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Find the unknown angle in each diagram.

Mathematics

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21 steps

Here are the solutions for the unknown angles in each diagram:

Part a· 2 steps

  1. Identify the type of triangle.

    The two tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal.

    b=70∘b = 70^\circ

  2. Use the sum of angles in a triangle.

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

    a+b+70∘=180∘a + b + 70^\circ = 180^\circ a+70∘+70∘=180∘a + 70^\circ + 70^\circ = 180^\circ a+140∘=180∘a + 140^\circ = 180^\circ a=180∘−140∘a = 180^\circ - 140^\circ a=40∘a = 40^\circ

    The unknown angles are a=40∘,b=70∘\boxed{a = 40^\circ, b = 70^\circ}.

Part b· 2 steps

  1. Find the interior angle qq.

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

    q+60∘+60∘=180∘q + 60^\circ + 60^\circ = 180^\circ q+120∘=180∘q + 120^\circ = 180^\circ q=180∘−120∘q = 180^\circ - 120^\circ q=60∘q = 60^\circ

  2. Find the exterior angle pp.

    An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

    p=60∘+60∘p = 60^\circ + 60^\circ p=120∘p = 120^\circ

    Alternatively, pp and qq form a linear pair, so p+q=180∘p + q = 180^\circ.

    p+60∘=180∘p + 60^\circ = 180^\circ p=180∘−60∘p = 180^\circ - 60^\circ p=120∘p = 120^\circ

    The unknown angles are p=120∘,q=60∘\boxed{p = 120^\circ, q = 60^\circ}.

Part c· 2 steps

  1. Identify the type of triangle.

    The tick marks on the sides and the angle markings indicate that all three sides are equal and all three angles are equal. This is an equilateral triangle.

  2. Determine the angles.

    In an equilateral triangle, each interior angle is 60∘60^\circ.

    x=60∘x = 60^\circ z=60∘z = 60^\circ

    The unknown angles are x=60∘,z=60∘\boxed{x = 60^\circ, z = 60^\circ}.

Part d· 2 steps

  1. Find the interior angle adjacent to the 130∘130^\circ exterior angle.

    Let this interior angle be yy. Angles on a straight line sum to 180∘180^\circ.

    y+130∘=180∘y + 130^\circ = 180^\circ y=180∘−130∘y = 180^\circ - 130^\circ y=50∘y = 50^\circ

  2. Use the sum of angles in a triangle.

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

    b+60∘+y=180∘b + 60^\circ + y = 180^\circ b+60∘+50∘=180∘b + 60^\circ + 50^\circ = 180^\circ b+110∘=180∘b + 110^\circ = 180^\circ b=180∘−110∘b = 180^\circ - 110^\circ b=70∘b = 70^\circ

    Alternatively, the exterior angle is equal to the sum of the two opposite interior angles.

    130∘=b+60∘130^\circ = b + 60^\circ b=130∘−60∘b = 130^\circ - 60^\circ b=70∘b = 70^\circ

    The unknown angle is b=70∘\boxed{b = 70^\circ}.

Part e· 2 steps

  1. Identify the type of triangle.

    The square symbol indicates a right angle, which is 90∘90^\circ.

  2. Use the sum of angles in a triangle.

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

    a+90∘+50∘=180∘a + 90^\circ + 50^\circ = 180^\circ a+140∘=180∘a + 140^\circ = 180^\circ a=180∘−140∘a = 180^\circ - 140^\circ a=40∘a = 40^\circ

    The unknown angle is a=40∘\boxed{a = 40^\circ}.

Part f· 1 step

  1. Use the sum of angles in a triangle.

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

    x+60∘+40∘=180∘x + 60^\circ + 40^\circ = 180^\circ x+100∘=180∘x + 100^\circ = 180^\circ x=180∘−100∘x = 180^\circ - 100^\circ x=80∘x = 80^\circ

    The unknown angle is x=80∘\boxed{x = 80^\circ}.

Part g· 3 steps

  1. Find angle zz.

    Angle zz and the 140∘140^\circ exterior angle form a linear pair.

    z+140∘=180∘z + 140^\circ = 180^\circ z=180∘−140∘z = 180^\circ - 140^\circ z=40∘z = 40^\circ

  2. Identify the type of triangle.

    The tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal, so x=yx = y.

  3. Use the sum of angles in a triangle.

    x+y+z=180∘x + y + z = 180^\circ x+x+40∘=180∘x + x + 40^\circ = 180^\circ 2x+40∘=180∘2x + 40^\circ = 180^\circ 2x=180∘−40∘2x = 180^\circ - 40^\circ 2x=140∘2x = 140^\circ x=140∘2x = \frac{140^\circ}{2} x=70∘x = 70^\circ

    Since x=yx = y, then y=70∘y = 70^\circ.
    The unknown angles are x=70∘,y=70∘,z=40∘\boxed{x = 70^\circ, y = 70^\circ, z = 40^\circ}.

Part h· 3 steps

  1. Find angle zz.

    Angle zz and the 80∘80^\circ exterior angle form a linear pair.

    z+80∘=180∘z + 80^\circ = 180^\circ z=180∘−80∘z = 180^\circ - 80^\circ z=100∘z = 100^\circ

  2. Identify the type of triangle.

    The tick marks on the sides indicate that it is an isosceles triangle. The angles opposite the equal sides are equal, so x=yx = y.

  3. Use the sum of angles in a triangle.

    x+y+z=180∘x + y + z = 180^\circ x+x+100∘=180∘x + x + 100^\circ = 180^\circ 2x+100∘=180∘2x + 100^\circ = 180^\circ 2x=180∘−100∘2x = 180^\circ - 100^\circ 2x=80∘2x = 80^\circ x=80∘2x = \frac{80^\circ}{2} x=40∘x = 40^\circ

    Since x=yx = y, then y=40∘y = 40^\circ.
    The unknown angles are x=40∘,y=40∘,z=100∘\boxed{x = 40^\circ, y = 40^\circ, z = 100^\circ}.

Part h(i)· 4 steps

  1. Find angle ww in triangle ADC.

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

    w+45∘+30∘=180∘w + 45^\circ + 30^\circ = 180^\circ w+75∘=180∘w + 75^\circ = 180^\circ w=180∘−75∘w = 180^\circ - 75^\circ w=105∘w = 105^\circ

  2. Find angle yy in triangle ABC.

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

    y+60∘+50∘=180∘y + 60^\circ + 50^\circ = 180^\circ y+110∘=180∘y + 110^\circ = 180^\circ y=180∘−110∘y = 180^\circ - 110^\circ y=70∘y = 70^\circ

  3. Find angle xx.

    Angle xx is the sum of the two angles at vertex A.

    x=45∘+60∘x = 45^\circ + 60^\circ x=105∘x = 105^\circ

  4. Find angle zz.

    Angle zz is the sum of the two angles at vertex C.

    z=30∘+50∘z = 30^\circ + 50^\circ z=80∘z = 80^\circ

    The unknown angles are x=105∘,y=70∘,z=80∘,w=105∘\boxed{x = 105^\circ, y = 70^\circ, z = 80^\circ, w = 105^\circ}.

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