Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Golden ratio wikipedia , lookup
History of geometry wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Integer triangle wikipedia , lookup
LESSON F5.3 – GEOMETRY III LESSON F5.3 GEOMETRY III 473 474 TOPIC F5 GEOMETRY Overview You have learned how to identify different types of polygons, including triangles, and how to find their perimeter and their area. In this lesson, you will learn more about triangles. You will study the properties of triangles in which two or three lengths or two or three angles have the same measure. You will then work with similar triangles, triangles with the same shape but different size. Also, you will study right triangles and the Pythagorean Theorem. Finally, you will learn more about parallelograms and parallel lines. Before you begin, you may find it helpful to review the following mathematical ideas which will be used in this lesson: To see these Review problems worked out, go to the Overview module of this lesson on the computer. Review 1 Use supplementary angles. ∠ABC and ∠DEF are supplementary angles. m∠ABC = 144˚. Find m∠DEF. Answer: 36° Review 2 Use vertical angles. ∠XZV and ∠UZV are vertical angles. m∠XZV = 44.6˚. Find m∠UZV. Answer: 44.6° Review 3 Square a whole number. Find the value of this expression: 7 2 + 8 2. Answer: 113 Review 4 Find the square root of a whole number. 2+ Find the value of this expression: 6 82. Answer: 10 Review 5 Work with a proportion. Let’s solve this problem using a proportion. Mary finishes 3 homework problems in 5 minutes. Working at the same rate, find x, the number of problems she will finish in 30 minutes. Here’s a proportion we can use x 3 to find x: = . What is the value of x? 5 30 Answer: 18 LESSON F5.3 GEOMETRY III OVERVIEW 475 Explain In Concept 1: Triangles and Parallelograms, you will learn about: CONCEPT 1: TRIANGLES AND PARALLELOGRAMS • The Sum of the Angle Measures of a Triangle The Sum of the Angle Measures of a Triangle • Congruent Triangles • Isosceles Triangles and Equilateral Triangles • Right Triangles and the Pythagorean Theorem In solving problems about angles in triangles, it is useful to know the total number of degrees in the angles of a triangle. The angle measures of a triangle add to 180 degrees. An example is shown in Figure 1. 65ß • Parallel Lines and Parallelograms 40ß 75ß Figure 1 40˚ + 65˚ + 75˚ = 180˚ You may find these Examples useful while doing the homework for this section. Example 1 1. In Figure 2, find the missing angle measure in ∆ABC. A ? We can name a triangle using its 3 vertices (corners). We can list the vertices in any order. “∆ABC” means “the triangle with vertices A, B, and C.” “∠A” means “the angle with vertex A.” 70ß 60ß B C Figure 2 Here’s one way to find the missing angle measure in this triangle: • Add the given angle measures. • Subtract that result from 180˚. So, the measure of ∠A is 50˚. 476 TOPIC F5 GEOMETRY 60˚ + 70˚ = 130˚ 180˚ – 130˚ = 50° 2. In Figure 3, find the missing angle measure in ∆DEF. Example 2 E 1 80 2 ˚ ? F 30˚ D Figure 3 Here’s one way to find the missing angle measure in this triangle: 1 2 1 2 • Add the given angle measures. 80 ˚ + 30˚ = 110 ˚ • Subtract that result from 180˚. 180˚ – 110 ˚ = 69 ˚ 1 2 1 2 1 2 So, the measure of ∠F is 69 ˚. 3. In Figure 4, the measures of two angles are shown. Find the measure of ∠S. Example 3 T 30˚ S 70˚ ? A Figure 4 Here’s one way to find the measure of ∠S. The measures of supplementary angles add to 180˚. Substitute 30˚ for m∠ T and 110˚ for m∠A. Subtract 30˚ and 110˚ from both sides. Solve for m∠S. m∠S + m∠ T + m∠A = 180˚ m∠S + 30˚ + 110˚ = 180˚ m∠S = 180˚ – 30˚ – 110˚ m∠S = 40˚ So, m∠S is 40˚. LESSON F5.3 GEOMETRY III EXPLAIN 477 “≅” means “is congruent to.” Congruent Triangles “AB” means “the line segment connecting vertices A and B.” Figures with the same shape and size are called congruent figures. See Figure 5. A 36ß 2 cm C J B 2 cm D K 36ß ∠J ∠K AB CD Figure 5 For two congruent triangles, we can match all 3 sides and all 3 angles. In Figure 6, matching marks show the corresponding sides and angles of the two congruent triangles. L ˘LMN ˘PQR P Q M R N Figure 6 The order of letters in the congruence relation matches the corresponding angles or vertices of each triangle. Since ∆LMN ∆PQR, ∠L corresponds to ∠P. ∠M corresponds to ∠Q. ∠N corresponds to ∠R. You may find these Examples useful while doing the homework for this section. Example 4 4. In Figure 7, ∆RUN is congruent to ∆FST. Which angle of ∆FST is congruent to ∠N? Figure 6 478 TOPIC F5 GEOMETRY One way is to use the order of the vertices in the given congruence. N is the third letter of ∆RUN. T is the third letter of ∆FST. So, ∠N is congruent to ∠ T. 5. In Figure 7, ∆RUN ∆FST. Which side of ∆FST is congruent to UN? Example 5 N R T U F S ˘RUN ˘FST Figure 7 One way is to use the order of the vertices in the given congruence. UN is congruent to ST. ∆RUN ∆FST Here are the other pairs of congruent sides: RU is congruent to FS. RN is congruent to FT. 6. ∆RUN ∆FST ∆RUN ∆FST The triangles in Figure 8 are congruent. Write a congruence relation for these triangles. J E Example 6 S T K I Figure 8 Here’s one way to write a congruence relation: • First, write one pair of corresponding vertices. • Then, write another pair of corresponding vertices. • Finally, write the last pair of corresponding vertices. ∆J ∆S ∆JE ∆SK ∆JET ∆SKI So, one congruence relation for these triangles is ∆JET ∆SKI. Any congruence relation that matches corresponding vertices is correct. Here are 3 more examples: ∆ETJ ∆KIS ∆JTE ∆SIK ∆TEJ ∆IKS LESSON F5.3 GEOMETRY III EXPLAIN 479 Example 7 7. The triangles in Figure 9 are congruent. Find the measure of ∠B. F C 50˚ N ? B 30˚ ∆FAN ∆CLB A L Figure 9 The congruence relation states that ∠B corresponds to ∠N. So, ∠B and ∠N have the same measure. Here’s one way to find the measure of ∠N. Add 50˚ and 30˚. Subtract that result from 180˚. 50˚ + 30˚ = 80˚ 180˚ – 80˚ = 100˚ So, ∠B and ∠N each have measure 100˚. Example 8 8. The triangles in Figure 10 are congruent. Find the length of RK. K N 3 ft 5 ft R I 4 ft S F ∆SRK ∆FIN Figure 10 The congruence relation states that RK corresponds to IN. So, RK and IN have the same length. The length of IN is 3 ft. So, the length of RK is also 3 ft. 480 TOPIC F5 GEOMETRY Isosceles Triangles and Equilateral Triangles There are several special types of triangles. An Isosceles triangle is one such type. An isosceles triangle has at least two sides that have the same length. In Figure 11, each triangle is an isosceles triangle. The word isosceles comes from two Greek words: ISOS + “equal” 82 cm “leg” So isosceles means having two or three “equal legs.” 60˚ 100 cm 76˚ SKELOS 100 cm 82 cm 52˚ 60˚ 52˚ 60˚ 100 cm 100 cm Isosceles Triangles Figure 11 In an isosceles triangle, the angles that are opposite the equal-length sides have the same measure. If a triangle has two angles with equal measures, then the opposite sides also have equal lengths, and the triangle is an isosceles triangle. Equilateral Triangles An Equilateral triangle is another special type of triangle. An equilateral triangle has three sides of equal length. In Figure 12, each triangle is an equilateral triangle. EQUI 60° 60° 100 cm 100 cm 60° 60° 120 cm In the word equilateral, the prefix “equi” means “equal” and “lateral” means “side.” + “equal” 120 cm LATERAL “side” So equilateral means having three “equal sides.” 60° 60° 100 cm 120 cm Equilateral Triangles Figure 12 In an equilateral triangle, all three angle measures are equal. Since the angle measures in a triangle add to 180˚, each angle in an equilateral triangle measures 180˚ ÷ 3. So, the measure of each angle is 60˚. If each angle of a triangle measures 60˚, then the triangle is an equilateral triangle. LESSON F5.3 GEOMETRY III EXPLAIN 481 You may find these Examples useful while doing the homework for this section. Example 9 9. In Figure 13, ∆PAT is an isosceles triangle. Which pair of angles must have the same measure? P 4 cm 2 cm A 4 cm T Figure 13 In ∆PAT, PA is the same length as AT. So the angles opposite these sides have the same measure. That is, m∠T = m∠P. Example 10 10. In Figure 14, ∆TOP is an isosceles triangle. Find the measure of ∠P. O T 50˚ ? P Figure 14 In an isosceles triangle, the angles across from the equal-length sides have equal measure. The matching marks on TO and OP indicate that these sides have equal lengths. So, ∠T and ∠P have the same measure, 50˚. Example 11 11. Find the length of SN in ∆SUN, as shown in Figure 15. N 70˚ S 3 cm 70˚ U Figure 15 Since two angles, ∠N and ∠U, of ∆SUN have equal measures, the triangle is isosceles. In an isosceles triangle, the sides opposite equal-measure angles have the same length. So SN has the same length as SU, 3 cm. 482 TOPIC F5 GEOMETRY 12. Sketch an isosceles triangle ∆SAW, with SA and SW of equal lengths. Figure 16 is one possible answer. Example 12 S W A Figure 16 13. In Figure 17, ∆FAX is an equilateral triangle. Find the length of FA. Example 13 X 6 cm F A Figure 17 In an equilateral triangle, all 3 sides have the same length. So FA has the same length as FX, 6 cm. (AX also has length 6 cm.) 14. In the triangle in Figure 18, find the measure of angle Q. Example 14 Q ? Figure 18 Since all 3 sides have matching marks, the 3 sides have equal lengths. That means the triangle is an equilateral triangle. All 3 angles have the same measure. 