 and t = 5 S r T t s R The hypotenuse is always opposite the right angle. In the given figure, the right angle is opposite side t, not side s. Use is the alternative formula for finding side s: s 2 = t 2 r 2. s 2 = t 2 r 2 t 2 = r 2 + s 2 5 2? = ? = = 25 s = r + t s s s = = = 34 s 2 = s 2 = 25 9 s 2 = 16 s = 4 s = 34 s Not using square units for area Find the area of a triangle whose base is 10 mm and height is 6mm. = ½bh = (½)(6)(10) rea is always given in square units. arry the units along with the measures to remind you to report the units as part of the answer. = ½bh = (½)(6mm)(10mm) = 30 mm 2 (mm)(mm) = mm 2 = 30 mm Not using an appropriate piee of information in a formula. Find the area of the given triangle: a b When a = 6, b = 10, and h = 8. = ½bh = (½)(6)(10) Make sure that you orretly identify eah piee of information before using it in a formula. learly labeling a drawing or sketh an help. a = 6 h = 8 b = 10 From the new drawing, the height is learly shown. = ½bh = (½)(6)(8) = 24 square units Using the area alulated, solve for h: = ½bh 24 = (½)(6)(h) 24 = 3h 24 3 = h h = 8, the height as given. = 30 square units

to establish the ongrueny of two triangles N a Triangle Have Three 90-Degree ngles? Of ourse not, but it is a fun optial illusion.

13 Setion 3.2 Triangles 185 Preparation Inventory efore proeeding, you should be able to: Find the area and perimeter of a right triangle Find the perimeter and area of a triangle, given its base and height. Use the Pythagorean Theorem to find the unknown side of a right triangle Find the area and perimeter of triangles using the Pythagorean Theorem Use the ongrueny rules to establish the ongrueny of two triangles N a Triangle Have Three 90-Degree ngles? Of ourse not, but it is a fun optial illusion. It was first reated by the Swedish artist Osar Reutersvärd in 1934, but later independently devised and made popular by the mathematiian Roger Penrose. He desribed it as impossibility in its purest form. You an learn more by searhing the internet for the Penrose triangle. Impossible Triangle sulpture by rian MaKay & hmad bas. It is loated at laisebrook roundabout, East Perth, Western ustralia. This Penrose triangle only appears to have three 90-degree angles when seen from ertain perspetives.

alulation of the measure of the third side alulating perimeters and areas of triangles orret and appropriate use of the perimeter formula for a triangle orret and appropriate use of the area formula

14 Setion 3.2 tivity Triangles Performane riteria Finding the third side of a right triangle when given the measures of two of its sides appropriate use of the Pythagorean Theorem to find the third side orret alulation of the measure of the third side alulating perimeters and areas of triangles orret and appropriate use of the perimeter formula for a triangle orret and appropriate use of the area formula for a triangle alulating the measures of sides and angles in two given triangles orret identifiation of orresponding parts orret alulation of missing angle or side measures orret determination of ongrueny ritial Thinking Questions 1. If two triangles are ongruent by the S-S-S rule, are the triangles always equilateral triangles? Explain. 2. If you know that two triangles have two pairs of equal orresponding angles, what an you determine about the third pair of angles? 3. Why an t angle-angle-angle be a rule to determine two ongruent triangles? 186

15 Setion 3.2 Triangles an a triangle have two obtuse angles? Explain your answer. 5. Why an you eliminate the negative square root when finding the square root in the Pythagorean Theorem? 6. Why is the height needed to measure the area of a triangle? 7. Given a triangle with the following side lengths: side a = 3 feet side b = 4 meters side = 2 yards, an you alulate the perimeter? If so, what units would your answer have? Explain your reasoning.

188 hapter 3 Geometry Tips for Suess If a figure

measurements Use graph paper to sketh more

same units Demonstrate Your Understanding 1.

16 188 hapter 3 Geometry Tips for Suess If a figure is not provided, make a quik sketh showing all measurements Use graph paper to sketh more aurately Make sure the measurements are in the same units Demonstrate Your Understanding 1. Find the perimeter of the following triangles: a) Problem Worked Solution Validation 18 in 22.5 in 13.5 in b) 18.5 m 6 m 6.8 m 20.7 m ) 12 m 15 m 9 m d) 10.8 m 17.3 m 9.6 m

17 Setion 3.2 Triangles Find the area of the following triangles: a) Problem Worked Solution Validation 18 in 22.5 in 13.5 in b) 18.5 m 6 m 6.8 m 20.7 m ) right triangular trat of land has side a = 41.6 miles and side b = 31.2 miles mi 41.6 mi d) 11 in 12 in

18 190 hapter 3 Geometry 3. Use the Pythagorean Theorem to solve the following problems. Round your answers to the nearest tenth, if neessary. a) Find. Problem Worked Solution Validation 11 in 12 in b) Find b. 15 in 20 in b ) Find a. a = 19.4 b = 13 d) Find and e. =? a = 7.5 e =? h = 4 d = 4.2

19 Setion 3.2 Triangles 191 Problem Worked Solution Validation e) Find the perimeter of the largest triangle in problem 3 d). f) Find the area of the largest triangle in problem 3 d). 4. re the triangles in the following pairs ongruent? Use what you know about triangles, inluding types of ongrueny and how to solve for unknown sides in right triangles, in order to determine if the triangles are ongruent with eah other. Provide an explanation for your answer(s). Triangle Pair Reason Type(s) of ongrueny a) 6.5 in 3.9 in 5.2 in in d

20 192 hapter 3 Geometry Triangle Pair Reason Type(s) of ongrueny b) 7 m 60 7 m 60 e m 60 E 7 m f ) w x 33

Worked Solution Identify Errors or Validate orret Proess Validation 1) The two right triangles below are ongruent. If 1 = 43, what is the measure of 2?

21 Setion 3.2 Triangles 193 Identify and orret the Errors In the seond olumn, identify the error(s) in the worked solution or validate its answer. If the worked solution is inorret, solve the problem orretly in the third olumn and validate your answer. Worked Solution Identify Errors or Validate orret Proess Validation 1) The two right triangles below are ongruent. If 1 = 43, what is the measure of 2? = 2 (orresponding parts) so 2 = 43 2) Find the measure of side b if = 22 in and a = 8 in. 2 = b 2 + a = b = b b 2 = 420 b 20.5 in 3) Find the measure of side a if = 2.5 in and b = 1.5 in. a 2 = 2 b 2 a 2 = a = a = 4 in

22 194 hapter 3 Geometry Worked Solution Identify Errors or Validate orret Proess Validation 4) Find the area of triangle DEF. D f = 5 E h = 4 d = 10.5 e = 8.5 F = ½bh = (½)(10.5)(5) = square units 5) right triangle has 1 = 26. What is the measure of 2? 1 2 The triangle is a right triangle, with one angle = 90. The three angles must add to 180, therefore: 180 ( ) = = 64 2 = 64

Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z.

1 Use measurements of the two triangles elow to find x and y. Are the triangles similar or ongruent? Explain. 1a Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x,