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Find the value of x in each triangle. Give your reasons.

Mathematics

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Answers

6 parts
a

65\text{65}

b

70\text{70}

c

59\text{59}

d

72\text{72}

e

70\text{70}

f

80\text{80}

Step-by-step solution

20 steps

Here are the values of xx for each triangle:

Part a· 2 steps

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. In an isosceles triangle, the angles opposite the equal sides are equal.

    The angle 65∘65^\circ is opposite one of the equal sides, and x∘x^\circ is opposite the other equal side.

    x=65x = 65

    Reason: Base angles of an isosceles triangle are equal.
    The value of xx is 65\boxed{\text{65}}.

Part b

70\text{70}

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. In an isosceles triangle, the angles opposite the equal sides are equal.

    The angle 70∘70^\circ is one of the base angles, and x∘x^\circ is the other base angle.

    x=70x = 70

    Reason: Base angles of an isosceles triangle are equal.
    The value of xx is 70\boxed{\text{70}}.

Part c

59\text{59}

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. In an isosceles triangle, the base angles (angles opposite the equal sides) are equal.

    Let the base angles be y∘y^\circ. The vertex angle is 62∘62^\circ.

  3. The sum of angles in a triangle is 180∘180^\circ.

    62∘+y∘+y∘=180∘62^\circ + y^\circ + y^\circ = 180^\circ 62+2y=18062 + 2y = 180 2y=180−622y = 180 - 62 2y=1182y = 118 y=1182y = \frac{118}{2} y=59y = 59
  4. The angle x∘x^\circ is one of the base angles.

    x=59x = 59

    Reason: Base angles of an isosceles triangle are equal, and the sum of angles in a triangle is 180∘180^\circ.
    The value of xx is 59\boxed{\text{59}}.

Part d

72\text{72}

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. In an isosceles triangle, the base angles (angles opposite the equal sides) are equal.

    Let the base angles be y∘y^\circ. The vertex angle is 36∘36^\circ.

  3. The sum of angles in a triangle is 180∘180^\circ.

    36∘+y∘+y∘=180∘36^\circ + y^\circ + y^\circ = 180^\circ 36+2y=18036 + 2y = 180 2y=180−362y = 180 - 36 2y=1442y = 144 y=1442y = \frac{144}{2} y=72y = 72
  4. The angle x∘x^\circ is one of the base angles.

    x=72x = 72

    Reason: Base angles of an isosceles triangle are equal, and the sum of angles in a triangle is 180∘180^\circ.
    The value of xx is 72\boxed{\text{72}}.

Part e

70\text{70}

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. In an isosceles triangle, the base angles (angles opposite the equal sides) are equal.

    Let the base angles be y∘y^\circ. The vertex angle is 40∘40^\circ.

  3. The sum of angles in a triangle is 180∘180^\circ.

    40∘+y∘+y∘=180∘40^\circ + y^\circ + y^\circ = 180^\circ 40+2y=18040 + 2y = 180 2y=180−402y = 180 - 40 2y=1402y = 140 y=1402y = \frac{140}{2} y=70y = 70
  4. The angle x∘x^\circ is one of the base angles.

    x=70x = 70

    Reason: Base angles of an isosceles triangle are equal, and the sum of angles in a triangle is 180∘180^\circ.
    The value of xx is 70\boxed{\text{70}}.

Part f

80\text{80}

  1. Identify the type of triangle.

    The markings on two sides indicate that it is an isosceles triangle.

  2. The angle 130∘130^\circ is an exterior angle.

    The interior angle adjacent to it forms a linear pair.

    Interior angle=180∘−130∘=50∘\text{Interior angle} = 180^\circ - 130^\circ = 50^\circ
  3. This interior angle (50∘50^\circ) is one of the base angles of the isosceles triangle.

    Since the base angles are equal, the other base angle is also 50∘50^\circ.

  4. The sum of angles in a triangle is 180∘180^\circ.

    The angle x∘x^\circ is the vertex angle.

    x∘+50∘+50∘=180∘x^\circ + 50^\circ + 50^\circ = 180^\circ x+100=180x + 100 = 180 x=180−100x = 180 - 100 x=80x = 80

    Reason: Angles on a straight line sum to 180∘180^\circ, base angles of an isosceles triangle are equal, and the sum of angles in a triangle is 180∘180^\circ.
    The value of xx is 80\boxed{\text{80}}.

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