180° ÷ 3 = 60˚ So, m∠Q = 60˚. “m∠Q” means the measure of angle Q. LESSON F5.3 GEOMETRY III EXPLAIN 483 Right Triangles and the Pythagorean Theorem Another special type of triangle is a right triangle. A right triangle has one right angle that measures 90˚, as shown in Figure 19. hypotenuse leg 90° leg Right Triangle Figure 19 In a right triangle, the two sides that form the right angle are called the legs of the triangle. The side opposite the right angle is called the hypotenuse. It’s the longest side. In a right triangle, the right angle is the largest angle. The other two angles are acute angles. The Pythagorean Theorem The Pythagorean Theorem describes an important relationship among the sides of a right triangle, as shown in Figure 21. Pythagorean Theorem hypotenuse c leg b leg a a2 + b2 = c 2 Figure 21 a, b, and c are the lengths of the sides, as shown in Figure 21. In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: a 2 + b 2 = c 2 The Pythagorean Theorem is true only for right triangles. This gives us a way to test whether a triangle is a right triangle. See Figure 22. c b a Figure 22 484 TOPIC F5 GEOMETRY • If a 2 + b 2 = c 2, then the triangle is a right triangle. • If a 2 + b 2 ≠ c 2, then the triangle is not a right triangle. (Here, a, b, and c are the lengths of the sides. The length of the longest side, the hypotenuse, is c.) A Pythagorean triple is a group of three whole numbers that can be used as the lengths of the sides of a right triangle. Figure 23 shows some examples of Pythagorean triples. 5 4 13 12 3 5 (3, 4, 5) (6, 8, 10) (9, 12, 15) (12, 16, 20) (5, 12, 13) (10, 24, 26) (15, 36, 39) (20, 48, 52) 21 29 20 (20, 21, 29) (40, 42, 58) (60, 63, 87) (80, 84, 116) Figure 23 15. In a given right triangle, the measure of one angle is 35˚. Find the measures of the other angles. Example 15 Figure 24 shows a right triangle with one angle whose measure is 35˚. You may find these Examples useful while doing the homework for this section. B 35ß A C Figure 24 One of the angles in a right triangle must be a right angle. It’s labeled ∠B. Since ∠B is a right angle, m∠B = 90˚. To find the measure of ∠A, add 90° and 35˚. 90˚ + 35˚ = 125˚ Subtract that result from 180˚. 180˚ – 125˚ = 55˚ So, m∠A = 55˚. 16. In a certain right triangle, the lengths of the sides are 9 cm, 12 cm, and 15 cm. Which of these lengths is the length of the hypotenuse? Example 16 The hypotenuse is the longest side, so the length of the hypotenuse is 15 cm, as shown in Figure 25. 15 cm 12 cm 9 cm Figure 25 LESSON F5.3 GEOMETRY III EXPLAIN 485 Example 17 17. In ∆TRI, the lengths of the sides are 4 ft, 5 ft, and 6 ft. Is ∆TRI a right triangle? See Figure 26. T 6 ft 5 ft R I 4 ft Figure 26 Here’s one way to find out whether ∆TRI is a right triangle: • Square the length of each side. • Add the two smaller squares and see if the result equals the largest square. Since 16 + 25 = 41 (not 36), ∆TRI is not a right triangle. Example 18 18. 4 2 = 4 4 = 16 5 2 = 5 5 = 25 6 2 = 6 6 = 36 Is 16 + 25 = 36? Is 41 = 36? No. In Figure 27, the triangle shown is a right triangle. Find the length of its hypotenuse. 8 ft 6 ft c Figure 27 Here’s one way to find the length of the hypotenuse. Use the Pythagorean Theorem. Square the given lengths. Add the squares. To find c, take the square root of 100. The length of the hypotenuse is 10 ft. Example 19 19. 78 cm a Figure 28 TOPIC F5 GEOMETRY = = = = = c2 c2 c2 c2 c In Figure 28, the triangle shown is a right triangle. Find a, the length of one of its legs. 72 cm 486 a2 + b2 62 + 82 36 + 64 100 10 Here’s one way to find the length of the leg. Use the Pythagorean Theorem. Square the given lengths. Add the squares. To get a 2 by itself, subtract 5184 from both sides. a2 + b2 2 a + 72 2 a 2 + 5184 – 5184 a2 = = = c2 78 2 6084 – 5184 = 900 To find a, take the square root of 900. The length of the leg is 30 cm. a = 30 Parallel Lines and Parallelograms You have learned about parallel lines. Now you will study some properties of the angles formed by a line that crosses two parallel lines. In Figure 29, line k and line n are parallel. Line t crosses lines k and n, and forms eight angles, which are shown. When line k is parallel to line n, we write k || n. t k 1 2 3 4 n 5 7 6 8 Figure 29 As shown in Figure 30, there are 4 pairs of vertical angles. Vertical angles are opposite angles. t k 1 2 34 n 56 7 8 ∠1 and ∠4 ∠2 and ∠3 are vertical angles are vertical angles ∠5 and ∠8 ∠6 and ∠7 are vertical angles are vertical angles Vertical angles have the same measure whether lines k and n are parallel or not. Figure 30 LESSON F5.3 GEOMETRY III EXPLAIN 487 Vertical Angles Vertical angles have the same measure. m∠1 = m∠4 m∠5 = m∠8 m∠2 = m∠3 m∠6 = m∠7 Corresponding angles are two angles on the same side of line t that are both “above” or both “below” lines k and n. There are 4 pairs of corresponding angles, as shown in Figure 31. t k 1 2 34 n 56 78 Corresponding angles form an “F.” This “F” may face in any direction. k k n n Caution! Line k and line n must be parallel for corresponding angles to have the same measure. ∠1 and ∠5 ∠2 and ∠6 are corresponding angles are corresponding angles ∠3 and ∠7 ∠4 and ∠8 are corresponding angles are corresponding angles Figure 31 Corresponding Angles Because line k || line n, corresponding angles have the same measure. m∠1 = m∠5 m∠3 = m∠7 m∠2 = m∠6 m∠4 = m∠8 Alternate interior angles are two angles on opposite sides of line t that are both “inside” line k and line n. 488 TOPIC F5 GEOMETRY There are 2 pairs of alternate interior angles, as shown in Figure 32. Alternate interior angles form a “Z.” This “Z” may face in any direction. t k k n k 1 2 34 n ∠3 and ∠6 n 56 78 ∠4 and ∠5 are alternate interior angles are alternate interior angles Caution! Lines k and n must be parallel for alternate interior angles to have the same measure. Figure 32 Alternate Interior Angles Because line k || line n, alternate interior angles have the same measure. m∠3 = m∠6 m∠4 = m∠5 Parallelograms Recall that a parallelogram is a polygon with 4 sides that has two pairs of parallel sides. See Figure 33. In a parallelogram, the opposite sides have the same length. Figure 33 LESSON F5.3 GEOMETRY III EXPLAIN 489 In Figure 34, all the marked angles with “(” or “)” have the same measure as angle 1. That’s because vertical angles have the same measure and corresponding angles formed by parallel lines have the same measure. vertical angles 2 1 4 3 corresponding angles Figure 34 Notice that angle 1 and angle 3 have the same measure. In the same way, angle 2 and angle 4 have the same measure. It follows that: Opposite angles of a parallelogram have the same measure. In Figure 35, since opposite sides of the parallelogram are parallel, the corresponding angles have the same measure. The measures of supplementary angles add to 180˚. 1 corresponding angles 2 3 4 supplementary angles Figure 35 It follows that: Consecutive angles of a parallelogram are supplementary angles. 490 TOPIC F5 GEOMETRY 20. In Figure 36, j || u. The measure of angle 1 is 80˚. Find the measures of the other angles. Example 20 t j You may find these Examples useful while doing the homework for this section. 80˚ 1 2 3 4 u 5 7 6 8 Figure 36 Here’s one way to find the measures of the other angles. ∠4 and ∠1 are vertical angles, so they have the same measure. ∠2 and ∠1 are supplementary angles, so their measures add to 180˚. 