1. **Problem statement:** Given triangle ABC with angles at B and C as 72° and 36° respectively, and points and lines as described, we need to: - Write the angles BCA and ACB. - Calculate angle ABC and deduce the nature of triangle ABC. - Given |AB|=4cm, |AL|=2cm, and L lies on line (xy) at point E, prove that (AE) is parallel to (BC). - Deduce the nature of quadrilateral AEBC. 2. **Step 1: Write angles BCA and ACB.** - Angle BCA is the angle at point C between points B and A. - Angle ACB is the same as BCA (notation difference), so BCA = 36° (given). 3. **Step 2: Calculate angle ABC and deduce the nature of triangle ABC.** - Sum of angles in triangle ABC is 180°. - Given angles: $\angle B = 72^\circ$, $\angle C = 36^\circ$. - Calculate $\angle A$: $$\angle A = 180^\circ - 72^\circ - 36^\circ = 72^\circ$$ - So angles are $\angle A = 72^\circ$, $\angle B = 72^\circ$, $\angle C = 36^\circ$. - Since two angles are equal, triangle ABC is isosceles with $AB = AC$. 4. **Step 3: Prove (AE) is parallel to (BC).** - Given $|AB|=4cm$, $|AL|=2cm$, and L lies on (xy) at point E. - Since $AL$ is half of $AB$, point E divides segment AB in ratio 1:2. - By Thales' theorem, if a line through E is parallel to BC, then: $$\frac{AE}{AB} = \frac{AL}{AC}$$ - Since $AB = AC$ (isosceles), and $AL = 2cm$, $AE = 2cm$, the ratios are equal. - Therefore, (AE) is parallel to (BC). 5. **Step 4: Deduce the nature of quadrilateral AEBC.** - Since (AE) is parallel to (BC), and AE and BC are segments in quadrilateral AEBC, - Quadrilateral AEBC has one pair of opposite sides parallel. - Therefore, AEBC is a trapezoid (trapezium). **Final answers:** - $\angle BCA = 36^\circ$, $\angle ACB = 36^\circ$ - $\angle ABC = 72^\circ$, triangle ABC is isosceles. - (AE) is parallel to (BC). - Quadrilateral AEBC is a trapezoid.