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Sergio039 [100]
3 years ago
9

In triangle KLM, if K is congruent to L, KL = 9x - 40, LM = 7x - 37, & KM = 3x + 23, find x & the measure of each angle.

Mathematics
1 answer:
Dennis_Churaev [7]3 years ago
7 0

Answer: x = 15, ∠K = 45.7°, ∠L = 45.7°, ∠M = 88.6°

<u>Step-by-step explanation:</u>

Since ∠K ≅ ∠L, then ΔKLM is an isoceles triangle with base KL

 KM ≅ LM

3x + 23 = 7x - 37

       23 = 4x - 37

       60 = 4x

        15 = x

KM = LM = 3x + 23

               = 3(15) + 23

               = 45 + 23

               = 68

KL = 9x - 40

    = 9(15) - 40

    = 135 - 40

    = 95

Next, draw a perpendicular bisector KN from K to KL. Thus, N is the midpoint of KL and ΔMNL is a right triangle.

  • Since N is the midpoint of KL and KL = 95, then NL = 47.5
  • Since ∠N is 90°, then NL is adjacent to ∠l and ML is the hypotenuse

Use trig to solve for ∠L (which equals ∠K):

cos ∠L = \frac{adjacent}{hypotenuse}

cos ∠L = \frac{47.5}{68}

      ∠L = cos⁻¹ (\frac{47.5}{68})

      ∠L = 45.7  

Triangle sum Theorem:

∠K + ∠L + ∠M = 180°    

45.7 + 45.7 + ∠M = 180

       91.4     + ∠M = 180

                      ∠M = 88.6

       

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Answer:

m<GFA = 110

Step-by-step explanation:

1. ABCD - parallelogram                          Definition of a parallelogram

                                                                   (AB ll CD) (AD ll BC)

2. m<B + m<C = 180                                    Consectuive angles in a

   110 + m<C = 180                                       parallelogram are supplementary

    m<C = 70                                    

3. m<GCB = 1/2 m<C                                  Definition of angles bisector

   m<GCB = 70

4. m<B = m<D = 110                                    Opposite angles in a

                                                                    parrallelogram are congruent

5. m<CDG = 1/2 m<D                                 Defintion of an angle bisector

  m<CDG = 55

6. m<GCB+m<CDG+m<CGD=180              Sum of anlges in a triangle (ΔCDG)

  70 + 55 + m<CGD = 180

   125 + m<CGD = 180

   m<CGD = 55

7. m<CGD + m<DGF = 180                            Linear pair, supplmentary angles

  55 + m<DGF = 180

  m<DGF = 125

8. m<C = m<A = 70                                     Opposite angles in a paralellogram

                                                                     are congruent

9. m<ADG = 1/2m<D                                       Definiton of an angle bisector

  m<ADG = 55

10.m<ADG+m<DFG+m<GFA+m<A=360        Sum of angles in quadrilateral

   55 + 125 + m<GFA + 70 = 360                    DGFA

    m<GFA + 250 = 360

   m<GFA = 110

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