180˚ – 80˚ = 100˚ ∠3 and ∠2 are vertical angles, so they have the same measure. ∠5 and ∠1 are corresponding angles, so they have the same measure. ∠6 and ∠2 are corresponding angles, so they have the same measure. ∠7 and ∠3 are corresponding angles, so they have the same measure. ∠8 and ∠4 are corresponding angles, so they have the same measure. m∠4 = m∠1 = 80˚ m∠2 = 100˚ m∠3 = m∠2 = 100˚ m∠5 = m∠1 = 80˚ m∠6 = m∠2 = 100˚ m∠7 = m∠3 = 100˚ m∠8 = m∠4 = 80˚ Here’s another way to find m∠6: Since ∠6 and ∠3 are alternate interior angles, m∠6 = m∠3 = 100˚. Similarly, since ∠4 and ∠5 are alternate interior angles, m∠4 = m∠5 = 80˚. LESSON F5.3 GEOMETRY III EXPLAIN 491 Example 21 21. Line segment BE is parallel to line segment CD. Find the missing angle measures in ∆ACD and ∆ABE. See Figure 37. C 40˚ B ? ? ? D E Figure 37 Here’s a way to find the measure of angle ABE. Since BE || CD, ∠ABE and ∠BCD are corresponding angles. Corresponding angles have the same measure. So, m∠ABE = 40˚. 30˚ A Here’s a way to find the measure of angle AEB. The angle measures of a triangle add to 180˚. Since 30˚ + 40˚ = 70˚ and 180˚ – 70˚ = 110˚, m∠AEB = 110˚. Here’s a way to find the measure of angle D. Since BE || CD, ∠D and ∠AEB are corresponding angles. Corresponding angles have the same measure. So, m∠D = 110˚. Example 22 22. Find the missing angle measures in parallelogram GRAM. See Figure 38. G R 125˚ ? ? ? A M Figure 38 In a parallelogram, opposite angles have the same measure. Since angle M and angle R are opposite angles, m∠M = 125˚. In a parallelogram, consecutive angles are supplementary. Angle G and angle R are consecutive angles. Since 180˚ – 125˚ = 55˚, m∠G = 55˚. Since angle A and angle G are opposite angles, the measure of angle A is also 55˚. Example 23 23. Find the missing side lengths in parallelogram RUSH. See Figure 39. ? R U ? H 2 cm 5 cm Figure 39 S In a parallelogram, opposite sides have the same length. Since US and RH are opposite sides, the length of RH is 2 cm. Since SH and RU are opposite sides, the length of RU is 5 cm. 492 TOPIC F5 GEOMETRY Explain In Concept 2: Similar Polygons, you will find a section on each of the following: CONCEPT 2: SIMILAR POLYGONS • Recognizing Similar Polygons You have learned that congruent objects have the same shape and the same size. Similar objects must also have the same shape, but they may or may not be the same size. An architect’s scale model is similar to the full-size building. The street map of a city is similar to the actual streets and blocks of the city. If you shrink a picture when you photocopy it, the two images are similar. If you enlarge a photograph, the original and the enlarged version are similar. • Writing a Similarity Statement • Using Shortcuts to Recognize Similar Triangles Recognizing Similar Polygons In Figure 40, the pentagons are similar polygons. They are also congruent. • Finding the Measures of Corresponding Angles of Similar Triangles • Finding the Lengths of Corresponding Sides of Similar Triangles D C A B H F G same shape same size ABCDE ~ FGHIJ Congruent objects must be similar. But not all similar objects are congruent. “~” means “is similar to” I J E Figure 40 In Figure 41, the hexagons are similar polygons, but they are not congruent. O N P U K T V M S Q L same shape different sizes R KLMNOP ~ QRSTUV Figure 41 For two polygons to be similar, both of the following must be true: • Corresponding angles have the same measure. • Lengths of corresponding sides have the same ratio. LESSON F5.3 GEOMETRY III EXPLAIN 493 In Figure 42, the quadrilaterals STAR and FIND are similar, since they satisfy both of the conditions above. D 15 cm R 10 cm S F 6 cm 4 cm 9 cm 6 cm T A 8 cm I 12 cm STAR ~ FIND N Figure 42 ST is the length of line segment ST. FI is the length of line segment FI, and so on. We also say that the lengths of corresponding sides are proportional. You may find these Examples useful while doing the homework for this section. Example 24 Corresponding angles have the same measure: Corresponding lengths have the same ratio: m∠S = m∠F m∠T = m∠I ST 4 2 = = FI 6 3 2 TA 8 = = IN 12 3 m∠A = m∠N m∠R = m∠D AR 6 2 = = ND 9 3 2 RS 10 = = DF 15 3 24. Are the polygons ABCD and EFGH in Figure 43 similar? B 3 cm 2 cm A C 2 cm 3 cm D 3 cm F G 140ß 40ß 2 cm 2 cm 40ß 140ß E H 3 cm Figure 43 The polygons are not similar. Corresponding lengths have the same ratio, but corresponding angles do not have the same measure: 2 2 • =1 3 =1 3 2 =1 2 3 =1 3 • All the angles in the rectangle measure 90˚. Two of the angles in the parallelogram measure 40˚, and two of the angles measure 140˚. Example 25 25. Are the polygons ABCD and EFGH in Figure 44 similar? B 3 cm 2 cm A C 2 cm 3 cm D Figure 44 494 TOPIC F5 GEOMETRY F 2 cm G 2 cm 2 cm E 2 cm H The polygons in Figure 44 are not similar. Corresponding angles have the same measure, but lengths of corresponding sides do not have the same ratio: • All the angles have the same measure, 90˚. 2 2 • =1 3 = 1.5 2 2 =1 2 3 = 1.5 2 Some lengths of corresponding sides have ratio 1 and some have ratio 1.5. Example 26 26. In Figure 45, triangle SUM is similar to triangle CAR. Find the missing angle measures. C R M S 20˚ 60˚ U ∆SUM ~ ∆CAR A Figure 45 Corresponding angles have the same measure. So, m∠C = m∠S = 20˚ and m∠U = m∠A = 60˚. You can use the fact that the angle measures in a triangle add to 180˚ to find the measure of the third angle in each triangle. • Add the angle measures you know. 20˚ + 60˚ = 80˚ • Subtract that result from 180˚. 180˚ – 80˚ = 100˚ So, m∠M = m∠R = 100˚. __ length of CA 27. In Figure 46, ∆SUM is similar to ∆CAR. Find this ratio: length of SU Example 27 C R M S 3 cm 2 cm U ∆SUM ~ ∆CAR A Figure 46 In similar triangles, corresponding lengths have the same ratio. __ __ 3 length of AR length of CA = = 2 length of UM length of SU LESSON F5.3 GEOMETRY III EXPLAIN 495 Writing a Similarity Statement In Figure 47, triangle FAN and triangle CLB are similar. F A N L C B Figure 47 The order of letters in the similarity statement ∆FAN ~ ∆CLB matches the corresponding angles or vertices of each polygon. ∠F corresponds to ∠C. ∠A corresponds to ∠L. ∠N corresponds to ∠B. ∆FAN ~ ∆CLB ∆FAN ~ ∆CLB ∆FAN ~ ∆CLB The order of the letters in the similarity statement also matches corresponding sides. ∆FAN ~ ∆CLB ∆FAN ~ ∆CLB ∆FAN ~ ∆CLB FA corresponds to CL. AN corresponds to LB. FN corresponds to CB. Example 28 28. Write a similarity statement for the triangles in Figure 48. You may find these Examples useful while doing the homework for this section. T B W C K A Figure 48 Here’s one way to write a similarity statement: • First, write one pair of corresponding vertices. • Then, write another pair of corresponding vertices. • Finally, write the last pair of corresponding vertices. ∆A ∆K ∆AB ∆KT ∆ABC ~ ∆KTW So, one similarity statement for these triangles is ∆ABC ~ ∆KTW. Here are other similarity statements: ∆BAC ~ ∆TKW ∆ACB ~ ∆KWT ∆CAB ~ ∆WKT 496 TOPIC F5 GEOMETRY 29. ∆SLY ~ ∆DOG Example 29 In ∆DOG, find the angle that has the same measure as angle Y. Use the order of the letters in the similarity statement. ∠Y corresponds to ∠G. ∆SLY ~ ∆DOG So, angle G has the same measure as angle Y. 30. ∆SLY ~ ∆DOG Example 30 The ratio of SL to DO is 4 to 3. Find the ratio of LY to OG. SL is the length of line segment SL. LY is the length of line segment LY, and so on. Use the order of the letters in the similarity statement. SL corresponds to DO. ∆SLY ~ ∆DOG LY corresponds to OG. ∆SLY ~ ∆DOG The lengths of corresponding sides have the same ratio. So the ratio of LY to OG is also 4 to 3. Shortcuts for Determining Similar Triangles You know that two polygons are similar if all their corresponding angles have the same measure and all their corresponding lengths have the same ratio. However, you can show that two triangles are similar by using less information. Here are two shortcuts. Shortcut #1 Two triangles are similar if two angles of one triangle have the same measure as two angles of the other triangle. This shortcut is sometimes called “Angle Angle” or “AA.” Shortcut #2 Two triangles are similar if all their corresponding lengths have the same ratio. 31. Are the triangles in Figure 49 similar? P 55ß Caution! These two shortcuts work only for triangles. Example 31 O You may find these Examples useful while doing the homework for this section. J T A 55ß M Figure 49 LESSON F5.3 GEOMETRY III EXPLAIN 497 Each triangle has an angle with measure 55°. Each triangle has an angle with measure 90˚. So by Shortcut #1, the triangles are similar. Example 32 32. Write a similarity statement for the triangles in Figure 49. ∠O corresponds to ∠A. ∠P corresponds to ∠M. One similarity statement for these triangles is ∆TOP ~ ∆JAM. Example 33 33. Are the triangles in Figure 50 similar? 