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Chapter 4 Congruent Triangles 4.1 Triangles and Angles Parts of Triangles Vertex Points joining the sides of a triangle Adjacent Sides Sides that share a common vertex Classification by Sides Equilateral Isosceles 3 congruent sides At least 2 congruent sides Scalene No congruent sides Classification by Angles Acute Equiangular 3 congruent angles Right 3 acute angles 1 right angle Obtuse 1 obtuse angle Parts of Isosceles Triangles Legs The sides that are congruent. Base The non-congruent side. Isosceles triangles Base angles are congruent. Vertex angle Base angles Base legs Parts of Right Triangles Hypotenuse The side that is opposite the right angle. It is always the longest side. Legs The sides that form the right angle Right Triangles hypotenuse leg leg Interior Angles The angles on the inside of a triangle. Triangle Sum Conjecture The sum of the measures of the angles in every triangle is 180. Example Find the measure of each angle. 2x + 10 x x+2 Exterior Angles The angles that are adjacent to the interior angles The exterior angles always add to equal 360° Definitions Exterior Angle Adjacent Interior Angle Remote Interior Angles Exterior Angles of a Triangle Use your straightedge to draw a triangle. Extend one side out as shown B b a A c x C Exterior Angles of a Triangle Trace angles a and b onto a transparency so that they are adjacent. How does this compare to angle x? B b a a b A c x C Triangle Exterior Angle Conjecture The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles B b a A c x=a+b C Example Find the missing measures 80° 53° Example Find the missing measures 60° 120° Example Page 199 #37 (2x – 8)° x° 31° 4.2 Congruence and Triangles Terms Congruent Corresponding angles Figures that are exactly the same size and shape are congruent The angles that are in corresponding positions are congruent Corresponding sides The sides that are in corresponding positions are congruent Naming Congruent Figures When a congruence statement is made it is important to match up corresponding parts. Third Angle Theorem If two angles in one triangle are equal to two angles in another triangle, then the third angles in each triangle are also equal. Examples 1 What is the measure of: (page 205) P ΔLMN ΔPQR Q N M P R 45° N Which side is congruent to 105° L M R segment QR Segment LN Example 2 Given ABC PQR, find the values of x and y. R (6y – 4)° Q A 85° B 50° P (10x + 5)° C 4.3 Proving Triangles Congruent SSS and SAS Warm-Up Complete the following statement B BIG A I R G T Definitions included angle An angle that is between two given sides. included side A side that is between two given angles. Example 1 Use the diagram. Name the included angle between the pair of given sides. JK and KL PKand LK KPand PL J K L P Triangle Congruence Shortcut SSS If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. Triangle Congruence Shortcuts SAS If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Example 2 Complete the congruence statement. Name the congruence shortcut used. U S T V W STW Example 3 Determine if the following are congruent. Name the congruence shortcut used. H L I M N HIJ LMN J Example 4 Complete the congruence statement. Name the congruence shortcut used. A B C X O XBO R Example 5 Complete the congruence statement. Name the congruence shortcut used. SPQ P T S Q 4.4 Proving Triangles Congruent ASA and AAS Triangle Congruence Shortcuts ASA If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Triangle Congruence Shortcuts SAA If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent. Example 1 Complete the congruence statement. Name the congruence shortcut used. Q U QUA D A Example 2 Complete the congruence statement. Name the congruence shortcut used. M R N Q P RMQ Example 3 Determine if the following are congruent. Name the congruence shortcut used. F B E A ABC FED C D 4.6 Isosceles, Equilateral, and Right Triangles Warm-Up 1 Find the measure of each angle. 60° a 90° 90° 30° b Warm-Up 2 Find the measure of each angle. 110 90 150 Isosceles triangles The base angles of an isosceles triangle are congruent. If a triangle has at least two congruent angles, then it is an isosceles triangle. If the sides are congruent then the base angles are congruent. Example 1 35° x Example 2 b 15° a Example 3 Find each missing measure m n 10 cm 63° p Equilateral Triangles If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. Hypotenuse-Leg (HL) If the hypotenuse and the leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent. Example 4 Find the value of x 2x in 12 in Example 5 Find the value of x and y. y x Example 6 Find the value of x and y. x° 75° x° y° Chapter 8 Similarity 8.1 Ratio and Proportion Ratios The ratio of a to b can be written as a/b a : b The denominator cannot be zero Simplifying Ratios Ratios should be expressed in simplified form 6:8 = 3:4 Before reducing, make sure that the units are the same. 12in : 3 ft 12in : 36 in 1: 3 Examples (page 461) Simplify each ratio 10. 