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10.2 Triangles

10.2 Triangles. C. 3. 5. l ∥ m. 4. m. 2. 1. B. A. l. Euclid postulated that only one line could be drawn through a point not on a line that is parallel to the first line. If transversals are constructed through A and C, and through B and C, triangle ABC will be formed.

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10.2 Triangles

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  1. 10.2 Triangles

  2. C 3 5 l ∥ m 4 m 2 1 B A l Euclid postulated that only one line could be drawn through a point not on a line that is parallel to the first line. If transversals are constructed through A and C, and through B and C, triangle ABC will be formed. ∡2 = ∡5 and ∡1 = ∡3 (they are alternate interior angles.)

  3. C 3 5 l ∥ m 4 m 2 1 B A l ∡1 = ∡3and ∡2 = ∡5 (they are alternate interior angles.) ∡3 + ∡4 + ∡5 = 180° (they combine to make a straight angle.) ∡1 + ∡4 + ∡2 = 180° (Substitute ∡1 for ∡3 and ∡2 for ∡5.) The sum of the measures of the angles in a triangle is 180°

  4. The sum of the measures of the angles in a triangle is 180° Find x. x° It is right here. 8 45° 35° 45° + 35° +x° = 180° x° = 100°

  5. The sum of the measures of the angles in a triangle is 180° Find the measures of the numbered angles 35° 1 2 6 25° 3 4 5 ∡1= 90° ∡4= 25° ∡2= 180° − 90° − 35° = 55° ∡5= 180° − 55° − 25° = 100° ∡3= 55° ∡6= 180° − 25° = 155°

  6. A triangle can be classified by its angles. One angle in a right triangle is 90°. The angles in an acute triangle are all less than 90°. One angle in an obtuse triangle is greater than 90°.

  7. A triangle can be classified by its sides. Two sides of an isosceles triangle are equal length. The angles opposite the equal sides will be equal. The sides of a scalene triangle are all different lengths All three sides of an equilateral triangle are the same length. The angles will all be 60°.

  8. Classify the triangles. 55° 8 45° 80° 8 This is an isosceles right triangle. The other two angles must be 45° This is an acute scalene triangle.

  9. Classify the triangles. 60° 60° 60° 40° 40° This is an obtuse isosceles triangle. The third angle must be 100° This is an equilateral triangle.

  10. C Similar Triangles D O G A T Similar triangles have the same shape but not necessarily the same size. △CAT is similar to △DOG △CAT ~△DOG The measure of corresponding angles in similar triangles will be equal. m∠C = m∠Dm∠A = m∠Om∠T = m∠G (List the vertices so that corresponding angles are in the same position.)

  11. C Similar Triangles D O G A T Similar triangles have the same shape but not necessarily the same size. △CAT ~△DOG The lengths of corresponding sides in similar triangles will be proportional.

  12. C Similar Triangles D △CAT ~△DOG 18 6 10 O G 75° A T The measure of corresponding angles in similar triangles will be equal. The lengths of corresponding sides in similar triangles will be proportional. m∠O = 75°

  13. Similar Triangles The measure of corresponding angles in similar triangles will be equal. The lengths of corresponding sides in similar triangles will be proportional. △ABE ~△DCE 15 20 A 10 D B C E

  14. Similar Triangles The measure of corresponding angles in similar triangles will be equal. The lengths of corresponding sides in similar triangles will be proportional. If the person is 4 feet and six inches tall, how high is the light? Measuring along the ground is easy. 4.5 25 10

  15. Similar Triangles The measure of corresponding angles in similar triangles will be equal. The lengths of corresponding sides in similar triangles will be proportional. A △ABC ~△EDC 9 20 D C B 12 15 E

  16. The Pythagorean Theorem The Pythagorean theorem can be used to find the length of AC. A △ABC ~△EDC 15 9 20 D C B 12 The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the hypotenuse. 15 92 + 122 = AC2 E 81 + 144 = AC2 225 = AC2

  17. Why does the square on the hypotenuse have the same area as the sum of the squares on the two legs? c2 b a a2 a a a a2 c b b2 b b2 b a b The area of the square with sides of length a + b is a2 + b2 + the area of four of the right triangles.

  18. Why does the square on the hypotenuse have the same area as the sum of the squares on the two legs? c2 b a a a a2 c b c2 b b b2 a a b The area of the square with sides of length a + b is c2 + the area of four of the right triangles.

  19. a2 b b a a b2 a a a c2 b b b b a2 + b2 = c2 a a b a b

  20. How long does an extension ladder need to be if it is to be used to get on a roof that is 20 feet high? It is suggested that the base of the ladder be 1 foot away from the building for each four feet of vertical height. 52 + 202 = c2 25 + 400 = c2 425 = c2 20.6155 ≈ c About 21 feet. Add another three feet to have something to hold on to while you’re getting off the ladder. 20 5

  21. How high does an extension ladder reach if it is 18 feet long and placed 4 feet from the building? 42 + b2 = 182 16 + b2 = 324 b2 = 308 b≈ 17.55 feet About 17 feet and 6⅝ inches. 18 4

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