16 cm H 12 cm T 20 cm E 15 cm W N 9 cm 12 cm O Figure 50 TH 16 4 = = WO 12 3 TE 20 4 = = WN 15 3 4 HE 12 = = ON 9 3 All corresponding lengths have the same ratio. So, by Shortcut #2, the triangles are similar. Example 34 34. Write a similarity statement for the triangles in Figure 50. TH corresponds to WO, so T corresponds to W and H corresponds to O. TE corresponds to WN, so E corresponds to N. One similarity statement for these triangles is ∆THE ~ ∆WON. 498 TOPIC F5 GEOMETRY Measures of Corresponding Angles of Similar Triangles To find missing angle measures in similar triangles, you can apply what you have learned about triangles. • The angle measures of a triangle add to 180˚. • In similar triangles, corresponding angles have the same measure. • In similar triangles, the order of the letters in a similarity statement matches the corresponding angles. 35. The triangles in Figure 51 are similar. ∆BUS ~ ∆DRV. Find the missing angle measures. Example 35 V B You may find these Examples useful while doing the homework for this section. S U ˘BUS ~ ˘DRV D 60ß R Figure 51 In ∆DRV, one way to find the measure of angle V is to use what you know about the sum of the angle measures in a triangle. • Add the given angle measures. • Subtract that result from 180˚. 60˚ + 90˚ = 150˚ 180˚ – 150˚ = 30˚ So, m∠V = 30˚. One way to find the measure of angle B in ∆BUS is to use the order of the letters in the similarity statement, ∆BUS ~ ∆DRV. Angle B corresponds to angle D. In similar triangles, corresponding angles have the same measure. So, m∠B = m∠D = 60˚. In the same way, angle S has the same measure as angle V. So, m∠S = m∠V =30˚. Lengths of Corresponding Sides of Similar Triangles To find missing lengths in similar triangles, you can apply what you have learned about similar triangles. • In similar triangles, lengths of corresponding sides have the same ratio. • In similar triangles, the order of the letters in a similarity statement matches the corresponding sides. LESSON F5.3 GEOMETRY III EXPLAIN 499 You may find these Examples useful while doing the homework for this section. Example 36 36. The triangles in Figure 52 are similar. ∆WAY ~ ∆OUT. Find x, the length of OT. Y 12 ft W 10 ft 14 ft A T x 5 ft O U ˘WAY ~ ˘OUT Figure 52 Here’s one way to find x. AY is the length of line segment AY. UT is the length of line segment UT, and so on. • Corresponding lengths have the same ratio. AY UT = WY OT • We know AY, UT, and WY. 10 5 = 12 x 10 x = 5 12 10x = 60 • Cross multiply. • To get x by itself, divide both sides by 10. 10x 60 = 10 10 x=6 So, x, the length of OT is 6 ft. Example 37 37. In Figure 53, triangle ABC and triangle ADE are similar. Find y, the length of DE. Notice that the triangles in this figure “overlap.” Look carefully at the similarity statement to find pairs of corresponding angles and corresponding lengths. E ˘ABC ~ ˘ADE y C 3 cm A 5 cm D B 20 cm Figure 53 500 TOPIC F5 GEOMETRY Here’s one way to find y. AB BC = AD DE 5 3 = 20 y • Corresponding lengths have the same ratio. •We know AB, AD, and BC. 5 y = 3 20 5y = 60 •Cross multiply. 60 5y = 5 5 •To get y by itself, divide both sides by 5. y = 12 So, y, the length of DE is 12 cm. 38. In Figure 54, triangle CRN is similar to triangle COB. Find x, the length of NB. B Example 38 ∆CRN ~ ∆COB x N 12 cm C 20 cm R 10 cm O Figure 54 Here’s one way to find x. Notice that here, x is the length of only a part of one side of ∆COB. • Corresponding lengths have the same ratio. • We know CR and CN. CO = 20 + 10 = 30 CB = x + 12 CR CN = CO CB 12 20 = 30 x + 12 • Cross multiply. 20(x + 12) = 30 12 20(x + 12) = 360 • To remove the parentheses, use the distributive property. • Subtract 240 from both sides. 20x + 240 = 360 • To get x by itself, divide both sides by 20. 20x + 240 – 240 = 360 – 240 20x = 120 120 20x = 20 20 x =6 So, x, the length of NB is 6 cm. LESSON F5.3 GEOMETRY III EXPLAIN 501 Explore This Explore contains two investigations. • Congruent Triangles • “Door to Door” Investigation 1: Congruent Triangles For this investigation, you will need a ruler and a protractor. You have learned that two triangles are congruent if all corresponding sides have the same length and all corresponding angles have the same measure. Often you don’t need to verify all of the above information to know that two triangles are congruent. In this investigation you will test six ways for using less information to prove that two triangles are congruent. Some of these ways are true and some are false. You have been introduced to these investigations in the Explore module of this lesson on the computer. You can complete them using the information given here. 1. True or False: Two triangles are congruent if they have one pair of congruent sides. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. c. See if you can draw another triangle (∆DEF) with both of these features: • One side of ∆DEF is congruent to one side of ∆ABC, and • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. 502 TOPIC F5 GEOMETRY 2. True or False: Two triangles are congruent if they have two pairs of congruent sides. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. c. See if you can draw another triangle (∆DEF) with both of these features: • Two sides of ∆DEF are congruent to two sides of ∆ABC, and • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. 3. True or False: Two triangles are congruent if they have three pairs of congruent sides. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. LESSON F5.3 GEOMETRY III EXPLORE 503 c. See if you can draw another triangle (∆DEF) with both of these features: • Three sides of ∆DEF are congruent to three sides of ∆ABC, and • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. 4. True or False: Two triangles are congruent if they have three pairs of congruent angles. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. c. See if you can draw another triangle (∆DEF) with both of these features: • Three angles of ∆DEF are congruent to three angles of ∆ABC, and • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. 504 TOPIC F5 GEOMETRY 5. True or False: Two triangles are congruent if they have two pairs of congruent sides and the angles formed by those sides are also congruent. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. c. See if you can draw another triangle (∆DEF) with both of these features: • Two sides of ∆DEF are congruent to two sides of ∆ABC, and • The angles formed by those sides are congruent. • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. 6. True and False: Two triangles are congruent if they have two pairs of congruent angles and the sides shared by those angles are also congruent. a. Draw a triangle and label its vertices A, B, and C. b. Measure the angles of ∆ABC with a protractor. Measure the sides with a ruler. Record your measurements on the triangle. LESSON F5.3 GEOMETRY III EXPLORE 505 c. See if you can draw another triangle (∆DEF) with both of these features: • Two angles of ∆DEF are congruent to two angles of ∆ABC, and • The sides shared by those angles are congruent. • ∆DEF is not congruent to ∆ABC. d. Based on your investigation in (c), do you think the conjecture is true? Discuss what you tried in (c) and how you decided whether the conjecture is true. Investigation 2: “Door to Door” 1. First, sketch a copy of the rectangle in Figure 55. Then sketch a larger rectangle which is similar to the first rectangle and has the property that corresponding lengths 2 of sides are in the ratio . 1 1 cm 3 cm Figure 55 a. Find the perimeter of each rectangle. b. What is the ratio of the perimeters? (Put the perimeter of the larger rectangle in the numerator.) How does the ratio of the perimeters compare to the ratio of the corresponding lengths? c. Sketch another larger rectangle similar to the rectangle in Figure 55, so that the 3 ratio of corresponding lengths is . 1 d. Find the perimeter of each rectangle. 506 TOPIC F5 GEOMETRY e. What is the ratio of the perimeters? (Put the perimeter of the larger rectangle in the numerator.) How does the ratio of the perimeters compare to the ratio of the corresponding lengths? f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by sketching some more similar shapes and finding the ratio of their perimeters. 2. Sketch a larger rectangle similar to the rectangle in Figure 56, so that the ratio of 2 their corresponding lengths is . 1 2 cm 3 cm Figure 56 a. Find the area of each rectangle. b. What is the ratio of the areas? (Put the area of the larger rectangle in the numerator.) How does the ratio of the areas compare to the ratio of the corresponding lengths? c. Sketch another larger rectangle similar to the rectangle in Figure 56, so that the 3 ratio of the corresponding lengths is . 