Using angle relationships

Real geometry problems give you only one or two angles and a few "marked parallels". Your job is to fill in the rest using the rules you have learned.

Idea

A practical recipe:

1. Mark every given angle on the diagram.
2. Identify parallels. Look for arrowheads on lines , they mean those lines are parallel.
3. Spot the relationship. Is the angle pair a linear pair, vertically opposite, corresponding, alternate, or co-interior? Each relationship gives an equation.
4. Write equations and solve. Use BODMAS and simple algebra to find the unknowns.
5. Verify by checking that all angles around each point sum to $360^\circ$ and each linear pair sums to $180^\circ$.

A common chain of deductions: given $\angle A$, find $\angle B$ vertically opposite (so $= \angle A$), find $\angle C$ corresponding to $\angle B$ (so $= \angle A$), find $\angle D$ as $180 - \angle C$ (so $= 180 - \angle A$). With practice, you can do this in your head.

Always draw a clean diagram. Cramped angles are the main cause of mistakes.

Worked examples

Example 1. Two parallel lines $l \parallel m$ are cut by a transversal. The angle at the top crossing on the right is $50^\circ$. Find the angle at the bottom crossing on the left, inside the parallels.

The $50^\circ$ is, say, $\angle 1$. The angle described is $\angle 6$ , alternate interior to... wait. Let us be precise: $\angle 1$ at top-right, $\angle 6$ at bottom-left inside. These are alternate interior angles , equal. So the answer is $50^\circ$.

Example 2. Lines $l$ and $m$ are parallel; transversal $t$ makes a co-interior angle pair $(x, 110^\circ)$. Find $x$.

Co-interior: $x + 110 = 180 \Rightarrow x = 70^\circ$.

Example 3. A street has two parallel kerbs. A pedestrian crossing makes an angle of $30^\circ$ with one kerb. What angle does it make with the other?

A transversal across parallels , corresponding angles equal $30^\circ$. So the crossing makes the same $30^\circ$ angle with the other kerb.

Example 4. In the figure, $\angle A = 4x$ and $\angle B = 5x$ form a linear pair. Find each.

$4x + 5x = 180 \Rightarrow 9x = 180 \Rightarrow x = 20$. So $\angle A = 80^\circ$ and $\angle B = 100^\circ$.

Try it yourself

1. Two parallel lines are cut by a transversal; one angle is $73^\circ$. Find the corresponding angle.
2. Co-interior angles are $(2x + 20)^\circ$ and $(3x + 10)^\circ$. Find $x$.
3. At an intersection, one angle is $135^\circ$. Find the other three.
4. Two adjacent angles form a linear pair. One is $5$ less than thrice the other. Find both.
5. Three lines meet at a point making angles $a, b, c$ with $a = 90^\circ$, $b = 130^\circ$. Find $c$.
6. Alternate exterior angles are equal , true or false?
7. Co-interior angles between two parallel lines are $(x + 20)^\circ$ and $(2x - 50)^\circ$. Find $x$ and the angles.
8. Two roads cross. The angle of crossing is $40^\circ$ on one side. Find all four angles.

Activity

Find a real intersection (room corner, road crossing). Estimate or measure the angles. Verify they obey the linear-pair and vertically-opposite rules.