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Chapter 4 Sections 1 and 2

Chapter 4 Sections 1 and 2. Classifying Triangles Measuring Angles in Triangles. Warm-Up. 1) Name a pair of consecutive interior angles. 2)  If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent.

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Chapter 4 Sections 1 and 2

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  1. Chapter 4Sections 1 and 2 Classifying Triangles Measuring Angles in Triangles

  2. Warm-Up 1) Name a pair of consecutive interior angles. 2)  If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent. 3) If line l is parallel to line AB, name a pair of supplementary angles. 4) If line AB represents the x-axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined. C 10 1 11 8 7 l 5 6 9 12 2 4 3 A B

  3. Warm-Up 1) Name a pair of consecutive interior angles. EX: <5 and <4 <6 and <3 2)  If line l is parallel to line AB, name a pair of congruent angles and state why they are congruent. EX: <1 and <2 because they are corresponding angles. < 9 and <3 because they are alternate interior angles. C 10 1 11 8 7 l 5 6 9 12 2 4 3 A B

  4. Warm-Up 3) If line l is parallel to line AB, name a pair of supplementary angles. EX: <9 and <2 <5 and <4 4) If line AB represents the x-axis and line AC represents the y-axis, is the slope of line CB positive, negative, zero, or undefined. Negative C 10 1 11 8 7 l 5 6 9 12 2 4 3 A B

  5. Vocabulary • Triangle- A three-sided polygon • Polygon- A closed figure in a plane that is made up of segments. • Acute Triangle- All the angles are acute. • Obtuse Triangle- One angle is obtuse. • Right Triangle- One angle is right. Leg Hypotenuse Leg

  6. Vocabulary Cont. Equlangular Triangle- An acute triangle in which all angles are congruent. Scalene Triangle- No two sides are congruent. Isosceles Triangle- At least two sides are congruent. Equilateral Triangle- All the sides are congruent. Angle Sum Theorem- The sum of the measures of the angles of a triangle is 180. Vertex angle Leg Base angle Base

  7. Vocabulary Cont. Auxiliary Line- A line or line segment added to a diagram to help in a proof. Third Angle Theorem- If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Exterior Angle- An angle that forms a linear pair with one of the angles of the polygon. Interior Angle- An angle inside a polygon. Exterior angle Interior angles

  8. Vocabulary Cont. Remote Interior Angles- The interior angles of the triangle not adjacent to a given exterior angle. Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Remote Interior angles Exterior angle Corollary- A statement that can be easily proven using a theorem. Corollary- The acute angles of a right triangle are complementary. Corollary- There can be at most one right or obtuse angle in a triangle.

  9. P Example 1: Triangle PQR is an equilateral triangle. One side measures 2x + 5 and another side measures x + 35. Find the length of each side. Since it is an equilateral triangle all the sides are congruent. 2x + 5 = x + 35 x + 5 = 35 x = 5 Plug 5 in for x in to either equation. x + 35 5 + 35 40 Each side of the triangle is 40. (2x + 5) (x + 35) R Q

  10. P Example 2: Triangle PQR is an isosceles triangle. <P is the vertex angle, PR = x + 7 , RQ = x – 1, and QP = 3x – 5. Find x, PR, RQ, and QP. Since it is an isosceles triangle and we know that <P is the vertex angle, PR is congruent to PQ. PR = PQ x + 7 = 3x - 5 7 = 2x - 5 12 = 2x 6 = x Plug 6 in for x in the equation for PR. PR = x + 7 PR = 6 + 7 PR = 13 = PQ (x + 7) (3x - 5) R Q (x - 1) Plug 6 in for x in the equation for RQ. RQ = x – 1 RQ = 6 – 1 RQ = 5