1 d. Find the area of each rectangle. LESSON F5.3 GEOMETRY III EXPLORE 507 e. What is the ratio of the areas? (Put the area of the larger rectangle in the numerator.) How does the ratio of the areas compare to the ratio of the corresponding lengths? f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by sketching some more similar shapes and finding the ratio of their areas. 3. Sketch a rectangular prism similar to the one in Figure 57, so that the ratio of their 2 corresponding lengths is . 1 2 cm 1 cm 3 cm Figure 57 a. Find the volume of each rectangular prism. b. What is the ratio of the volumes? (Put the volume of the larger prism in the numerator.) How does the ratio of the volumes compare to the ratio of the corresponding lengths? c. Sketch another rectangular prism similar to the one in Figure 57, so that the ratio 3 of corresponding lengths is . 1 d. Find the volume of each prism. 508 TOPIC F5 GEOMETRY e. What is the ratio of the volumes? (Put the volume of the larger prism in the numerator.) How does the ratio of the volumes compare to the ratio of the corresponding lengths? f. Look back at your results in (b) and (e). Do you see a pattern? Test your ideas by sketching some more similar prisms and finding the ratio of their volumes. LESSON F5.3 GEOMETRY III EXPLORE 509 Homework Concept 1: Triangles and Parallelograms The Sum of the Angle Measures of a Triangle 4. In Figure 61, find the measure of ∠E. P For help working these types of problems, go back to Examples 1–3 in the Explain section of this lesson. 1. 40˚ E In Figure 58, find the measure of ∠W. ? 75˚ H K Figure 61 65˚ ? W 5. In Figure 62, find the measure of ∠Q. 75˚ R E 30˚ Figure 58 2. ? Q 100˚ In Figure 59, find the measure of ∠T. W T Figure 62 ? 50˚ 30˚ O 6. In Figure 63, find the measure of ∠M. U U Figure 59 3. 50° In Figure 60, find the measure of ∠N. Y 50˚ M ? 70° A Figure 63 7. ? 60˚ N L In Figure 64, find the measure of ∠I. Q 40˚ Figure 60 65˚ ? I Figure 64 510 TOPIC F5 GEOMETRY B 8. 13. In Figure 70, find the measure of ∠Z. In Figure 65, find the measure of ∠S. S Z ? ? 100˚ 50˚ D F Figure 65 9. O In Figure 66, find the measure of ∠A. 66 32 ˚ 80˚ T Figure 70 14. In Figure 71, find the measure of ∠W. A D ? 42.5˚ P 2 66 3 ˚ 1 33 3 ˚ T Figure 66 E ? 32.5˚ 10. In Figure 67, find the measure of ∠R. W Figure 71 15. In Figure 72, find the measure of ∠B. T B 105˚ ? A 42.5˚ ? R Figure 67 H 11. In Figure 68, find the measure of ∠D. 22.2˚ 129˚ X Figure 72 16. In Figure 73, find the measure of ∠T. D ? A U 22.2˚ 28.8˚ H Figure 68 30˚ 12. In Figure 69, find the measure of ∠T. R T 80 21 ˚ ? 69 21 ˚ C 69 21 ˚ ? T Figure 73 Y Figure 69 LESSON F5.3 GEOMETRY III HOMEWORK 511 17. In Figure 74, find the measure of ∠BAT. 21. In Figure 78, find the measure of ∠N. N T ? ? B 70˚ 70˚ Figure 74 U Figure 78 18. In Figure 75, find the measure of ∠AGS. G 72˚ ? 40˚ R A 22. In Figure 79, find the measure of ∠M. M S ? A Figure 75 C 118˚ 19. In Figure 76, find the measure of ∠GLO. 94˚ L Y Figure 79 118˚ ? 23. In Figure 80, find the measure of ∠Y. G O Figure 76 20. In Figure 77, find the measure of ∠EVM. V 96˚ 96˚ H E ? ? 38˚ Y Figure 80 24. In Figure 81, find the measure of ∠E. D M E 70˚ Figure 77 30˚ K Figure 81 512 TOPIC F5 GEOMETRY ? E Congruent Triangles For help working these types of problems, go back to Examples 4–8 in the Explain section of this lesson. 28. In Figure 85, ∆DEF is congruent to ∆GHI. Which angle of ∆DEF is congruent to ∠I? D F 25. In Figure 82, ∆ABC is congruent to ∆DEF. Which angle of ∆DEF is congruent to ∠B? I C H G F A E D Figure 85 29. In Figure 86, ∆ABC is congruent to ∆DEF. Which side of ∆DEF is congruent to AB? B E Figure 82 C 26. In Figure 83, ∆BCD is congruent to ∆EFG. Which angle of ∆EFG is congruent to ∠D? A F D B G B E E Figure 86 30. In Figure 87, ∆BCD is congruent to ∆EFG. Which side of ∆BCD is congruent to EG? C D F G Figure 83 B E 27. In Figure 84, ∆CDE is congruent to ∆FGH. Which angle of ∆CDE is congruent to ∠F? C C F D F Figure 87 D H 31. In Figure 88, ∆CDE is congruent to ∆FGH. Which side of ∆CDE is congruent to GH? E F C G Figure 84 H D E G Figure 88 LESSON F5.3 GEOMETRY III HOMEWORK 513 32. In Figure 89, ∆DEF is congruent to ∆GHI. Which side of ∆GHI is congruent to DF? 36. In Figure 93, the triangles are congruent. Write a congruence relation for these triangles. E D F H G I I H G F D E Figure 93 Figure 89 37. In Figure 94, ∆CDE is congruent to ∆FGH. Find the measure of ∠G. 33. In Figure 90, the triangles are congruent. Write a congruence relation for these triangles. C A 40˚ D E F C B E D ? ∆CDE ≅ ∆FGH Figure 90 H 30˚ F G 34. In Figure 91, the triangles are congruent. Write a congruence relation for these triangles. F Figure 94 38. In Figure 95, ∆DEF is congruent to ∆GHI. Find the measure of ∠H. C D I G E B Figure 91 35. In Figure 92, the triangles are congruent. Write a congruence relation for these triangles. ? G F D F 108° C H H D 48° E E ∆DEF ≅ ∆GHI Figure 95 G Figure 92 514 TOPIC F5 GEOMETRY 42. In Figure 99, ∆DEF is congruent to ∆GHI. Find the measure of ∠I. 39. In Figure 96, ∆EFG is congruent to ∆HIJ. Find the measure of ∠I. G I ? 24˚ ∆EFG ≅ ∆HIJ E H G 94˚ I F D F 108° ? 48° H J ∆DEF ≅ ∆GHI E Figure 99 Figure 96 43. In Figure 100, ∆EFG is congruent to ∆HIJ. Find the measure of ∠H. 40. In Figure 97, ∆FGH is congruent to ∆IJK. Find the measure of ∠H. G G 24˚ E F 94˚ F ∆FGH ≅ ∆IJK ? I I H ? J H ∆EFG ≅ ∆HIJ 38° J 58° Figure 100 K Figure 97 41. In Figure 98, ∆CDE is congruent to ∆FGH. Find the measure of ∠F. C 40˚ E F ? 30˚ H D ∆CDE ≅ ∆FGH G Figure 98 LESSON F5.3 GEOMETRY III HOMEWORK 515 47. In Figure 104, ∆KLM is congruent to ∆NOP. Find the length of LM. 44. In Figure 101, ∆FGH is congruent to ∆IJK. Find the measure of ∠F. G M N 10 cm L O 6 cm K P ? F ∆KLM ≅ ∆NOP Figure 104 I 48. In Figure 105, ∆STU is congruent to ∆VWX. Find the length of SU. H ∆FGH ≅ ∆IJK U S 58° 38° J K ∆STU ≅ ∆VWX 7.5 cm Figure 101 45. In Figure 102, ∆SRK is congruent to ∆FIN. Find the length of SR. T V W 5 cm K N X 3 ft 5 ft Figure 105 R I 4 ft S F ∆SRK ≅ ∆FIN Isosceles Triangles and Equilateral Triangles For help working these types of problems, go back to Examples 9–14 in the Explain section of this lesson. Figure 102 46. In Figure 103, ∆CDE is congruent to ∆FGH. Find the length of FG. 49. In Figure 106, ∆RIT is an isosceles triangle. Which pair of angles must have the same measure? D T 6 in C 4 in 6 ft E 6 ft ∆CDE ≅ ∆FGH H R 3 ft Figure 106 F G Figure 103 516 TOPIC F5 GEOMETRY I 50. In Figure 107, ∆BSU is an isosceles triangle. Which pair of angles must have the same measure? 54. In Figure 111, ∆JET is an isosceles triangle. Find the measure of ∠E. J S 8 cm 4 cm B ? E 35˚ T Figure 111 8 cm U Figure 107 51. In Figure 108, ∆WAY is an isosceles triangle. Which pair of angles must have the same measure? 55. In Figure 112, ∆WHO is an isosceles triangle. Find the measure of ∠W. W ? Y 4 ft H 6 ft W 55˚ O 4 ft Figure 112 56. In Figure 113, ∆SAM is an isosceles triangle. Find the measure of ∠S. A Figure 108 52. In Figure 109, ∆TOY is an isosceles triangle. Which pair of angles must have the same measure? 3 cm T S ? Y 2 cm M 2 cm O Figure 109 27˚ 53. In Figure 110, ∆BAT is an isosceles triangle. Find the measure of ∠T. B A Figure 113 57. In Figure 114, find the length of BA. B A 70˚ ? ? T Figure 110 A 70˚ 8 cm 70˚ T Figure 114 LESSON F5.3 GEOMETRY III HOMEWORK 517 58. In Figure 115, find the length of JT. 65. In Figure 122, ∆ASK is an equilateral triangle. Find the length of AS. J 6 ft E A ? 35˚ 35˚ T 6 ft Figure 115 59. In Figure 116, find the length of HO. S K Figure 122 W 66. In Figure 123, ∆PAL is an equilateral triangle. Find the length of AL. 55˚ 3 ft H L ? P 55˚ O 5 cm Figure 116 A Figure 123 60. In Figure 117, find the length of AM. 67. In Figure 124, ∆JAM is an equilateral triangle. Find the length of AM. S 4 cm 27˚ M 8 cm M J ? 27˚ A A Figure 117 Figure 124 61. Sketch an isosceles triangle, ∆PUT, with PU and PT of equal lengths. 68. In Figure 125, ∆BDU is an equilateral triangle. Find the length of BU. 62. Sketch an isosceles triangle, ∆CAN, with CA and AN of different lengths. 