Shape and Space - Angle Relationships (Complementary, Supplementary, Vertically Opposite)

Find the value of $x$ if $x$ and $54^{\circ}$ are complementary angles.

1. Since the angles are complementary, their sum is $90^{\circ}$.
2. Write the equation: $x + 54^{\circ} = 90^{\circ}$.
3. Subtract $54^{\circ}$ from both sides: $x = 90^{\circ} - 54^{\circ}$.
4. Calculate: $x = 36^{\circ}$.

We use the property that complementary angles must add up to a right angle ($90^{\circ}$).

Two lines intersect to form an 'X'. If one angle is $125^{\circ}$, find the measure of its vertically opposite angle and the measure of an adjacent supplementary angle.

1. Vertically opposite angles are equal, so the opposite angle is $125^{\circ}$.
2. Adjacent angles on a straight line are supplementary, so their sum is $180^{\circ}$.
3. Let the adjacent angle be $y$: $y + 125^{\circ} = 180^{\circ}$.
4. Solve for $y$: $y = 180^{\circ} - 125^{\circ} = 55^{\circ}$.

This problem applies two rules: opposite angles in an intersection are equal, and angles on a straight line total $180^{\circ}$.

1. Since the angles are on a straight line, they are supplementary.
2. Write the equation: $2y + 112^{\circ} = 180^{\circ}$.
3. Subtract $112^{\circ}$ from both sides: $2y = 68^{\circ}$.
4. Divide by 2: $y = 34^{\circ}$.

Supplementary angles must sum to $180^{\circ}$. Solving for $y$ involves isolating the variable through subtraction and division.

1. Complementary angles sum to $90^{\circ}$.
2. Write the equation: $3k + 42^{\circ} = 90^{\circ}$.
3. Subtract $42^{\circ}$ from both sides: $3k = 48^{\circ}$.
4. Divide by 3: $k = 16^{\circ}$.

Because the outer rays form a right angle ($90^{\circ}$), the sum of the inner angles equals $90^{\circ}$.