  11. Example 3: Given triangle STU with vertices S(2,3), T(4,3), and U(3,-2), use the distance formula to prove triangle STU is isosceles. If it is isosceles two of the sides have the same length. The distance formula is d=√((x2 – x1)2 + (y2 – y1)2) ST d=√((x2 – x1)2 + (y2 – y1)2) d=√((2– 4)2+ (3– 3)2) d=√((-2)2+ (0)2) d=√(4 + 0) d=√(4) d= 2 SU d=√((x2 – x1)2 + (y2 – y1)2) d=√((2– 3)2+ (3– -2)2) d=√((2 – 3)2 + (3 + 2)2) d=√((-1)2+ (5)2) d=√(1 + 25) d=√(26) Since TU and SU are congruent but ST is not this is an isosceles triangle. TU d=√((x2 – x1)2 + (y2 – y1)2) d=√((4 – 3)2+ (3– -2)2) d=√((4 – 3)2 + (3 + 2)2) d=√((1)2+ (5)2) d=√(1 + 25) d=√(26)

  12. Example 4: Given triangle STU with vertices S(2,6), T(4,-5), and U(-3,0), use the distance formula to prove triangle STU is scalene. If it is scalene none of the sides are congruent. The distance formula is d=√((x2 – x1)2 + (y2 – y1)2) ST d=√((x2 – x1)2 + (y2 – y1)2) d=√((2– 4)2+ (6 – -5)2) d=√((2 – 4)2+ (6 + 5)2) d=√((-2)2+ (11)2) d=√(4 + 121) d=√(125) SU d=√((x2 – x1)2 + (y2 – y1)2) d=√((2– -3)2+ (6– 0)2) d=√((2 + 3)2 + (6 -0)2) d=√((5)2+ (6)2) d=√(25 + 36) d=√(61) Since none of the sides are congruent, this is a scalene triangle. TU d=√((x2 – x1)2 + (y2 – y1)2) d=√((4 – -3)2+ (-5 – 0)2) d=√((4 + 3)2 + (-5 -0)2) d=√((7)2+ (-5)2) d=√(49 + 25) d=√(74)

  13. Example 5: A surveyor has drawn a triangle on a map. One angle measures 42 degrees and another measures 53 degrees. Find the measure of the third angle. According to the angle sum theorem, all the angles in a triangle add up to 180. 180 = 42 + 53 + x 180 = 95 + x 85 = x So the third angle is 85 degrees. 42 x 53

  14. Example 6: A surveyor has drawn a triangle on a map. One angle measures 41 degrees and another measures 74 degrees. Find the measure of the third angle. According to the angle sum theorem, all the angles in a triangle add up to 180. 180 = 41 + 74 + x 180 = 115 + x 65 = x So the third angle is 65 degrees. 41 x 74

  15. Example 7: Find the measure of each numbered angle in the figure if line lis parallel to line m. l 5 60 2 m<1 <1 and 135 are supplementary. 180 = m<1 + 135 45 = m<1 m<5 <1 and <5 are alternate interior angles so they are congruent. m<1 = m<5 45 = m<5 m<2 <5, <2, and 60 are supplementary. 180 = m<5 + m<2 + 60 180 = 45 + m<2 + 60 180 = 105 m<2 75 = m<2 135 1 3 4 m m<3 All the angles in a triangle add up to 180. 180 = m<1 + m<2 + m<3 180 = 45 + 75 + m<3 180 = 120 + m<3 60 = m<3 m<4 <3 and <4 are supplementary. 180 = m<3 + m<4 180 = 60 + m<4 120 = m<4

  16. Example 8: Find x, y, and m<ABC A Find x According to the exterior angle theorem the remote interior angles are add up to the exterior angle. 3x – 22 = 80 + x 2x -22 = 80 2x = 102 x = 51 Find y All the angles in a triangle add up to 180. 180 = 80 + x + y 180 = 80 + 51 + y 180 = 131 + y 49 = y 80 (3x – 22) x y B C m<ABC Plug 51 in for x in the equation for m<ABC. m<ABC = 3x – 22 m<ABC = 3(51) – 22 m<ABC = 153 – 22 m<ABC = 131

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