63. Sketch an isosceles triangle, ∆RAT, with all sides of equal lengths. D 2 in B U 64. Sketch an isosceles triangle, ∆SLO, with LO and SO of equal lengths. Figure 125 69. In Figure 126, find the measure of angle S. A S ? K Figure 126 518 TOPIC F5 GEOMETRY 70. In Figure 127, find the measure of angle A. 79. In a certain right triangle, the lengths of the sides are 25 in, 60 in, and 65 in. Find the lengths of the legs. L 5 cm 80. In a certain right triangle, the lengths of the sides are 5m, 12m, and 13m. Find the lengths of the legs. P 5 cm 5 cm 81. In ∆STW, the lengths of the sides are 3 cm, 4 cm, and 5 cm. Is ∆STW a right triangle? ? A Figure 127 71. In Figure 128, find the measure of angle M. J ? M 82. In ∆LUG, the lengths of the sides are 5 ft, 11 ft, and 13 ft. Is ∆LUG a right triangle? 83. In ∆PTA, the lengths of the sides are 9 cm, 10 cm, and 12 cm. Is ∆PTA a right triangle? 84. In ∆TIP, the lengths of the sides are 10 ft, 24 ft, and 26 ft. Is ∆TIP a right triangle? 85. In ∆BIN, the lengths of the sides are 10 cm, 12 cm, and 24 cm. Is ∆BIN a right triangle? A Figure 128 72. In Figure 129, find the measure of angle U. 86. In ∆COW, the lengths of the sides are 15 in, 36 in, and 39 in. Is ∆COW a right triangle? 87. In ∆IRT, the lengths of the sides are 60 cm, 63 cm, and 87 cm. Is ∆IRT a right triangle? D 88. In ∆MTA, the lengths of the sides in are 5 ft, 7 ft, and 9 ft. Is ∆MTA a right triangle? B ? U Figure 129 89. The triangle in Figure 130 is a right triangle. Find c, the length of its hypotenuse. 3 in Right Triangles and the Pythagorean Theorem 4 in c For help working these types of problems, go back to Examples 15– 19 in the Explain section of this lesson. 73. In a right triangle, the measure of one angle is 28˚. Find the measures of the other angles. 74. In a right triangle, the measure of one angle is 80˚. Find the measures of the other angles. 75. In a right triangle, the measure of one angle is 30˚. Find the measures of the other angles. 76. In a right triangle, the measure of one angle is 45˚. Find the measures of the other angles. Figure 130 90. The triangle in Figure 131 is a right triangle. Find c, the length of its hypotenuse. c 10 cm 24 cm Figure 131 77. In a certain right triangle, the lengths of the sides are 60 cm, 63 cm, and 87 cm. Find the length of the hypotenuse. 78. In a certain right triangle, the lengths of the sides are 15 ft, 20 ft, and 25 ft. Find the length of the hypotenuse. LESSON F5.3 GEOMETRY III HOMEWORK 519 91. The triangle in Figure 132 is a right triangle. Find c, the length of its hypotenuse. 16 cm 96. The triangle in Figure 137 is a right triangle. Find b, the length of one of its legs. 12 cm 15 ft 9 ft c Figure 132 b Figure 137 92. The triangle in Figure 133 is a right triangle. Find c, the length of its hypotenuse. Parallel Lines and Parallelograms c 18 ft For help working these types of problems, go back to Examples 20–23 in the Explain section of this lesson. 97. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 2. 24 ft Figure 133 t 93. The triangle in Figure 134 is a right triangle. Find b, the length of one of its legs. k 3 in 75˚ 1 3 b 5 in r 5 7 Figure 134 94. The triangle in Figure 135 is a right triangle. Find a, the length of one of its legs. 52 cm a 48 cm 6 8 Figure 138 98. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 3. 99. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 4. Figure 135 95. The triangle in Figure 136 is a right triangle. Find a, the length of one of its legs. a 87 cm 2 4 100. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 5. 101. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 6. 63 cm 102. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 7. 103. In Figure 138, k || r. The measure of angle 1 is 75˚. Find the measure of angle 8. Figure 136 104. In Figure 138, k || r. Are angles 1 and 5 supplementary angles, corresponding angles, or vertical angles? 520 TOPIC F5 GEOMETRY 105. In Figure 139, line segment AB is parallel to line segment ED. Find the measure of angle A. 112. In Figure 141, line segment KL is parallel to line segment MN. Find the measure of angle JLK. 113. In Figure 142, the quadrilateral is a parallelogram. Find the measure of angle Q. A E U Q 40˚ B C D Figure 139 106. In Figure 139, line segment AB is parallel to line segment ED. Find the measure of angle C. 110˚ D 107. In Figure 140, line segment TU is parallel to line segment VW. Find the measure of angle STU. Figure 142 114. In Figure 142, the quadrilateral is a parallelogram. Find the measure of angle U. V 65˚ T A 115. In Figure 142, the quadrilateral is a parallelogram. Find the measure of angle A. 116. In Figure 143, the quadrilateral is a parallelogram. Find the measure of angle F. S 45˚ W U F L Figure 140 108. In Figure 140, line segment TU is parallel to line segment VW. Find the measure of angle SUT. P 45˚ I 109. In Figure 140, line segment TU is parallel to line segment VW. Find the measure of angle W. 110. In Figure 141, line segment KL is parallel to line segment MN. Find the measure of angle M. Figure 143 117. In Figure 144, the quadrilateral is a parallelogram. Find the length of SH. P 4 cm M U 10 cm K H J 45˚ S Figure 144 100˚ L Figure 141 N 111. In Figure 141, line segment KL is parallel to line segment MN. Find the measure of angle JKL. 118. In Figure 144, the quadrilateral is a parallelogram. Find the length of PH. LESSON F5.3 GEOMETRY III HOMEWORK 521 123. Are the polygons in Figure 148 similar polygons? 119. In Figure 145, the quadrilateral is a parallelogram. Find the length of DE. 6 ft D E 4 ft 4 ft 6 ft 100˚ 80˚ 100˚ 3 ft K 80˚ 3 ft S 8 ft 8 ft 100˚ 80˚ Figure 145 3 ft 100˚ 120. In Figure 145, the quadrilateral is a parallelogram. Find the length of ES. 80˚ 6 ft Figure 148 124. Are the polygons in Figure 149 similar polygons? Concept 2: Similar Polygons 3 ft 3 ft 80˚ 100˚ Recognizing Similar Polygons 4 ft 4 ft 4 ft 4 ft 100˚ 80˚ For help working these types of problems, go back to Examples 24–27 in the Explain section of this lesson. 3 ft 3 ft Figure 149 121. Are the polygons in Figure 146 similar polygons? 125. Are the rectangles in Figure 150 similar rectangles? 6 ft 6 ft 80˚ 8 ft 8 ft 8 ft 4 cm 100˚ 8 ft 3 cm 3 cm 3 cm 2 cm 2 cm 3 cm 100˚ 4 cm 80˚ 6 ft 6 ft Figure 150 Figure 146 126. Are the triangles in Figure 151 similar triangles? 122. Are the rectangles in Figure 147 similar rectangles? 6 ft 3 cm 3 ft 8 ft 8 ft 4 ft 4 ft 60˚ 60˚ 3 cm 60° 60˚ 3 cm 4 cm 60˚ 6 ft Figure 147 522 TOPIC F5 GEOMETRY 60˚ 4 cm Figure 151 3 ft 4 cm 133. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle L. 127. Are the hexagons in Figure 152 similar hexagons? 6 cm 4 cm 120˚ 120˚ 4 cm 120˚ 4 cm 120˚ 4 cm 120˚ G 6 cm 6 cm 120˚ 120˚ 120˚ 4 cm 120˚ 4 cm K 120˚ 6 cm 6 cm 120˚ 100˚ I 120˚ L 6 cm H Figure 152 128. Are the pentagons in Figure 153 similar pentagons? 5 ft 108˚ 108˚ 108˚ 5 ft 108˚ 5 ft 108˚ 3 ft 108˚ 135. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle H. 108˚ 3 ft 3 ft 108˚ ∆GHI ~ ∆JKL Figure 155 134. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle G. 5 ft 3 ft J 25˚ 108˚ 108˚ 3 ft 5 ft Figure 153 129. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle D. 136. In Figure 155, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle K. 137. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF. Find this ratio: ___ length of BC length of EF D 8 cm B D E 10 cm B 80˚ E A A 35˚ C C ∆ABC ~ ∆DEF F ∆ABC ~ ∆DEF Figure 154 130. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF Find the measure of angle B. 131. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle C. 132. In Figure 154 , the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle F. F Figure 156 138. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF. Find this ratio: ___ length of DF length of AC 139. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF. Find this ratio: ___ length of EF length of BC 140. In Figure 156, the triangles are similar. ∆ABC ~ ∆DEF. Find this ratio: ___ length of AC length of DF LESSON F5.3 GEOMETRY III HOMEWORK 523 141. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL. Find this ratio: __ length of HI length of KL 147. In Figure 158, the triangles are similar. Complete this similarity statement: ∆KAF ~ ∆_____ 148. In Figure 158, the triangles are similar. Complete this similarity statement: ∆KFA ~ ∆_____ 149. In Figure 159, the triangles are similar. Complete this similarity statement: ∆USA ~ ∆_____ I J 3 ft U 4 ft L K G A ∆GHI ~ ∆JKL S H W R Figure 157 142. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL. Find this ratio: __ length of JK length of GH Figure 159 143. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL. Find this ratio: __ length of KL length of HI 144. In Figure 157, the triangles are similar. ∆GHI ~ ∆JKL. Find this ratio: __ length of GH length of JK Writing a Similarity Statement For help working these types of problems, go back to Examples 28–30 in the Explain section of this lesson. 