16 students 24 students 12. 22 feet 52 feet 18. 60 cm 1m Examples (page 461) Simplify each ratio 20. 2 mi 3000 ft 24. 20 oz. 4 lb There are 5280 ft in 1 mi. There are 16 oz in 1 lb. Examples (page 461) Find the width to length ratio 14. 16 mm 20 mm 16. 12 in. 2 ft Using Ratios Example 1 The perimeter of the isosceles triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of the sides and the base of the triangle. L N M Using Ratios Example 2 The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles 4x 8x 3x Using Ratios Example 3 The ratios of the side lengths of ΔQRS to the corresponding side lengths of ΔVTU are 3:2. Find the unknown lengths. Q U T 2 cm V S 18 cm R Proportions Proportion Ratio = Ratio Fraction = Fraction Solving Proportions Cross multiply Let the means equal the extremes Properties of Proportions Cross Product Property a c If , then ad bc b d Reciprocal Property a c b d If , then b d a c Solving Proportions Example 1 9 6 14 x Solving Proportions Example 2 s5 s 4 10 Solving Proportions Example 3 A photo of a building has the measurements shown. The actual building is 26 ¼ ft wide. How tall is it? 2.75 in 1 7/8 in 8.2 Problem solving in Geometry with Proportions Properties of Proportions a c a b If , then b d c d a c ab cd If , then b d b d Example 1 Tell whether the statement is true or false A. s 15 s 3 If , then 10 t t 2 B. 3 5 3 x 5 y If , then x y x y Example 2 In the diagram MQ LQ MN LP Find the length of LQ. M 6 N 15 13 Q L 5 P Geometric Mean Geometric Mean The geometric mean between two numbers a and b is the positive number x such that a x x b Example 3 Find the geometric mean between 35 and 175. Example 4 You are building a scale model of your uncle’s fishing boat. The boat is 62 ft long and 23 ft wide. The model will be 14 in. long. How wide should it be? 8.3 Similar Polygons Similar Polygons Polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional. Similar figures are dilations of each other. (They are reduced or enlarged by a scale factor.) The symbol for similar is Example 1 Determine if the sides of the polygon are proportional. 8m 12 m 6m 8m 6m Example 2 Determine if the sides of the polygon are proportional. 15 m 9m 5m 3m 12 m 4m Example 3 Find the missing measurements. HAPIE NWYRS 6 H A P 5 AP = EI = SN = YR = E I 4 18 W 24 Y N S 21 R Example 4 Find the missing measurements. QUAD SIML S 12 L 65º A D 8 125º Q 20 25 I 95º U M QD = MI = mD = mU = mA = 8.4/8.5 Similar Triangles Similar Triangles To be similar, corresponding sides must be proportional and corresponding angles are congruent. Similarity Shortcuts AA Similarity Shortcut If two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar. Similarity Shortcuts SSS Similarity Shortcut If three sides in one triangle are proportional to the three sides in another triangle, then the triangles are similar. Similarity Shortcuts SAS Similarity Shortcut If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Similarity Shortcuts We have three shortcuts: AA SAS SSS Example 1 9 g 6 4 7 10.5 Example 2 k 32 h 50 24 30 Example 3 36 42 m 24 1. A flagpole 4 meters tall casts a 6 meter shadow. At the same time of day, a nearby building casts a 24 meter shadow. How tall is the building? 4 m 6m 24m 2. Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers? 3. A 10 meter rope from the top of a flagpole reaches to the end of the flagpole’s 6 meter shadow. How tall is the nearby football goalpost if, at the same moment, it has a shadow of 4 meters? 10m 6m 4m 4. Private eye Samantha Diamond places a mirror on the ground between herself and an apartment building and stands so that when she looks into the mirror, she sees into a window. The mirror is 1.22 meters from her feet and 7.32 meters from the base of the building. Sam’s eye is 1.82 meters above the ground. How high is the window? 1.82 1.22 7.32 8.6 Proportions and Similar Triangles Proportions Using similar triangles missing sides can be found by setting up proportions. Theorem Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Q T RT RU If TU || QS ,then . TQ US R S U Theorem Converse of the Triangle Proportionality Theorem Q If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT RU T If TQ R S U US , thenTU || QS . Example 1 In the diagram, segment UY is parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is the length of segment YX? U V W Y X Example 2 Given the diagram, determine whether segment PQ is parallel to segment TR. Q 9.75 P R 9 26 T 24 S Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Example 3 In the diagram, 1 2 3, AB =6, BC=9, EF=8. What is x? C 9 B 6 A 1 D 3 2 8 x E F Example 4 In the diagram, LKM MKN. Use the given side lengths to find the length of segment MN. 15 L N M 3 17 K 5. Juanita, who is 1.82 meters tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks 12.20 meters along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 meters from the end of the shadows. How tall is the tree? 6.10 1.82 12.20