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Find the value of x, y, and angles in problems 1.3, 1.4, 1.5, and 1.6.

Mathematics

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Here are the solutions for the problems:

Part 1.3· 2 steps

  1. The angles 2x2x and x+30∘x+30^\circ form a linear pair on a straight line, so their sum is 180∘180^\circ.

    2x+(x+30∘)=180∘2x + (x+30^\circ) = 180^\circ

  2. Combine like terms and solve for xx.

    3x+30∘=180∘3x + 30^\circ = 180^\circ 3x=180∘−30∘3x = 180^\circ - 30^\circ 3x=150∘3x = 150^\circ x=150∘3x = \frac{150^\circ}{3} x=50∘x = 50^\circ

    The value of xx is \boxed{\text{50^\circ}}.

Part 1.4· 1 step

  1. The angles xx and 55∘55^\circ are vertically opposite angles.

    Vertically opposite angles are equal.

    x=55∘x = 55^\circ

    The value of xx is \boxed{\text{55^\circ}}.

Part 1.5· 3 steps

  1. The angles ∠ABD\angle ABD and ∠CBE\angle CBE are vertically opposite angles.

    Since ∠ABD=70∘\angle ABD = 70^\circ, then xx must be equal to 70∘70^\circ.

    x=70∘x = 70^\circ

  2. The angles ∠ABD\angle ABD and ∠ABE\angle ABE form a linear pair, so their sum is 180∘180^\circ.

    70∘+y=180∘70^\circ + y = 180^\circ

  3. Solve for yy.

    y=180∘−70∘y = 180^\circ - 70^\circ y=110∘y = 110^\circ

    The values are x = \boxed{\text{70^\circ}} and y = \boxed{\text{110^\circ}}.

Part 1.6· 2 steps

  1. The angles x+70∘x+70^\circ and 100∘100^\circ form a linear pair on a straight line, so their sum is 180∘180^\circ.

    (x+70∘)+100∘=180∘(x+70^\circ) + 100^\circ = 180^\circ

  2. Combine like terms and solve for xx.

    x+170∘=180∘x + 170^\circ = 180^\circ x=180∘−170∘x = 180^\circ - 170^\circ x=10∘x = 10^\circ

    The value of xx is \boxed{\text{10^\circ}}.

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