lel lines, alternate angles are equal. Look for a Z-shape. Line $AB$ is parallel to line $CD$. A transversal cuts them. An interior angle below $AB$ on the left of the transversal is 34°. Angle $x$ is above $CD$ on the right of the transversal. 1. The two angles are inside the parallel lines and on opposite sides of the transversal. 2. This is a Z-shape, so the angles are alternate angles. 3. Therefore $x = 34^\circ$ because alternate angles are equal. ## 5. Co-interior angles: the C-shape Co-interior angles are inside the parallel lines and on the same side of the transversal. Co-interior angles add to 180°. Look for a C-shape. Two parallel lines are cut by a transversal. One co-interior angle is 118°. The other co-interior angle is $x$. 1. The two angles are inside the parallel lines and on the same side of the transversal. 2. This is a C-shape, so they are co-interior angles. 3. Co-interior angles add to 180°, so $x = 180^\circ - 118^\circ = 62^\circ$. Corresponding and alternate angles are equal, but co-interior angles add to 180°. If you see a C-shape, do not copy the angle across unchanged. ## 6. Combining with triangles In harder questions, you often need more than one fact. Work one angle at a time and give a reason each time. An isosceles triangle has two equal sides. The angles opposite those equal sides are equal. Line $AB$ is parallel to line $CD$. Points E and F lie on $AB$, and point G lies on $CD$. The triangle EFG is isosceles with $EG = FG$. The outside angle AEG is 112°. Find $\angle FGD$. 1. Angle AEG and angle GEF lie on a straight line, so $\angle GEF = 180^\circ - 112^\circ = 68^\circ$. 2. Since $EG = FG$, triangle EFG is isosceles. The base angles are equal, so $\angle EFG = 68^\circ$. 3. Line $EF$ is parallel to line $GD$, and $FG$ is a transversal. 4. Therefore $\angle FGD = 68^\circ$ because alternate angles are equal. Angle chasing means finding unknown angles one by one. Write the new angle size on the diagram, then write the reason next to your working. ## 7. Parallelograms: hidden parallel lines A parallelogram is a four-sided shape with both pairs of opposite sides parallel. In parallelogram $ABCD$, this means $AB \parallel CD$ and $AD \parallel BC$. A parallelogram question is usually a parallel-lines question in disguise. Sometimes a side is extended, or a diagonal is drawn. A diagonal is a line joining opposite corners. ABCD is a parallelogram. Point E lies on the same straight line as B and C. Angle DAB is 126° and angle AEB is 38°. Find $\angle BAE$. 1. In a parallelogram, opposite sides are parallel, so $AD \parallel BC$. 2. Since E lies on the same straight line as B and C, line $BE$ is also parallel to line $AD$. 3. Angles DAB and ABE are co-interior angles, so they add to 180°. Therefore $\angle ABE = 180^\circ - 126^\circ = 54^\circ$. 4. In triangle ABE, angles add to 180°, so $\angle BAE = 180^\circ - 54^\circ - 38^\circ = 88^\circ$. 1. Mark every known angle on the diagram before calculating. 2. Write a reason next to each new angle: straight line, vertically opposite, corresponding, alternate, co-interior, triangle total, isosceles or parallelogram. 3. If two lines are marked as parallel, look for F, Z or C shapes before doing harder triangle work. - Which angle rule gives equal angles in an F-shape? - Which angle rule gives angles that add to 180° in a C-shape? - In an isosceles triangle, where are the equal angles compa