145. In Figure 158, the triangles are similar. Complete this similarity statement: ∆AFK ~ ∆_____ T A F Y P Figure 158 146. In Figure 158, the triangles are similar. Complete this similarity statement: ∆FAK ~ ∆_____ TOPIC F5 GEOMETRY 150. In Figure 159, the triangles are similar. Complete this similarity statement: ∆ASU ~ ∆_____ 151. In Figure 159, the triangles are similar. Complete this similarity statement: ∆SUA ~ ∆_____ 152. In Figure 159, the triangles are similar. Complete this similarity statement: ∆UAS ~ ∆_____ 153. ∆LAW ~ ∆YER Find the angle in ∆LAW that has the same measure as angle Y. 154. ∆LAW ~ ∆YER Find the angle in ∆LAW that has the same measure as angle E. 155. ∆LAW ~ ∆YER Find the angle in ∆LAW that has the same measure as angle R. 156. ∆LAW ~ ∆YER Find the angle in ∆YER that has the same measure as angle A. 157. ∆PUT ~ ∆DWN Find the angle in ∆DWN that has the same measure as angle P. K 524 E 158. ∆PUT ~ ∆DWN Find the angle in ∆DWN that has the same measure as angle T. 159. ∆PUT ~ ∆DWN Find the angle in ∆DWN that has the same measure as angle U. 170. Are the triangles in Figure 161 similar? T 160. ∆PUT ~ ∆DWN Find the angle in ∆PUT that has the same measure as angle D. 161. ∆LAW ~ ∆YER The ratio of LA to YE is 7 to 3. Find the ratio of AW to ER. P 162. ∆LAW ~ ∆YER The ratio of LA to YE is 7 to 3. Find the ratio of LW to YR. 42˚ S W 163. ∆LAW ~ ∆YER The ratio of LA to YE is 7 to 3. Find the ratio of ER to AW. 164. ∆LAW ~ ∆YER The ratio of LA to YE is 7 to 3. Find the ratio of YR to LW. 165. ∆CAT ~ ∆NIP The ratio of CT to NP is 9 to 4. Find the ratio of CA to NI. 166. ∆CAT ~ ∆NIP The ratio of CT to NP is 9 to 4. Find the ratio of AT to IP. 42˚ Figure 161 171. Are the triangles in Figure 162 similar? A C E 167. ∆CAT ~ ∆NIP The ratio of CT to NP is 9 to 4. Find the ratio of NI to CA. 168. ∆CAT ~ ∆NIP The ratio of CT to NP is 9 to 4. Find the ratio of IP to AT. D F 55˚ 55˚ B F D Figure 162 Shortcuts for Recognizing Similar Triangles For help working these types of problems, go back to Examples 31– 34 in the Explain section of this lesson. 172. Are the triangles in Figure 163 similar? A T 45˚ 169. Are the triangles in Figure 160 similar? 45˚ A B C C E 42˚ 35˚ V 35˚ Figure 163 B F M D Figure 160 LESSON F5.3 GEOMETRY III HOMEWORK 525 173. Are the triangles in Figure 164 similar? 177. Complete this similarity statement for the triangles in Figure 168: ∆IHG ~ ∆_____ G G 25˚ K 25° 55˚ J 25˚ K 25° 55˚ J 100˚ 100˚ I 100˚ 100˚ I L L H Figure 164 H Figure 168 174. Are the triangles in Figure 165 similar? 178. Complete this similarity statement for the triangles in Figure 169: ∆BAC ~ ∆_____ N 52˚ A B 100˚ C 28˚ E E 35˚ 100˚ H 35˚ C 25˚ B U Figure 165 175. Are the triangles in Figure 166 similar? B F D 35˚ 80˚ A 65˚ 35˚ D Figure 169 80˚ E 179. Complete this similarity statement for the triangles in Figure 170: ∆TWP ~ ∆_____ F C T Figure 166 176. Are the triangles in Figure 167 similar? G P 48˚ I J 42˚ 94˚ 38˚ H K 38˚ 48˚ S W L Figure 167 42˚ F Figure 170 526 TOPIC F5 GEOMETRY D 180. Complete this similarity statement for the triangles in Figure 171: ∆CBA ~ ∆_____ A 184. Complete this similarity statement for the triangles in Figure 175: ∆FDC ~ ∆_____ D C E F P 53˚ N 55˚ 55˚ 36˚ B 53˚ C F D 181. Complete this similarity statement for the triangles in Figure 172: ∆BAC ~ ∆_____ B 185. Are the triangles in Figure 176 similar? 15 cm D 35˚ 80˚ 65˚ 9 cm 12 cm 80˚ 35˚ L Figure 175 Figure 171 A 36˚ E F C 25 cm 20 cm Figure 172 182. Complete this similarity statement for the triangles in Figure 173: ∆GIH ~ ∆_____ 15 cm Figure 176 G I 48˚ 186. Are the triangles in Figure 177 similar? J Y 24 ft 94˚ W 38˚ 38˚ K H 48˚ 20 ft L Figure 173 28 ft 183. Complete this similarity statement for the triangles in Figure 174: ∆ABE ~ ∆_____ 12 ft E O D 36˚ F 53˚ B Figure 174 10 ft 14 ft U Figure 177 53˚ A 91˚ A T 36˚ C LESSON F5.3 GEOMETRY III HOMEWORK 527 190. Complete this similarity statement for the triangles in Figure 181: ∆WAY ~ ∆_____ 187. Are the triangles in Figure 178 similar? Y 24 ft 7 in 6 in W 20 ft 4 in 3 in 28 ft 5 in 5 in A T Figure 178 12 ft 10 ft 188. Are the triangles in Figure 179 similar? O U 14 ft 24 cm Figure 181 191. Complete this similarity statement for the triangles in Figure 182: ∆CJW ~ ∆_____ 10 cm 26 cm 24 cm C 13 cm 12 cm W M 10 cm 26 cm 5 cm Figure 179 J 189. Complete this similarity statement for the triangles in Figure 180: ∆AKE ~ ∆_____ 15 cm W E C 12 cm 9 cm T 25 cm A 15 cm E Figure 180 528 G 192. Complete this similarity statement for the triangles in Figure 183: ∆BCA ~ ∆_____ 12 in K 5 cm Figure 182 A 20 cm 13 cm 12 cm TOPIC F5 GEOMETRY U 7 in 10 in R B 14 in C Figure 183 6 in 5 in V Measurements of Corresponding Angles of Similar Triangles For help working these types of problems, go back to Example 35 in the Explain section of this lesson. 201. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure of angle B. V B 193. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV. Find the measure of angle V. C V R T 58˚ 32˚ U ∆ABC ~ ∆UTV A Figure 186 36˚ Y E ∆RYE ~ ∆CBV 202. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure of angle U. C B Figure 184 203. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure of angle C. 194. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV. Find the measure of angle Y. 204. In Figure 186, ∆ABC is similar to ∆UTV. Find the measure of angle V. 195. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV. Find the measure of angle C. 205. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO. Find the measure of angle B. 196. In Figure 184, the triangles are similar. ∆RYE ~ ∆CBV. Find the measure of angle R. O 197. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure of angle B. P G B C 54˚ R 81˚ A L ∆BCA ~ ∆RLO T Figure 187 H 48˚ 54˚ L ∆GHL ~ ∆TBP B Figure 185 198. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure of angle P. 206. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO. Find the measure of angle A. 207. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO. Find the measure of angle O. 208. In Figure 187, the triangles are similar. ∆BCA ~ ∆RLO. Find the measure of angle L. 199. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure of angle G. 200. In Figure 185, ∆GHL is similar to ∆TBP. Find the measure of angle T. LESSON F5.3 GEOMETRY III HOMEWORK 529 209. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of of angle B. For help working these types of problems, go back to Examples 36–38 in the Explain section of this lesson. D B E 65˚ A 35˚ C Lengths of Corresponding Sides of Similar Triangles 217. The triangles in Figure 190 are similar. ∆WAY ~ ∆OUT. Find x, the length of UT. F Y 12 ft ∆ABC ~ ∆DEF W 10 ft Figure 188 14 ft 210. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle C. A T 211. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle D. 6 ft O 212. In Figure 188, the triangles are similar. ∆ABC ~ ∆DEF. Find the measure of angle E. 213. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle G. y U Figure 190 218. The triangles in Figure 190 are similar. ∆WAY ~ ∆OUT. Find y, the length of OU. 219. The triangles in Figure 191 are similar. ∆CJW ~ ∆EGM. Find x, the length of CJ. G K J 55˚ C 24 cm W x 100˚ I M 26 cm L H ∆WAY ~ ∆OUT x ∆GHI ~ ∆JKL J y 13 cm Figure 189 214. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle H. 215. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle J. 216. In Figure 189, the triangles are similar. ∆GHI ~ ∆JKL. Find the measure of angle L. 530 TOPIC F5 GEOMETRY ∆CJW ~ ∆EGM E 5 cm G Figure 191 220. The triangles in Figure 191 are similar. ∆CJW ~ ∆EGM. Find y, the length of EM. 221. The triangles in Figure 192 are similar. ∆AKE ~ ∆WTC. Find x, the length of AK. 15 cm W 225. The triangles in Figure 194 are similar. ∆ABC ~ ∆ADE. Find x, the length of BC. A C A y 12 cm 5 cm T B 25 cm C x ∆ABC ~ ∆ADE ∆AKE ~ ∆WTC x 20 cm K E 15 cm Figure 192 222. The triangles in Figure 192 are similar. ∆AKE ~ ∆WTC. Find y, the length of TC. 223. The triangles in Figure 193 are similar. ∆BKR ~ ∆DPC. Find x, the length of PC. 5 cm D B y C x D Figure 194 226. The triangles in Figure 195 are similar. ∆JKL ~ ∆JMN. Find x, the length of MN. 6 ft P 10 cm 6 cm E 12 cm J 2 ft L N 1 ft K R 8 cm ∆BKR ~ ∆DPC Figure 193 224. The triangles in Figure 193 are similar. ∆BKR ~ ∆DPC. Find y, the length of DP. K x ∆JKL ~ ∆JMN M Figure 195 LESSON F5.3 GEOMETRY III HOMEWORK 531 227. The triangles in Figure 196 are similar. ∆ABC ~ ∆ADE. Find x, the length of AB. 229. The triangles in Figure 198 are similar. ∆ABC ~ ∆ADE. Find x, the length of BC. A A x ∆ABC ~ ∆ADE ∆ABC ~ ∆ADE 15 cm 20 cm C B 6 cm 20 cm C B D E 12 cm Figure 196 x D Figure 198 228. The triangles in Figure 197 are similar. ∆JKL ~ ∆JMN. Find x, the length of KL. 230. The triangles in Figure 199 are similar. ∆PQR ~ ∆PST. Find x, the length of QR. 6 ft J 12 ft L 4 ft E 12 cm N 8 ft P x R T x 3 ft 6 ft ∆JKL ~ ∆JMN ∆PQR ~ ∆PST K Q M S Figure 197 Figure 199 231. The triangles in Figure 200 are similar. ∆PQR ~ ∆PST. Find x, the length of PR. 12 ft P x R T 2 ft Q 6 ft ∆PQR ~ ∆PST S Figure 200 532 TOPIC F5 GEOMETRY 232. The triangles in Figure 201 are similar. ∆TUV ~ ∆TMN. Find x, the length of UT. 18 ft 8 ft P U 3 ft M 235. The triangles in Figure 204 are similar. ∆PQR ~ ∆PST. Find x, the length of RT. x x R T 4 ft T Q V 9 ft 12 ft ∆PQR ~ ∆PST S ∆TUV ~ ∆TMN Figure 204 N Figure 201 233. The triangles in Figure 202 are similar. ∆ABC ~ ∆ADE. Find x, the length of BD. 236. The triangles in Figure 205 are similar. ∆PQR ~ ∆PST. Find x, the length of RT. 8 cm P R x T A 4 cm ∆PQR ~ ∆PST 6 cm Q S ∆ABC ~ ∆ADE Figure 205 15 cm 237. The triangles in Figure 206 are similar. ∆JKL ~ ∆JMN. Find x, the length of LJ. M B C 9 cm ∆JKL ~ ∆JMN x D 18 ft E 12 cm K Figure 202 234. The triangles in Figure 203 are similar. ∆FGH ~ ∆FIJ. Find x, the length of GI. I x G 5 cm 6 ft N 8 ft L x J Figure 206 F 3 cm 6 cm H ∆FGH ~ ∆FIJ J Figure 203 LESSON F5.3 GEOMETRY III HOMEWORK 533 238. The triangles in Figure 207 are similar. ∆PEA ~ ∆POD. Find x, the length of EO. O 240. The triangles in Figure 209 are similar. ∆PEA ~ ∆POD. Find x, the length of PA. O ∆PEA ~ ∆POD x E E 6 cm 18 ft P 10 cm A 5 cm D 239. The triangles in Figure 208 are similar. ∆JKL ~ ∆JMN. Find x, the length of NL. M ∆JKL ~ ∆JMN K 9 cm 6 cm x L 4 cm J Figure 208 534 P x A Figure 209 Figure 207 N ∆PEA ~ ∆POD 9 ft TOPIC F5 GEOMETRY 15 ft D Evaluate Take this Practice Test to prepare for the final quiz in the Evaluate module of this lesson on the computer. Practice Test 1. One of the angles in a right triangle measures 10˚. Find the measure of the other acute angle of the triangle. 2. In Figure 210, ∆ABC is congruent to ∆DEF. length AB = length BC m∠A = 71˚ Find the measure of ∠E. E B F C 71˚ D A Figure 210 3. In Figure 211: ∆JKL is an equilateral triangle. length LK = 24 cm h = 123 cm Find x, the length of JM. K h J x M Figure 211 L LESSON F5.3 GEOMETRY III EVALUATE 535 4. In Figure 212: line AB is parallel to line CD line AC is parallel to line BD length AB = 8.1 cm length AC = 10.2 cm m∠ACD = 54˚ Find length BD, m∠ABD, and m∠BDC. A C D B Figure 212 5. The length of each side of a given pentagon measures 2 in. A larger pentagon which is similar to the given pentagon has a side that measures 2.4 in. Find the perimeter of the larger pentagon. 6. Triangle ABC is similar to triangle DEF. AB = 14 cm BC = 15 cm AC = 8 cm DE = 42 cm Find x, the length of EF. (Hint: Sketch the two triangles, making sure to match corresponding sides.) 7. In Figure 213: AB is parallel to DE AB = 130 cm BC = 117 cm DE = 390 cm Find y, the length of DC. E B 117 cm 390 cm 130 cm C A y D Figure 213 536 TOPIC F5 GEOMETRY 8. In Figure 214: ∆CED ~ ∆AEB EC = 21 cm CD = 18.9 cm AB = 37.8 cm Find x, the length of EA. 37.8 cm A 18.9 cm C B D 21 cm E Figure 214 LESSON F5.3 GEOMETRY III EVALUATE 537 538 TOPIC F5 GEOMETRY Topic F5 Cumulative Review These problems cover the material from this and previous topics. You may wish to do these problems to check your understanding of the material before you move on to the next topic, or to review for a test. 1 2 1 4 1. Find: 180° – (48 ° + 22 °) 2. In Figure F5.1, find the measure of F. S 30° U 100° ? F Figure F5.1 3. In Figure F5.2, find the measure of angle A. (Hint: ∆PLA is an equilateral triangle.) L 2 ft P 2 ft 2 ft ? A 4. 5. Figure F5.2 Find: 22.5 cm + 10.2 cm + 22.5 cm + 10.2 cm In Figure F5.3, name a point, a line segment, a ray, and a line. P R Q Figure F5.3 1 3 6. Use the order of operations to do this calculation: (4.4 + 4.6) + 4 7. Melissa is building a fence around her rectangular garden. The garden is 16 yards long and 10 yards wide. How many yards of fence must she build in order to completely enclose the garden? 8. Find: 180° – (32.4° + 27.8°) 9. Name the polygon that has 8 sides. 1 10. Maurice is painting the ceiling in his living room. The ceiling is 5 yards by 3 6 yards. How many square yards will he have to paint to give the entire ceiling one coat of paint? 11. Find the area of the parallelogram shown in Figure F5.4. 5 ft 4 ft 5 ft 3 ft Figure F5.4 TOPIC F5 CUMULATIVE REVIEW 539 12. Give the degree measure of a right angle. 13. If an angle measures 100.2°, is it an acute, obtuse, right, or straight angle? 14. In Figure F5.6, find the length of ZX. (Hint: ∆XYZ is an isosceles triangle.) Z 5 cm ? Y 35° Figure F5.6 35° X 15. The most popular item of a pennant manfacturer is the classic triangular felt pennant. If this pennant is 16.5 inches long from the center of its base to its tip and its base is 8 inches long, how many square inches of felt are needed to make the pennant? 16. In Figure F5.7, f || g. The measure of angle 8 is 75°. Find the measure of angle 2. t f 1 3 2 4 g 5 7 6 8 75° Figure F5.7 17. In Figure F5.8, find the measure of Y. H 96° ? 58° Y Figure F5.8 18. In a right triangle, the measure of one angle is 62°. Find the measures of the other angles. 19. The quadrilateral in Figure F5.9 is a parallelogram. Find the measure of angle L. F 135° P ? I Figure F5.9 1 3 20. If an angle measures 33 °, is it an acute, obtuse, right, or straight angle? 540 TOPIC F5 GEOMETRY L 21. Find the area of the trapezoid shown in Figure F5.10. 10 ft 6 ft 8 ft Figure F5.10 22. If mU = 82.5° and mV = 97.5°, are angles U and V complementary, supplementary, or neither? 23. The radius of a circle is 18 inches. Find its circumference and its area. 24. In Figure F5.11, what is the measure of BAC? C D 74° A B E Figure F5.11 25. ∆ABC is congruent to ∆DEF. Which angle of ∆DEF is congruent to F? 26. The lengths of the sides of a certain right triangle are 80 cm, 84 cm, and 116 cm. Find the length of the hypotenuse. 27. Bud is covering the surface of a gift box with gold foil. The box is a rectangular prism that is 4 inches tall, 5 inches wide, and 8 inches long. How many square inches of foil are needed to cover the entire surface of the box with no overlap? 28. Al is building a trunk to store bedding and other items. He figures that a trunk with a volume of 9 cubic feet should be large enough. If he builds the trunk so that it is 2 feet high and 3 feet long, how wide should he make the trunk so that it has a volume of 9 cubic feet? 29. ∆ABC is congruent to ∆DEF. Which side of ∆DEF is congruent to BC? 30. The radius of a cylinder is 2 inches. The height of the cylinder is 6 inches. What is the surface area of the cylinder? 31. The radius of a cylinder is 2 inches. The height of the cylinder is 6 inches. What is the volume of the cylinder? Use 3.14 to approximate . Round your answer to two decimal places. 32. Find the value of this expression: –122 – 82 33. The lengths of the sides of ∆USA are 9 ft, 10 ft, and 12 ft. Is ∆USA a right triangle? 34. The diameter of the base of a cone is 3 centimeters. The height of the cone is 12 centimeters. What is the volume of the cone? Use 3.14 to approximate . Round your answer to two decimal places. TOPIC F5 CUMULATIVE REVIEW 541 35. Are the polygons in Figure F5.12 12 ft similar polygons? 8 ft 16 ft 6 ft Figure F5.12 36. ∆FAR ~ ∆OUT. (Triangle FAR is similar to triangle OUT.) Find the angle in ∆OUT that has the same measure as angle A. 37. ∆FAR ~ ∆OUT. The ratio of FA to OU is 8 to 3. Find the ratio of AR to UT. 38. Complete this similarity statement for the triangles in Figure F5.13: ∆SRC ~ ∆_____ R U 10 cm 12 cm 7 cm R S C 14 cm 5 cm 6 cm V Figure F5.13 39. The triangles in Figure F5.14 are similar. ∆GHI ~ ∆JKL. Find the measure of angle G. G K 55° H J 100° I L ∆GHI ~ ∆ JKL Figure F5.14 2 12 40. Find the missing number, x, that makes this proportion true: = . 5 x 41. The triangles in Figure F5.15 are similar. ∆PQR ~ ∆PST. Find x, the length of PR. P x 6 ft R Q ∆PQR ~ ∆PST Figure F5.15 542 TOPIC F5 GEOMETRY T 1 ft 3 ft S 42. The triangles in Figure. F5.16 are similar. ∆JKL ~ ∆JMN Find x, the length of NL. M K 18 cm ∆JKL ~ ∆JMN 12 cm N x L 8 cm J Figure F5.16 43. The volume of a sphere is 36 cubic feet. What is the radius of the sphere? TOPIC F5 CUMULATIVE REVIEW 543 544 TOPIC F5 GEOMETRY