LESSON 18.1: Geometry Review 1.1: ANGLES Angle Measure Angles are measured using two different systems: degree or radian. We will be using degrees, minutes, and seconds to measure angles. 1 minute = 1 degree 1 degree = 60 minutes 60 1 minute = 60 seconds 1 second = 1 minute 60 1° = 60' Symbolically: 1” = 60” 1 degree = 60 × 60 seconds 1 second = 1 degrees 3600 = 3600 seconds ° 1" = 1 1° = 3600" 3600 Example 1: ° ° a) 36.5°= 36° + 0.5 x 60’ b) 68°45’18” = 68° + 45 + 18 60 3600 = 36°30’ = 68° + 0.75° + 0.005° = 68.755° Special Angles Acute Angle: An angle is acute Obtuse Angle: An angle is Right Angle: A right angle if it measures between 0° and obtuse if it measures between measures exactly 90°. 90° 90° and 180°. In a diagram, the square marking at an angle indicates a right angle. Straight Angle: an angle which Complementary Angles: Two Supplementary Angles: Two measures exactly 180°. angles are complementary if angles are supplementary if they they sum to 90°. sum to 180°. ∠ABC = 180° ∠ x + ∠ y = 90° ∠ α + ∠ β = 180° C B y A x α β Chapter 1: Geometry Review Page 1 Example 2: a) If ∠A = 23° and ∠B = 67° then ∠A and ∠B are complementary since ∠A + ∠B = 90° b) If ∠X = 15°, ∠Y = 65° and ∠Z = 10°, even though ∠X + ∠Y + ∠Z = 90° they are NOT complementary angles. A complementary angle refers only to pairs of angles. c) What is the complement of the supplement of the angle 145°? Solution: Supplement of 145° = (180 – 145)° = 35° Complement of 35° = (90 – 35)° = 55° The complement of the supplement of the angle 145° is 55° d) Twice the supplement of a certain angle is 5 times its complement. Find the angle. Solution: Let the angle be x Then: supplement of x = (180 – x) complement of x = (90 – x) and: 2 (supplement of x) = 5 (complement of x) Supplement of x Hence: (180 – x) x 2()180 − x = 5(90 − x) (90 – x) 360 − 2x = 450 − 5x Complement of x − 2x + 5x = 450 − 360 3x = 90 x = 30 Hence the angle is 30°. Chapter 1: Geometry Review Page 2 Exercises 18.1.1: 1. Convert to degrees and minutes: (a) 57.5° (b) 114.2° (c) 34.75° 2. Convert to decimal degrees: (a) 74°15’ (b) 17°48’ (c) 56°60’ 3. State whether each angle given is acute, obtuse, right, or straight. (a) 97°55’ (b) 48.6° (c) 180° (d) 88.6° (e) 90° (f) the supplement of 45° (g) the complement of 45° (h) the complement of any acute angle (i) the supplement of any acute angle (j) the supplement of a right angle 4. What angle has the same measure as its supplement? 5. What angle has the same measure as its complement? 6. Find the complement of each angle: (a)74° (b)56° (c)34.5° (d)88.6° (e)17°25’ (f)6°34’ 7. Find the supplement of each angle: (a)104° (b)8° (c)132.2° (d)88.6° (e)97°55’ (f)16°14’ 8. What angle has a supplement that measures three times its complement? 9. What angle has a supplement that measures ten times its complement? 10. Find the supplement of the complement of the angle measuring 40°. 11. Find the complement of the supplement of the angle measuring 120°. 12. Find the complement of a right angle. Chapter 1: Geometry Review Page 3 18.1.2: TRIANGLES General A triangle has three angles, or vertices, and three sides. Each vertex of the triangle is named using capital letters: A, B, C, etc. The angle-sum of a triangle is 180°. That is, the sum of the 3 angles of the triangle is 180°. The longest side of the triangle is always opposite the largest angle of the triangle; the shortest side is opposite the smallest angle. Example 1: a) b) 65° 67° x 90° 35° 78° Since x° + 65° + 90° = 180° 67° + 35° + 78° = 180° x° + 155° = 180° x° + 155° = 180° x° = 25° The third angle of the triangle is 25° c) d) β 63°43’ 44.8° 46°39’ β x Since 44.8° + β + β = 180° 46°39’ + 63°43’ + x = 180° 44.8° + 2β = 180° 110°22’ + x = 180° 2β = 180° − 44.8° 2β = 135.2° ⎛Since 46°39' + 63°43' = 109°82' ⎞ ⎜ ⎟ 135.2° ⎝ = 110°22' ⎠ β = 2 x = 180° - 110°22’ β = 67.6° The third angle of the triangle is 67.6° ⎛Since 180° −110°22' = 179°60'−110°22' ⎞ ⎜ ⎟ ⎝ = 69°38' ⎠ x = 69°38' Chapter 1: Geometry Review Page 4 The Components of a Triangle Median: A line from the vertex Altitude: A line from the vertex Angle Bisector: A line from the of an angle to the midpoint of of an angle perpendicular to the vertex of an angle to the the opposite side. opposite side. opposite side which bisects that angle. AD is a median of ∆ABC. XW is an altitude of ∆XYZ. PT is an angle bisector of ∆PQR. A Y P W B Q Z D T X C R Naming the Sides and Angles of a Triangle Angles: There are three common systems of naming the angles of a triangle: A 1. Using the vertices, with the vertex of the angle in the center. α 2. Using a number or letter (often a Greek letter) inside the angle. 