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

Here are the solutions for the problems:

1.3 Step 1: The angles $2x$ and $x+30^\circ$ form a linear pair on a straight line, so their sum is $180^\circ$. $2x + (x+30^\circ) = 180^\circ$ Step 2: Combine like terms and solve for $x$. $3x + 30^\circ = 180^\circ$ $3x = 180^\circ - 30^\circ$ $3x = 150^\circ$ $x = \frac{150^\circ}{3}$ $x = 50^\circ$ The value of $x$ is $\boxed{50^\circ}$.

1.4 Step 1: The angles $x$ and $55^\circ$ are vertically opposite angles. Vertically opposite angles are equal. $x = 55^\circ$ The value of $x$ is $\boxed{55^\circ}$.

1.5 Step 1: The angles $\angle ABD$ and $\angle CBE$ are vertically opposite angles. Since $\angle ABD = 70^\circ$, then $x$ must be equal to $70^\circ$. $x = 70^\circ$ Step 2: The angles $\angle ABD$ and $\angle ABE$ form a linear pair, so their sum is $180^\circ$. $70^\circ + y = 180^\circ$ Step 3: Solve for $y$. $y = 180^\circ - 70^\circ$ $y = 110^\circ$ The values are x = \text{70^\circ} and y = \text{110^\circ}.

1.6 Step 1: The angles $x+70^\circ$ and $100^\circ$ form a linear pair on a straight line, so their sum is $180^\circ$. $(x+70^\circ) + 100^\circ = 180^\circ$ Step 2: Combine like terms and solve for $x$. $x + 170^\circ = 180^\circ$ $x = 180^\circ - 170^\circ$ $x = 10^\circ$ The value of $x$ is $\boxed{10^\circ}$.

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