Skill: Apply properties of quadrilaterals to determine unknown measures

A quadrilateral’s classification tells you which relationships must be true. Do not rely only on how the figure looks; use the stated properties.

For example, in a parallelogram:

• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary, so their measures add to $180^\circ$.

Worked example

$ABCD$ is a parallelogram. Suppose

$$m\angle A=3x+10$$

and

$$m\angle B=5x-30.$$

Find $x$ and the measures of all four angles.

Step 1: Choose the property

Angles $A$ and $B$ are consecutive angles in a parallelogram, so they are supplementary:

$$m\angle A+m\angle B=180^\circ.$$

Step 2: Substitute the expressions

$$(3x+10)+(5x-30)=180$$

Step 3: Solve for $x$

Combine like terms:

$$8x-20=180$$

Add $20$ to both sides:

$$8x=200$$

Divide by $8$:

$$x=25$$

Step 4: Find the angle measures

Substitute $x=25$ into each expression:

$$m\angle A=3(25)+10=85^\circ$$ $$m\angle B=5(25)-30=95^\circ$$

Since opposite angles in a parallelogram are congruent,

$$m\angle C=m\angle A=85^\circ$$

and

$$m\angle D=m\angle B=95^\circ.$$

Therefore,

$$\boxed{x=25}$$

and the angle measures are

$$\boxed{\angle A=85^\circ,\ \angle B=95^\circ,\ \angle C=85^\circ,\ \angle D=95^\circ}.$$

Check: consecutive angles add to $180^\circ$, since $85^\circ+95^\circ=180^\circ$.