3. According to the vertex of the triangle at that angle. C In the diagram the angle at the vertex A can be called ∠BAC, ∠ α, or ∠ A. B Sides: There are two common ways to name the sides of a triangle: 1. According to two vertices it connects. The sides of ∆ABC (diagram above) are AB, BC and AC. 2. According to the vertex of the angle opposite the side. Each side of the triangle is named using the lower case letter corresponding to the letter of the vertex of the triangle opposite that side. B In the triangle ∆ABC the vertices of the triangle are A, B, and C and the sides of this triangle must be a, b, and c, such that: c The side opposite the vertex A is called a. a The side opposite the vertex B is called b. The side opposite the vertex C is called c. A b C We often use this second method in trigonometry because the name of each side of a triangle indicates its specific relationship to the angles of the triangle. Chapter 1: Geometry Review Page 5 Classifying Triangles 1. Types of Triangles Defined by Angle Size Acute Angled Triangle: All angles of the triangle are acute. Obtuse Angled Triangle: One angle of the triangle is obtuse. Acute Triangle Obtuse Triangle Right Triangle: One angle of the triangle a right angle. A The other two angles of the triangle are acute and complementary. hypotenuse The side of the triangle opposite the right angle is called the c hypotenuse. The other sides are called the legs of the b triangle. The hypotenuse is always the longest side of the triangle. Pythagoras’ Theorem states the important relationship C a B between the lengths of the sides of a right triangle. 2. Types of Triangles Defined by Side Length Scalene Triangle: A triangle with three sides of different lengths. apex Isosceles Triangle: A triangle with two sides of equal length. Properties: ; The angles opposite to the equal sides are congruent. ; The third (unequal) angle of the triangle is called the apex. A line from the apex to the midpoint of the opposite side of the triangle bisects the apex angle from which it passes and is perpendicular to the opposite side. Any altitude of the triangle from the apex to the base is also a base median and an angle bisector. Equilateral Triangle: A triangle with three sides of equal length. Properties: All properties of the isosceles triangle, and: 60° ; All angles measure 60°. ; Any altitude of the triangle is also a median and an angle bisector. 60° ; Any median or altitude of the triangle divides the triangle into 2 congruent right triangles, with angles of 30°, 60°, and.90°. We will use this special property in the study of 60° Trigonometry. 30° 30° 60° 60° Chapter 1: Geometry Review Page 6 Relationships between Triangles Congruent Triangles: Two triangles are congruent if their three sides and the three corresponding angles are congruent. Example 2: The two triangles ∆ABC and ∆RPQ are congruent, C written ∆ABC ≅ ∆RPQ, since x three corresponding angles are equal: b ∠ A = ∠ R, ∠ B = ∠ P, ∠ C = ∠ Q, a R and the corresponding sides are equal: p A q a = r, b = p, and c = q. c x Q B (Note: the order in which the vertices are written when we name the triangles reflect the corresponding angles.) r P Similar Triangles Two triangles are similar if their angles are equal. The three corresponding pairs of sides are then in fixed proportion. Example 3: The triangles ∆DEF and ∆ZXY are similar, written ∆ABC ≈ E ∆ZXY, since ∠ D = ∠ Z, ∠ F = ∠ Y, then (using the angle sum of the triangle) we must have ∠ E = 50° = ∠ X. 1.7 cm 2.0 cm So three corresponding angles are equal. and: 75° ED 1.7 1 55° F = = D 1.8 cm XZ 5.1 3 DF 1.8 1 = = YZ 5.4 3 Y EF 2.0 1 = = XY 6.0 3 55° so the sides are in fixed proportion of 1:3. 5.4 cm 6.0 cm 75° X Z 5.1 cm Chapter 1: Geometry Review Page 7 Exercise 18.1.2: 1.